diff --git a/M2/Macaulay2/packages/=distributed-packages b/M2/Macaulay2/packages/=distributed-packages
index 1f89d2ebaa2..9f96961d7ea 100644
--- a/M2/Macaulay2/packages/=distributed-packages
+++ b/M2/Macaulay2/packages/=distributed-packages
@@ -250,6 +250,7 @@ OnlineLookup
 MergeTeX
 Probability
 Isomorphism
+DirectSummands
 CodingTheory
 WhitneyStratifications
 JSON
diff --git a/M2/Macaulay2/packages/DirectSummands.m2 b/M2/Macaulay2/packages/DirectSummands.m2
new file mode 100644
index 00000000000..3ab4c5d01a3
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands.m2
@@ -0,0 +1,668 @@
+---------------------------------------------------------------------------
+-- PURPOSE : to compute direct summands of modules and coherent sheaves
+--
+-- UPDATE HISTORY : created Oct 2023
+--
+-- TODO :
+-- 1. implement over Dmodules and other non-commutative rings
+-- 2. implement diagonalize for matrices (and later, complexes)
+-- 3. find a way to restrict and pass End to the summands
+-- 4. make summands work over ZZ (currently rank fails)
+-- 5. compute a general endomorphism without computing End
+-- 6. speed up computing eigenvalues over ZZ/p and GF(q)
+-- 7. once an indecomposable summand is found, call summands(N, M)
+-- 8. add option to split over a field extension
+-- 9. add Atlantic City, Las Vegas, or Monte Carlo style strategies?
+---------------------------------------------------------------------------
+newPackage(
+    "DirectSummands",
+    Version => "0.3",
+    Date => "April 20th 2025",
+    Headline => "decompositions of modules and coherent sheaves",
+    Authors => {
+	{ Name => "Devlin Mallory", Email => "malloryd@math.utah.edu", HomePage => "https://math.utah.edu/~malloryd/"},
+	{ Name => "Mahrud Sayrafi", Email => "mahrud@umn.edu",         HomePage => "https://math.umn.edu/~mahrud/"}
+	},
+    Keywords => { "Commutative Algebra" },
+    PackageImports => {
+	"Isomorphism",     -- for isIsomorphic
+	"Polyhedra",       -- for coneFromVData and coneComp
+	"PushForward",     -- only for frobenius.m2
+	"RationalPoints2", -- for rationalPoints in findIdempotents
+	"Truncations",     -- for effGenerators
+	"LocalRings",      -- for the local examples
+	"Varieties",       -- for the geometric examples
+	},
+    AuxiliaryFiles => true,
+    DebuggingMode => false
+    )
+
+export {
+    -- methods
+    "isIndecomposable",
+    "directSummands", "summands" => "directSummands",
+    "findProjectors",
+    "findIdempotents", "findIdem" => "findIdempotents",
+    "findSplitInclusion",
+    "generalEndomorphism",
+    "isomorphismTally",
+    "tallySummands",
+    -- frobenius methods
+    "frobeniusMap",
+    "frobeniusRing",
+    "frobeniusPullback",
+    "frobeniusPushforward",
+    "frobeniusTwist",
+    "potentialExtension",
+    "extendGroundField" => "changeBaseField",
+    "changeBaseField"
+    -- symbols
+    }
+
+importFrom_Core {
+    "raw", "rawReshape",
+    "rawNumberOfColumns",
+    "rawNumberOfRows",
+    "tryHooks",
+    "sortBy",
+    }
+
+-----------------------------------------------------------------------------
+-* Code section *-
+-----------------------------------------------------------------------------
+
+-- defined here and used in idempotents.m2 and homogeneous.m2
+DirectSummandsOptions = new OptionTable from {
+    Limit             => null, -- used in directSummands(Module, Module)
+    Strategy          => 3,    -- Strategy is a bitwise sum of the following:
+    -- 1  => use degrees of generators as heuristic to peel off line bundles first
+    -- 2  => check generators of End_0 as heuristic for finding idempotents
+    -- 4  => use splitComponentsBasic, skips computing splitting maps
+    -- 8  => precompute Homs before looking for idempotents
+    -- 16 => use summandsFromIdempotents even in graded case
+    "Splitting"       => null,
+    Tries             => null, -- see defaultNumTries below
+    Verbose           => false -- whether to print extra debugging info
+}
+
+-- the default number of attempts for randomized algorithms
+-- e.g. 145 tries in char 2, 10 in char 32003, and 1 in char 0
+defaultNumTries = p -> ceiling(0.1 + 100 / log p)
+--apply({2, 32003, 0}, defaultNumTries)
+
+-- helpers for computing Frobenius pushforwards of modules and sheaves
+-- TODO: move to PushForward package?
+load "./DirectSummands/pushforward2.m2"
+load "./DirectSummands/frobenius.m2"
+-- helpers for finding random idempotents of a module for the local case
+load "./DirectSummands/idempotents.m2"
+-- helpers for finding random projectors of a module for the graded case
+load "./DirectSummands/homogeneous.m2"
+
+-----------------------------------------------------------------------------
+-- Things to move to the Core
+-----------------------------------------------------------------------------
+
+-- return the submatrix with given degrees of target and source
+submatrixByDegrees(Matrix, Sequence) := (m, degs) -> (
+    (tar, src) := degs;
+    col := if src =!= null then positions(degrees source m, deg -> member(deg, src));
+    row := if tar =!= null then positions(degrees target m, deg -> member(deg, tar));
+    submatrix(m, row, col))
+
+-- rankWarn = true
+-- rank' = M -> if groundField ring M =!= ZZ then rank M else try rank M else (
+--     if rankWarn then ( rankWarn = false;
+-- 	printerr "warning: rank over integers is not well-defined; returning zero");
+--     0)
+
+-- this defines sorting on modules and sheaves
+CoherentSheaf ? CoherentSheaf :=
+Module ? Module := (M, N) -> if rank M != rank N then rank M ? rank N else degrees M ? degrees N
+
+position(ZZ, Function) := o -> (n, f) -> position(0..n-1, f)
+-- TODO: this is different from position(List,List,Function)
+position' = method()
+position'(VisibleList, VisibleList, Function) := (B, C, f) -> for b in B do for c in C do if f(b, c) then return (b, c)
+position'(ZZ,          ZZ,          Function) := (B, C, f) -> position'(0..B-1, 0..C-1, f)
+
+scan' = method()
+scan'(VisibleList, VisibleList, Function) := (B, C, f) -> for b in B do for c in C do f(b,c)
+
+char SheafOfRings := O -> char variety O
+
+GF'ZZ'ZZ = memoize (lookup(GF, ZZ, ZZ)) (options GF)
+GF'Ring  = memoize (lookup(GF, Ring))   (options GF)
+GF(ZZ, ZZ) := GaloisField => opts -> GF'ZZ'ZZ
+GF(Ring)   := GaloisField => opts -> GF'Ring
+
+-- TODO: what generality should this be in?
+-- WANT:
+--   R ** ZZ/101 to change characteristic
+--   R ** S to change coefficient ring
+-- TODO: can you change the ground field but keep the tower structure?
+QuotientRing   ** GaloisField :=
+PolynomialRing ** GaloisField := (R, L) -> R.cache#(map, L, R) ??= (
+    -- TODO: in general we may want to keep part of the ring tower
+    A := first flattenRing(R, CoefficientRing => null);
+    quotient sub(ideal A, L monoid A))
+
+changeBaseMap = method()
+changeBaseMap(Ring, Ring) := (L, S) -> S.cache#(map, L, S) ??= map(quotient sub(ideal S, L monoid S), S)
+
+changeBaseField = method()
+-- TODO: add more automated methods, e.g. where minors of the presentation factor
+changeBaseField(ZZ, Module) :=
+changeBaseField(ZZ, CoherentSheaf)   := (e, M) -> changeBaseField(GF(char ring M, e), M)
+changeBaseField(GaloisField, Module) := (L, M) -> M.cache#(symbol changeBaseField, L) ??= (
+    S := first flattenRing(ring M,
+	CoefficientRing => null);
+    K := coefficientRing S;
+    if class K =!= GaloisField then return (
+        R0 := quotient sub(ideal S, L monoid S);
+	M' := directSum apply(cachedSummands M,
+	    N -> coker sub(presentation N, R0));
+        M.cache#(symbol changeBaseField, L) = M');
+    -- don't needlessly create new rings:
+    if K.order === L.order then return M;
+    i0 := map(L, K);
+    LS := L(monoid S);
+    i1 := map(LS, ring ideal S, gens LS | { i0 K_0 });
+    R := quotient i1 ideal S;
+    i := map(R, S, i1);
+    directSum apply(cachedSummands M, N -> i ** N))
+
+changeBaseField(Ring, Module) := (L, M) -> M.cache#(symbol changeBaseField, L) ??= (
+    S := first flattenRing(ring M,
+	CoefficientRing => null);
+    psi := changeBaseMap(L, S);
+    directSum apply(cachedSummands M,
+	N -> coker(psi ** presentation N)))
+
+changeBaseField(Ring, Matrix) := (L, f) -> f.cache#(symbol changeBaseField, L) ??= (
+    S := first flattenRing(ring f,
+	CoefficientRing => null);
+    psi := changeBaseMap(L, S);
+    psi ** f)
+
+-- TODO: come up with a better way to extend ground field of a variety
+-- TODO: does this also need to be used for frobenius pushforward?
+sheaf' = (X, M) -> try sheaf(X, M) else (
+    if instance(X, ProjectiveVariety) then sheaf(Proj ring M, M) else
+    if instance(X,     AffineVariety) then sheaf(Spec ring M, M) else
+    error "extension of the coefficient field of the base variety is not implemented")
+
+changeBaseField(GaloisField, CoherentSheaf) := (L, F) -> sheaf'(variety F, changeBaseField(L, module F))
+
+importFrom_Varieties "flattenModule"
+prune' = method()
+prune' Module := M0 -> M0.cache.prune' ??= (
+    M := prune M0;
+    R := ring M;
+    if isHomogeneous M0
+    or instance(R, LocalRing) then return M;
+    S := ambient R;
+    MS := flattenModule M;
+    Sm := localRing(S, ideal gens S);
+    MSm := prune(MS ** Sm);
+    R ** liftUp MSm)
+
+nonzero = x -> select(x, i -> i != 0)
+nonnull = x -> select(x, i -> i =!= null)
+
+checkRecursionDepth = () -> if recursionDepth() > recursionLimit - 20 then printerr(
+    "Warning: the recursion depth limit may need to be extended; use `recursionLimit = N`")
+
+-----------------------------------------------------------------------------
+-- things to move to Isomorphism package
+-----------------------------------------------------------------------------
+
+module Module := identity
+
+-- TODO: speed this up
+-- TODO: implement isIsomorphic for sheaves
+-- TODO: add strict option
+tallySummands = L -> tally (
+    opts := Homogeneous => all(L, isHomogeneous);
+    L  = new MutableList from module \ L;
+    b := new MutableList from #L : true;
+    for i to #L-2 do if b#i then for j from i+1 to #L-1 do if b#j then (
+	if isIsomorphic(L#i, L#j, opts)
+	then ( b#j = false; L#j = L#i ));
+    new List from L)
+
+isomorphismTally = method()
+isomorphismTally List := L -> (
+    if not uniform L then error "expected list of elements of the same type";
+    if not (class L_0 === Module or class L_0 === CoherentSheaf ) then error "expected list of modules or sheaves";
+    opts := Homogeneous => all(L, isHomogeneous);
+    --L = new MutableList from L;
+    j := 0;
+    while j < #L list (
+	i := j + 1;
+	c := 1;
+	while i < #L do (
+	    if isIsomorphic(L#j, L#i, opts)
+	    then (
+		L = drop(L, {i, i});
+		c = c + 1)
+	    else i = i + 1);
+	j = j + 1;
+	(L#(j-1), c)))
+
+-----------------------------------------------------------------------------
+-- methods for finding general endomorphisms of degree zero
+-----------------------------------------------------------------------------
+
+importFrom_Truncations { "effGenerators" }
+
+coneComp = (C, u, v) -> (
+    --if u == v                            then symbol== else
+    if contains(C, matrix vector(v - u)) then symbol <= else
+    if contains(C, matrix vector(u - v)) then symbol > else incomparable)
+
+-- TODO: add this as a strategy to basis
+smartBasis = (deg, M) -> (
+    -- TODO: try splitting coker {{a, b^3}, {-b^3, a}} over ZZ/32003[a..b]/(a^2+b^6)
+    if M == 0 then return map(M, 0, 0);
+    if instance(deg, ZZ) then deg = {deg};
+    degs := if #deg == 1 then select(unique degrees M, d -> d <= deg) else (
+	-- FIXME: in the multigraded case sometimes just using basis is faster:
+	return basis(deg, M);
+        -- in the multigraded case, coneMin and coneComp can be slow
+	-- but for sufficiently large modules they are still an improvement
+        -- TODO: make them faster
+        C := coneFromVData effGenerators ring M;
+        --elapsedTime compMin(C, unique degrees M) -- TODO: this is not the right thing
+        select(unique degrees M, d -> coneComp(C, d, deg) == symbol <=));
+    if degs === {deg} then return map(M, , submatrixByDegrees(gens cover M, (, degs)));
+    M' := subquotient(ambient M,
+	if M.?generators then submatrixByDegrees(gens M, (, degs)),
+	if M.?relations  then relations M);
+    M'.cache.homomorphism = M.cache.homomorphism;
+    basis(deg, M')) -- caching this globally causes issues!
+
+-- matrix of (degree zero) generators of End M
+-- TODO: rename this
+-- also see gensHom0
+gensEnd0 = M -> M.cache#"End0" ??= (
+    -- TODO: need to pass options from Hom + choose the coefficient field
+    zdeg := if isHomogeneous M then degree 0_M;
+    if 0 < debugLevel then stderr << " -- computing End module ... " << flush;
+    A := Hom(M, M,
+	DegreeLimit       => zdeg,
+	MinimalGenerators => false);
+    if 0 < debugLevel then stderr << "done!" << endl;
+    if isHomogeneous M
+    then smartBasis(zdeg, A)
+    else inducedMap(A, , gens A))
+
+-- give a random vector in a module over a local ring
+localRandom = (M, opts) -> (
+    R := ring M;
+    -- TODO: which coefficient ring do we want?
+    K := try coefficientRing R else R;
+    v := random(cover M ** K, module K, opts);
+    -- TODO: sub should be unnecessary, but see https://github.com/Macaulay2/M2/issues/3638
+    vector inducedMap(M, , generators M * sub(v, R)))
+
+random(ZZ,   Module) :=
+random(List, Module) := Vector => o -> (d, M) -> vector map(M, , random(cover M, (ring M)^{-d}, o))
+random       Module  := Vector => o ->     M  -> (
+    if isHomogeneous M then random(degree 1_(ring M), M, o) else localRandom(M, o))
+
+generalEndomorphism = method(Options => options random)
+generalEndomorphism Module := Matrix => o -> M0 -> (
+    R := ring M0;
+    -- TODO: avoid this hack for local rings
+    M := if instance(R, LocalRing) then liftUp M0 else M0;
+    B := gensEnd0 M;
+    r := if isHomogeneous M
+    then random(source B, o)
+    else localRandom(source B, o);
+    R ** homomorphism(B * r))
+-- the sheaf needs to be pruned to prevent missing endomorphisms
+generalEndomorphism CoherentSheaf := SheafMap => o -> F -> (
+    sheaf generalEndomorphism(module prune F, o))
+
+-- overwrite two existing hooks, to be updated in Core
+addHook((quotient, Matrix, Matrix), Strategy => Default,
+    -- Note: this strategy only works if the remainder is zero, i.e.:
+    -- homomorphism' f % image Hom(source f, g) == 0
+    (opts, f, g) -> (
+	opts = new OptionTable from {
+	    DegreeLimit       => opts.DegreeLimit,
+	    MinimalGenerators => opts.MinimalGenerators };
+	map(source g, source f, homomorphism(homomorphism'(f, opts) // Hom(source f, g, opts)))))
+
+addHook((quotient', Matrix, Matrix), Strategy => Default,
+    -- Note: this strategy only works if the remainder is zero, i.e.:
+    -- homomorphism' f % image Hom(g, target f) == 0
+    (opts, f, g) -> (
+	opts = new OptionTable from {
+	    DegreeLimit       => opts.DegreeLimit,
+	    MinimalGenerators => opts.MinimalGenerators };
+	map(target f, target g, homomorphism(homomorphism'(f, opts) // Hom(g, target f, opts)))))
+
+importFrom_Core {"Hooks", "HookPriority"}
+Matrix.Hooks#(quotient,  Matrix, Matrix).HookPriority = drop(Matrix.Hooks#(quotient,  Matrix, Matrix).HookPriority, -1)
+Matrix.Hooks#(quotient', Matrix, Matrix).HookPriority = drop(Matrix.Hooks#(quotient', Matrix, Matrix).HookPriority, -1)
+
+-- left inverse of a split injection
+-- TODO: figure out if we can ever do this without computing End source g
+leftInverse = inverse' = method(Options => options Hom)
+leftInverse Matrix := opts -> g -> g.cache.leftInverse ??= quotient'(id_(source g), g, opts)
+
+-- right inverse of a split surjection
+-- FIXME: inverse may fail for a general split surjection:
+-- c.f. https://github.com/Macaulay2/M2/issues/3738
+-- TODO: figure out if we can ever do this without computing End target g
+rightInverse = method(Options => options Hom)
+rightInverse Matrix := opts -> g -> g.cache.rightInverse ??= quotient(id_(target g), g, opts)
+
+-- given N and a split injection inc:N -> M,
+-- we use precomputed endomorphisms of M
+-- to produce a general endomorphism of N
+generalEndomorphism(Matrix, Nothing, Matrix) := Matrix => o -> (N, null, inc) -> (
+    -- assert(N === source inc);
+    inv := leftInverse(inc,
+	DegreeLimit => degree 0_N,
+	-- FIXME: setting this to false sometimes
+	-- produces non-well-defined inverses
+	MinimalGenerators => true);
+    inc * generalEndomorphism(target inc, o) * inv)
+
+-- given N and a split surjection pr:M -> N,
+-- we use precomputed endomorphisms of M
+-- to produce a general endomorphism of N
+generalEndomorphism(Module, Matrix) := Matrix => o -> (N, pr) -> (
+    -- assert(N === target pr);
+    -- TODO: currently this still computes End_0(N)
+    -- figure out a way to compute the inverse without doing so
+    inv := rightInverse(pr,
+	DegreeLimit => degree 0_N,
+	-- FIXME: setting this to false sometimes
+	-- produces non-well-defined inverses
+	MinimalGenerators => true);
+    pr * generalEndomorphism(source pr, o) * inv)
+
+generalEndomorphism(Module, Matrix, Nothing) := Matrix => o -> (N, pr, null) -> generalEndomorphism(N, pr, o)
+generalEndomorphism(Module, Matrix, Matrix)  := Matrix => o -> (N, pr, inc) -> (
+    -- assert(N === target pr and source pr === target inc and N === source inc);
+    pr * generalEndomorphism(source pr, o) * inc)
+
+-----------------------------------------------------------------------------
+-- helpers for splitting and caching projection maps
+-----------------------------------------------------------------------------
+
+findSplitInclusion = method(Options => { Tries => 50 })
+--tests if M is a split summand of N
+findSplitInclusion(Module, Module) := opts -> (M, N) -> (
+    h := for i to opts.Tries - 1 do (
+        b := homomorphism random Hom(M, N, MinimalGenerators => false);
+        c := homomorphism random Hom(N, M, MinimalGenerators => false);
+        if isIsomorphism(c * b) then break b);
+    if h === null then return "not known" else return h)
+
+-- helper for splitting a free module and setting the split surjections
+splitFreeModule = (M, opts) -> components directSum apply(R := ring M; -degrees M, deg -> R^{deg})
+
+-- helper for splitting free summands by observing the degrees of generators
+splitFreeSummands = (M, opts) -> M.cache#"FreeSummands" ??= (
+    directSummands(R := ring M; apply(-unique degrees M, d -> R^{d}), M, opts))
+
+-- helper for splitting a module with known components
+-- (in particular, the components may also have summands)
+-- TODO: can we sort the summands here?
+splitComponents = (M, comps, splitfunc) -> (
+    n := #comps;
+    c := -1; -- summand counter
+    projs := if 1 < n then apply(n, i -> try M^[i]) else { id_M };
+    flatten apply(comps, projs, (N, p) -> (
+	L := splitfunc N;
+	-- Projection maps to the summands
+	try if #L > 1 then apply(#L, i ->
+	    M.cache#(symbol ^, [c += 1]) = N^[i] * p)
+	else M.cache#(symbol ^, [c += 1]) = p;
+	-- Inclusion maps are computed on-demand
+	L)))
+
+splitComponentsBasic = (M, ends, opts) -> (
+    -- the coker of each idempotent gives a summand, while
+    L1 := prune' \ apply(ends, coker);
+    -- the image of their composition is the complement.
+    L2 := prune' \ { image product ends };
+    -- TODO: call something like summands(L, M) here?
+    flatten apply(nonzero join(L1, L2), summands_opts))
+
+-- these are computed on-demand from the cached split surjection
+oldinclusions = lookup(symbol_, Module, Array)
+Module _ Array := Matrix => (M, w) -> M.cache#(symbol _, w) ??= (
+    if not M.cache#?(symbol ^, w) then oldinclusions(M, w)
+    else rightInverse(M.cache#(symbol ^, w),
+	DegreeLimit       => degree 0_M,
+	MinimalGenerators => true))
+
+-- TODO: can we return cached summands from the closest subfield?
+cachedSummands = M -> M.cache.summands ?? components M
+
+-----------------------------------------------------------------------------
+-- directSummands
+-----------------------------------------------------------------------------
+
+-- Note: M may need to be extended to a field extensions
+directSummands = method(Options => DirectSummandsOptions)
+directSummands Module := List => opts -> M -> M.cache.summands ??= (
+    checkRecursionDepth();
+    strategy := opts.Strategy;
+    R := ring M;
+    if prune' M == 0 then return { M };
+    -- Note: End does not work for WeylAlgebras or AssociativeAlgebras yet, nor does basis
+    if not isCommutative R and not isWeylAlgebra R then error "expected a commutative base ring";
+    -- Note: rank does weird stuff if R is not a domain
+    if opts.Verbose then printerr("splitting module of rank: ", toString rank M);
+    if rank cover M <= 1 then return M.cache.summands = { M.cache.isIndecomposable = true; M };
+    if isDirectSum  M    then return M.cache.summands = splitComponents(M, components M, directSummands_opts);
+    if isFreeModule M    then return M.cache.summands = splitFreeModule(M, opts);
+    if strategy & 1 == 1 then return M.cache.summands = (
+	splitComponents(M, splitFreeSummands(M, opts),
+	    directSummands_(opts, Strategy => strategy - 1)));
+    -- TODO: where should indecomposability check happen?
+    -- for now it's here, but once we figure out random endomorphisms
+    -- without computing Hom, this would need to move.
+    -- Note: this may return null if it is inconclusive
+    if strategy & 2  == 2  and isIndecomposable M === true then { M } else
+    if strategy & 16 != 16 and isHomogeneous M
+    then summandsFromProjectors(M, opts)   -- see DirectSummands/homogeneous.m2
+    else summandsFromIdempotents(M, opts)) -- see DirectSummands/idempotents.m2
+
+directSummands CoherentSheaf := List => opts -> F -> apply(
+    directSummands(module prune F, opts), N -> sheaf(variety F, N))
+
+-----------------------------------------------------------------------------
+
+-- tests whether L (perhaps a line bundle) is a summand of M
+-- Limit => N _recommends_ stopping after peeling N summands of L
+-- FIXME: it's not guaranteed to work, e.g. on X_4 over ZZ/2
+-- TODO: cache projection/inclusion maps
+-- TODO: cache this
+directSummands(Module, Module) := List => opts -> (L, M) -> (
+    checkRecursionDepth();
+    if ring L =!= ring M then error "expected objects over the same ring";
+    if rank L  >  rank M
+    or rank L  >= rank M and isFreeModule M then return {M};
+    limit := opts.Limit ?? numgens M;
+    tries := opts.Tries ?? defaultNumTries char ring M;
+    if 1 < #cachedSummands M then return flatten apply(cachedSummands M, N -> directSummands(L, N, opts));
+    if 1 < limit then (
+	N := M;
+	M.cache#(symbol ^, [0]) = id_M;
+	if opts.Verbose then stderr << " -- splitting summands of degree " << degrees L << ": " << flush;
+	comps := new MutableList from {M};
+	for i to limit - 1 do (
+	    pr := M.cache#(symbol ^, [#comps-1]);
+	    LL := directSummands(L, N, opts, Limit => 1);
+	    if #LL == 1 then break;
+	    M.cache#(symbol ^, [#comps])   = N^[1] * pr;
+	    M.cache#(symbol ^, [#comps-1]) = N^[0] * pr;
+	    comps#(#comps-1) = L; -- === LL#0
+	    comps##comps = N = LL#1);
+	if opts.Verbose then stderr << endl << flush;
+	return toList comps
+    );
+    zdeg := degree 0_M;
+    -- TODO: can we detect multiple summands of L at once?
+    -- perhaps find a projector onto a summand with several copies?
+    gensHom0 := (N, M) -> (
+	H := Hom(N, M,
+	    DegreeLimit => zdeg,
+	    MinimalGenerators => false);
+	smartBasis(zdeg, H));
+    h := catch if isFreeModule L then (
+	-- TODO: for the case of line bundle summands,
+	-- is it faster if we compute all of Hom and check
+	-- for line bundles of any degree all at once?
+	C := gensHom0(M, L); if numcols C == 0 then return {M};
+	-- Previous alternative:
+	-- h := for i from 0 to numcols C - 1 do (
+	--     isSurjective(c := homomorphism C_{i}) ...)
+	for i to tries - 1 do (
+	    c := homomorphism(C * random source C);
+	    if isSurjective c then throw (c, rightInverse c)))
+    else (
+        -- we look for a composition L -> M -> L which is the identity
+	B := gensHom0(L, M); if numcols B == 0 then return {M};
+	C  = gensHom0(M, L); if numcols C == 0 then return {M};
+        -- attempt to find a random isomorphism
+        for i to tries - 1 do (
+	    b := homomorphism(B * random source B);
+	    c := homomorphism(C * random source C);
+            --TODO: change isIsomorphism to isSurjective?
+	    if isIsomorphism(c * b) then throw (c, b));
+	-- TODO: is it worth doing the following lines? when does the random strategy above fail?
+	printerr "summands got to this part, it's probably useful!";
+	if opts.Strategy & 8 == 8 then (
+	    -- precomputing the Homs can sometimes be a good strategy
+	    -- TODO: confirm that this injectivity check is worthwhile and not slow
+	    Bhoms := select(apply(numcols B, i -> homomorphism B_{i}), isInjective);
+	    Choms := select(apply(numcols C, i -> homomorphism C_{i}), isSurjective);
+	    scan'(Choms, Bhoms, (c, b) -> if isIsomorphism(c * b) then throw (c, b)))
+	-- but sometimes too memory intensive
+	else scan'(numcols C, numcols B, (ci, bi) ->
+	    if isSurjective(c := homomorphism C_{ci})
+	    and isInjective(b := homomorphism B_{bi})
+	    and isIsomorphism(c * b) then throw (c, b))
+    );
+    (pr, inc) := if h =!= null then h else return {M};
+    if opts.Verbose then stderr << concatenate(rank L : ".") << flush;
+    N = prune' coker inc;
+    iso := inverse N.cache.pruningMap;
+    M.cache#(symbol ^, [0]) = pr;
+    M.cache#(symbol ^, [1]) = iso * inducedMap(coker inc, M);
+    {L, N})
+
+directSummands(CoherentSheaf, CoherentSheaf) := List => opts -> (L, G) -> apply(
+    directSummands(module prune L, module prune G, opts), N -> sheaf(variety L, N))
+
+-- attempt to peel off summands from a given list of modules
+directSummands(List, CoherentSheaf) :=
+directSummands(List, Module) := List => opts -> (Ls, M) -> sort (
+    if 1 < #cachedSummands M then flatten apply(cachedSummands M, N -> directSummands(Ls, N, opts))
+    else fold(Ls, {M}, (L, LL) -> splitComponents(M, LL, directSummands_(opts, L))))
+
+-----------------------------------------------------------------------------
+-- isIndecomposable
+-----------------------------------------------------------------------------
+
+-- returns false if an easy decomposition can be found
+-- (but does _not_ run the full directSummands algorithm)
+-- returns true if the module is certifiably indecomposable
+-- returns null for non-conclusive results
+isIndecomposable = method(Options => { Strategy => null, Verbose => false })
+isIndecomposable CoherentSheaf := o -> F -> isIndecomposable(module prune F, o)
+isIndecomposable Module := o -> M -> M.cache.isIndecomposable ??= tryHooks(
+    (isIndecomposable, Module), (o, M), (o, M) -> (
+	if 1 < debugLevel then printerr("isIndecomposable was inconclusive. ",
+	    "Try extending the field or pruning the sheaf.")))
+
+-- this strategy checks if:
+-- * M has only one degree zero endomorphism, namely identity, or
+-- * all degree zero endomorphisms of M are zero mod maximal ideal
+-- if a non-identity idempotent is found, it is cached in M
+addHook((isIndecomposable, Module), Strategy => "IdempotentSearch", (opts, M) -> (
+	(idemp, certified) := findBasicIdempotents(M, opts);
+	if #idemp > 0 then ( if opts.Verbose then printerr "module is decomposable!";  false ) else
+	if certified  then ( if opts.Verbose then printerr "module is indecomposable!"; true )
+    ))
+
+-- TODO
+--addHook((isIndecomposable, Module), Strategy => TorsionFree, (opts, M) -> (
+--	if isTorsionFree M and isDomain M
+--	and degree M // degree R <= 1 then return true))
+
+-----------------------------------------------------------------------------
+-* Development section *-
+-----------------------------------------------------------------------------
+
+--directSummands Matrix  := List => opts -> f -> apply(directSummands(coker f,  opts), presentation)
+-* TODO: not done yet
+diagonalize = M -> (
+    m := presentation M;
+    elapsedTime A := End M; -- most time consuming step
+    elapsedTime B := basis(degree 1_(ring M), A);
+    h = generalEndomorphism M;
+    N0 = image homomorphism B_{idem};
+    N = prune N0;
+    psi = inverse N.cache.pruningMap; -- map N0 --> N
+    phi = map(N0, M, homomorphism B_{idem}); -- map M --> N0
+    f = psi * phi; -- map M --> N
+    h' = f * h * inverse f;
+    source h' == N;
+    target h' == N;
+    h'
+    )
+*-
+
+--directSummands Complex := List => opts -> C -> () -- TODO: should be functorial
+
+-----------------------------------------------------------------------------
+-* Test section *-
+-----------------------------------------------------------------------------
+
+load "./DirectSummands/tests.m2"
+
+-----------------------------------------------------------------------------
+-* Documentation section *-
+-----------------------------------------------------------------------------
+
+beginDocumentation()
+
+load "./DirectSummands/docs.m2"
+
+end--
+
+restart
+check needsPackage "DirectSummands"
+
+uninstallPackage "DirectSummands"
+restart
+viewHelp installPackage "DirectSummands"
+viewHelp directSummands
+viewHelp
+
+end--
+restart
+debug needsPackage "DirectSummands"
+
+
+R = kk[x,y,z];
+n = 1000
+d = {100}
+elapsedTime smartBasis(0, Hom(R^n, R^d));
+elapsedTime smartBasis(0, Hom(R^n, R^d, DegreeLimit => 0));
+
+
+--------
+-- summand of 4th syzygy of residue field of ring defined by
+-- ideal(y*z,x*z,y^3,x*y^2+z^3,x^2*y,x^3) is indecomposable,
+-- but the current method doesn't really show that definitively
diff --git a/M2/Macaulay2/packages/DirectSummands/bench.m2 b/M2/Macaulay2/packages/DirectSummands/bench.m2
new file mode 100644
index 00000000000..63596081191
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/bench.m2
@@ -0,0 +1,59 @@
+restart
+debug needsPackage "DirectSummands"
+debugLevel = 1
+errorDepth = 2
+
+kk = ZZ/5
+R = quotient Grassmannian(1,3, CoefficientRing => kk)
+X = Proj R
+elapsedTime M = module frobeniusPushforward(1, OO_X); -- <1s in char 2 & 3
+rank M
+elapsedTime L = summands(M, Verbose => true);
+tally apply(summands M, N -> (rank N, degrees N))
+
+N = last M.cache#"FreeSummands";
+elapsedTime projs = findProjectors N;
+elapsedTime L0 = flatten(summands \ prune \ coker \ projs);
+elapsedTime L = summands(keys tallySummands L0, N, Verbose => true);
+M.cache.summands = drop(M.cache#"FreeSummands", -1) | L;
+
+-* Timing results:
+-----------------------------------
+ q   | F_* | Tot | End | misc
+-----------------------------------
+ 2   | <1s | <1s | <1s | noether
+ 4   | ~6s | 33m |  8m | Fields
+ 3   | <1s | 10s | <1s | noether
+ 5   | 43s | ~3h | ~3h | noether (once End is computed, pretty fast with some manual work)
+ 7   |     |     |     |
+ 9   |     |     |     |
+-----------------------------------
+field	q	rk	free summands		non-free summands
+-----------------------------------
+ZZ/2	2^1	 16	1:0  14:1   1:2
+ZZ/2	2^2	256	1:0  99:1  99:2  1:3	28:rk 2 bundles
+ZZ/3	3^1	 81	1:0  44:1  20:2		8: rk 2 bundles
+ZZ/3	3^2
+ZZ/5	5^1	625	1:0 190:1 300:2  6:3	64:rk 2 bundles
+ZZ/7	7^1
+*-
+
+
+kk = ZZ/2
+R = quotient Grassmannian(2,5, CoefficientRing => kk)
+X = Proj R
+elapsedTime M = module frobeniusPushforward(1, OO_X); -- ~20min in char 2
+rank M
+elapsedTime L = summands(M, Verbose => true);
+tally apply(L, N -> (rank N, degrees N))
+
+-* Timing results:
+-----------------------------------
+ q   | F_* | Tot | End | misc
+-----------------------------------
+ 2   | 20m |     |     | Fields
+-----------------------------------
+field	q	rk	free summands		non-free summands
+-----------------------------------
+ZZ/2	2^1	512	(312)			(200)
+*-
diff --git a/M2/Macaulay2/packages/DirectSummands/docs.m2 b/M2/Macaulay2/packages/DirectSummands/docs.m2
new file mode 100644
index 00000000000..0ced42fb882
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/docs.m2
@@ -0,0 +1,93 @@
+-- TODO:
+-- [x] example that is indecomposable
+-- [ ] example that decomposes after field extension
+-- [ ] example over ZZ, ZZ[i]/(i^2+1), QQ, QQ[i]/(i^2+1), GF, RR, CC?
+-- [ ] graded example
+-- [ ] local example
+-- [ ] example of passing hints of summands
+-- [ ] example of isIndecomposable
+
+doc ///
+Node
+  Key
+    DirectSummands
+  Headline
+    decompositions of graded modules and coherent sheaves
+  Description
+    Text
+      As an example, we prove the indecomposability of the @wikipedia "Horrocks–Mumford bundle"@ on $\PP^4$.
+    Example
+      needsPackage "BGG"
+      S = ZZ/32003[x_0..x_4];
+      E = ZZ/32003[e_0..e_4, SkewCommutative => true];
+      alphad = map(E^5, E^{-2,-2}, transpose matrix{
+	      { e_1*e_4, -e_0*e_2, -e_1*e_3, -e_2*e_4,  e_0*e_3},
+	      {-e_2*e_3, -e_3*e_4,  e_0*e_4, -e_0*e_1, -e_1*e_2}});
+      alpha = syz alphad;
+      alphad = beilinson(alphad, S);
+      alpha = beilinson(alpha, S);
+      FHM = prune homology(alphad, alpha)
+      assert(2 == rank FHM)
+      -- initially ~30s for End(FHM), ~110s for basis; ~35s in ZZ/2; now down to ~1s total!
+      assert({FHM} == summands FHM)
+      assert FHM.cache.isIndecomposable
+  Acknowledgement
+    The authors thank the organizers of the @HREF{"https://aimath.org/pastworkshops/macaulay2efie.html",
+	"Macaulay2 workshop at AIM"}@, where significant progress on this package was made.
+  Subnodes
+    directSummands
+
+Node
+  Key
+    directSummands
+    (directSummands, Module)
+    (directSummands, CoherentSheaf)
+  Headline
+    computes the direct summands of a graded module or coherent sheaf
+  Usage
+    summands M
+  Inputs
+    M:{Module,CoherentSheaf}
+  Outputs
+    :List
+      containing modules or coherent sheaves which are direct summands of $M$
+  Description
+    Text
+      This function attempts to find the indecomposable summands of a module or coherent sheaf $M$.
+    Example
+      S = QQ[x,y]
+      M = coker matrix{{x,y},{x,x}}
+      L = summands M
+      assert isIsomorphic(M, directSum L)
+  SeeAlso
+    findIdempotents
+///
+
+-- Template:
+///
+Node
+  Key
+  Headline
+  Usage
+  Inputs
+  Outputs
+  Consequences
+    Item
+  Description
+    Text
+    Example
+    CannedExample
+    Code
+    Pre
+  ExampleFiles
+  Contributors
+  References
+  Caveat
+  SeeAlso
+--    Tree
+--    CannedExample
+--  Contributors
+--  References
+--  Caveat
+--  SeeAlso
+///
diff --git a/M2/Macaulay2/packages/DirectSummands/frobenius.m2 b/M2/Macaulay2/packages/DirectSummands/frobenius.m2
new file mode 100644
index 00000000000..008f0cb714e
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/frobenius.m2
@@ -0,0 +1,213 @@
+--needsPackage "PushForward"
+--needsPackage "Polyhedra" -- for lattice points
+--needsPackage "Complexes"
+
+myPushForward = (f, M) -> (
+    directSum apply(cachedSummands M,
+	N -> pushFwd(f, N))
+    -- doesn't work for matrices:
+    -- pushForward(f, M)
+    -- pushForward(f, M, UseHilbertFunction => false)
+    )
+
+-----------------------------------------------------------------------------
+-* Frobenius pushforwards *-
+-----------------------------------------------------------------------------
+
+protect FrobeniusRing
+protect FrobeniusFormation
+frobeniusRing = method(TypicalValue => Ring)
+frobeniusRing(ZZ, Ring) := (e, R) -> (
+    if not R.?cache then R.cache = new CacheTable;
+    (Rp0, e0) := if R.cache.?FrobeniusFormation then R.cache.FrobeniusFormation else (R, 0);
+    if Rp0.cache#?(symbol FrobeniusRing, e0 + e) then Rp0.cache#(symbol FrobeniusRing, e0 + e)
+    else Rp0.cache#(symbol FrobeniusRing, e0 + e) = (
+	Rpe := newRing(Rp0, Degrees => (char Rp0)^(e0 + e) * degrees Rp0);
+	Rpe.cache = new CacheTable;
+	Rpe.cache.FrobeniusFormation = (Rp0, e0 + e);
+	Rpe)
+    )
+
+-- FIXMEEE:
+quotient Ideal := QuotientRing => opts -> (cacheValue symbol quotient) (I -> (ring I) / I)
+--ideal Ring := (cacheValue symbol ideal) (R -> ideal map(R^1, R^0, 0))
+
+-- FIXME: this might forget some information
+RingMap ** Module := Module => (f, M) -> directSum apply(cachedSummands M, N -> tensor(f, N))
+
+-- TODO: should also work if S is a finite field
+-- defined as a QuotientRing rather than GaloisField
+frobeniusTwistMap = (e, S) -> (
+    k := coefficientRing S;
+    if char S == 0 or k_0 == 1 then return map(S, S);
+    a := k_0;
+    p := char k;
+    map(S, S, gens S | {a^(p^e)}))
+
+/// -- TODO: add as test:
+  R = QQ[x,y] -- or ZZ/p
+  assert(R === frobeniusTwist(1, R))
+///
+
+protect FrobeniusTwist
+frobeniusTwist = method()
+frobeniusTwist(ZZ, Ring) := Ring => (e, S) -> (
+    k := coefficientRing S;
+    if char S == 0 or k_0 == 1 then return S;
+    if not S.?cache then S.cache = new CacheTable;
+    -- FIMXE: towers of pushforwards
+    S.cache#(symbol FrobeniusTwist, e) ??= (
+	F := frobeniusTwistMap(e, ring ideal S);
+	quotient F ideal S))
+frobeniusTwist(ZZ, Ideal) := Ideal => (e, I) -> frobeniusTwist(e, module I)
+frobeniusTwist(ZZ, Module) := Module => (e, M) -> (
+    S := ring M;
+    R := frobeniusTwist(e, S);
+    F := frobeniusTwistMap(e, S);
+    -- TODO: is this correct?
+    F ** M ** R)
+frobeniusTwist(ZZ, Matrix) := Matrix => (e, f) -> (
+    map(frobeniusTwist(e, target f), frobeniusTwist(e, source f),
+	frobeniusTwistMap(e, ring f) ** f))
+
+--TODO: maybe export "frobeniusMap"
+frobeniusMap = method(TypicalValue => RingMap)
+frobeniusMap(Ring, ZZ) := (R, e) -> frobeniusMap(e, R)
+frobeniusMap(ZZ, Ring) := (e, R) -> (
+    map(Re := frobeniusTwist(e, R), frobeniusRing(e, R),
+	apply(gens Re, g -> g^((char R)^e))))
+
+decomposeFrobeniusPresentation = (e, f) -> decomposePushforwardPresentation((char ring f)^e, f)
+
+protect FrobeniusPushforward
+frobeniusPushforward = method()
+--frobeniusPushforward(Thing, ZZ)   := (T, e) -> frobeniusPushforward(e, T)
+frobeniusPushforward(ZZ, Ring)    := (e, R) -> frobeniusPushforward(e, module R)
+frobeniusPushforward(ZZ, Ideal)   := (e, I) -> frobeniusPushforward(e, quotient I)
+-- TODO: cache in a way that the second pushforward is the same as applying pushforward twice
+frobeniusPushforward(ZZ, Module)  := (e, M) -> M.cache#(FrobeniusPushforward, e) ??= (
+    f := presentation myPushForward(
+	frobeniusMap(e, ring M),
+	frobeniusTwist(e, M));
+    if not isHomogeneous f then coker f
+    else directSum apply(decomposeFrobeniusPresentation(e, f), coker))
+
+frobeniusPushforward(ZZ, Matrix)  := (e, f) -> f.cache#(FrobeniusPushforward, e) ??= (
+    g := myPushForward(
+	frobeniusMap(e, ring f),
+	frobeniusTwist(e, f));
+    if not isHomogeneous g then g
+    else directSum decomposeFrobeniusPresentation(e, g))
+
+frobeniusPushforward(ZZ, SheafMap)  := (e, f) -> f.cache#(FrobeniusPushforward, e) ??= (
+    --if not(isFreeModule module source f and isFreeModule module target f) then error "expected a map between free modules";
+    g := myPushForward(
+	frobeniusMap(e, ring matrix f),
+	frobeniusTwist(e, matrix f));
+    if not isHomogeneous g then g
+    else Fg := first decomposeFrobeniusPresentation(e, g);
+    R := ring matrix f;
+    p := char R;
+    targetPres := presentation target Fg;
+    (tPrestardegs, tPressrcdegs) := toSequence(-degrees targetPres // p^e);
+    Fgtarget := sheaf coker map(R^tPrestardegs, R^tPressrcdegs, sub(targetPres, R));
+    sourcePres := presentation source Fg;
+    (sPrestardegs, sPressrcdegs) := toSequence(-degrees sourcePres // p^e);
+    Fgsource := sheaf coker map(R^sPrestardegs, R^sPressrcdegs, sub(sourcePres, R));
+    sheaf map(module Fgtarget, module Fgsource, sub(cover Fg, R)))
+
+--frobeniusPushforward(ZZ, Complex) := (e, C) -> () -- TODO
+
+frobeniusPushforward(ZZ, SheafOfRings)  := (e, O) -> (
+    X := variety O;
+    X.cache.FrobeniusPushforward   ??= new MutableHashTable;
+    X.cache.FrobeniusPushforward#e ??= frobeniusPushforward(e, O^1))
+frobeniusPushforward(ZZ, CoherentSheaf) := (e, N) -> N.cache#(FrobeniusPushforward, e) ??= if e == 1 then (
+    R := ring variety N;
+    p := char R;
+    FN := first components frobeniusPushforward(e, module N);
+    -- slow alternative:
+    -- FN = myPushForward(frobeniusMap(e, R), module N);
+    -- prune sheaf image basis(p^e * (max degrees FN // p^e), FN)
+    Fmatrix := sub(presentation FN, R);
+    (tardegs, srcdegs) := toSequence(-degrees Fmatrix // p^e);
+    -- TODO: how long does this take? is it worth caching?
+    sheaf prune coker map(R^tardegs, R^srcdegs, Fmatrix)) else (
+    frobeniusPushforward(1, frobeniusPushforward(e-1, N)))
+
+protect FrobeniusPullback
+frobeniusPullback = method()
+--frobeniusPullback(Thing, ZZ)  := (T, e) -> frobeniusPullback(e, T)
+frobeniusPullback(ZZ, Module) := (e, M) -> M.cache#(FrobeniusPullback, e) ??= (
+	R := ring M;
+	p := char R;
+	F := frobeniusMap(R, e);
+	R0 := source F;
+	A := presentation M;
+	A0 := sub(A, R0);
+	coker(F ** map(R0^(-(p^e) * degrees target A0), , A0)))
+frobeniusPullback(ZZ, CoherentSheaf) := (e, F) -> sheaf frobeniusPullback(e, module F)
+
+end--
+restart
+needsPackage "DirectSummands"
+needsPackage "NormalToricVarieties"
+
+-- Two cubics on P^2_(ZZ/2)
+X = toricProjectiveSpace(2, CoefficientRing => ZZ/2)
+S = ring X
+I = ideal(x_0^3+x_1^3+x_2^3)
+J = ideal(x_0^3+x_0^2*x_1+x_1^3+x_0*x_1*x_2+x_2^3)
+
+R = quotient I
+assert(rank \ summands frobeniusPushforward(1, OO_(Proj R)) == {2})
+assert(rank \ summands frobeniusPushforward(1, R) == {2,2})
+--M = coker frobeniusPushforward(char S, I) -- TODO: consolidate with toric version
+
+R = quotient J
+assert(rank \ summands frobeniusPushforward(1, OO_(Proj R)) == {1,1})
+assert(rank \ summands frobeniusPushforward(1, R) == {1, 1, 2}) -- FIXME: this is not correct
+--M = coker frobeniusPushforward(char S, J) -- TODO: consolidate with toric version
+
+--
+R = quotient I
+M = frobeniusPushforward(1, R);
+N1 = frobeniusPushforward(2, R)
+N2 = frobeniusPushforward(1, M)
+assert(N1 == N2) -- FIXME: why is this different?
+N2' = prune coker frobeniusPushforward(1, presentation M)
+assert(N2 == N2')
+
+
+--
+S=(ZZ/2)[x_0..x_2,y_0..y_2,Degrees=>{{1,0},{1,0},{1,0},{0,1},{0,1},{0,1}}];
+J=ideal(x_0*y_0+x_1*y_1+x_2*y_2);
+B=S/J;
+Y=Proj B;
+ frobeniusPushforward(B,1)
+--why is this giving 0? (if the degrees are standard it doesn't)
+--this is now fixed with the new code for decomposeFrobeniusPresentation -DM
+
+
+---
+restart
+needs "frobenius.m2"
+debug PushForward
+S=(ZZ/2)[x_0,x_1,x_2,y_0,y_1,y_2,Degrees=>{{1,0},{1,0},{1,0},{0,1},{0,1},{0,1}}]
+S0=(ZZ/2)[x_0,x_1,x_2]**(ZZ/2)[y_0,y_1,y_2];
+S0=tensor((ZZ/2)[x_0,x_1,x_2], (ZZ/2)[y_0,y_1,y_2], DegreeMap => null)
+e=1
+errorDepth=1
+target presentation myPushForward(frobeniusMap(e, ring S^1), frobeniusTwist(e, S^1))
+target presentation myPushForward(frobeniusMap(e, ring S0^1), frobeniusTwist(e, S0^1))
+
+g' = g
+peek g
+
+degrees (pushAuxHgs g')_0
+degrees (pushAuxHgs g'')_0
+--why are the degrees right for S but not S0?
+
+RB = RA = S0
+tensor(RB, RA, Join => false)
+tensor(RB, RA, Join => true)
diff --git a/M2/Macaulay2/packages/DirectSummands/homogeneous.m2 b/M2/Macaulay2/packages/DirectSummands/homogeneous.m2
new file mode 100644
index 00000000000..5b2ded09b4c
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/homogeneous.m2
@@ -0,0 +1,113 @@
+needsPackage "RationalPoints2"
+
+findProjectors = method(Options => DirectSummandsOptions)
+findProjectors Module := opts -> M -> (
+    R := ring M;
+    p := char R;
+    F := groundField R;
+    K := quotient ideal gens R;
+    n := numgens M;
+    L := null;
+    -- this is used in generalEndomorphism
+    -- to avoid recomputing the Hom module
+    (pr, inc) := opts#"Splitting" ?? (id_M, id_M);
+    -- TODO: sort the degrees to make finding eigenvalues faster?
+    -- degs := unique sort degrees M;
+    tries := opts.Tries ?? defaultNumTries p;
+    for c to tries - 1 do (
+	f := generalEndomorphism(M, pr, inc); -- about 20% of computation
+	if f == 0 then continue;
+	-- eigenvalues of f are necessarily over the field,
+	-- and we can prove that f can be diagonalized over R
+	-- (i.e. without passing to frac R), hence we can
+	-- compute the eigenvalues by going to the field
+	f0 := sub(K ** cover f, F);
+	-- finding eigenvalues would be faster if the matrix
+	-- was put in Jordan form first, but this is easier...
+	-- TODO: computing eigenvalues over coefficient field
+	-- would significantly speed up this step
+	eigen := eigenvalues'' f0; -- about 25% of computation
+	projs := select(for y in eigen list (f - y * id_M)^n,
+	    g -> not zero g and not isInjective g);
+	if #projs < 1 then (
+	    -- to be used as a suggestion in the error
+	    -- TODO: is there any way to tell if the module is indecomposable here?
+	    -- e.g. based on the characteristic polynomial factoring completely
+	    -- but having a single root only? (= End_0(M) has only one generator?)
+	    -- TODO: expand for inexact fields
+	    if L === null and not instance(F, InexactField) then L = extField { char f0 };
+	    continue);
+	return projs
+    );
+    -- TODO: skip the "Try using" line if the field is large enough, e.g. L === K
+    -- TODO: if L is still null, change the error
+    error("no idempotent found after ", tries, " attempts. ",
+	"Try using changeBaseField with ", toString L))
+
+-- TODO: can this be useful?
+findBasicProjectors = M -> (
+    R := ring M;
+    F := groundField R;
+    K := quotient ideal gens R;
+    n := numgens M;
+    B := gensEnd0 M;
+    for c to numcols B - 1 do (
+	f := homomorphism B_{c};
+	if f == id_M then return;
+	f0 := sub(K ** cover f, F);
+	eigen := eigenvalues' f0;
+	if #eigen > 1 then return for y in eigen list (f - y * id_M)^n);
+    {})
+
+-- this algorithm does not depend on finding idempotents,
+-- hence it is distinct from the Meat-Axe algorithm.
+summandsFromProjectors = method(Options => options findProjectors)
+summandsFromProjectors Module := opts -> M -> (
+    if not isHomogeneous M then error "expected homogeneous module";
+    if opts.Verbose then printerr "splitting summands using projectors";
+    if rank cover M <= 1 or prune' M == 0 then return {M};
+    -- TODO: if M.cache.Idempotents is nonempty, should we use it here?
+    -- maps M -> M whose (co)kernel is a (usually indecomposable) summand
+    projs := try findProjectors(M, opts) else return {M};
+    summandsFromProjectors(M, projs, opts))
+
+-- keep close to summandsFromIdempotents
+-- this algorithm is more efficient as it has a significant
+-- chance of splitting the module in a single iteration.
+summandsFromProjectors(Module, Matrix) := opts -> (M, pr) -> summandsFromProjectors(M, {pr}, opts)
+summandsFromProjectors(Module, List) := opts -> (M, ends) -> (
+    checkRecursionDepth();
+    -- in some examples, we use barebones splitComponentsBasic
+    if opts.Strategy & 4 == 4 or not isHomogeneous M
+    then return splitComponentsBasic(M, ends, opts);
+    -- maps from kernel summands and to cokernel summands
+    injs  := apply(ends, h -> inducedMap(M, ker h));
+    projs := apply(ends, h -> inducedMap(coker h, M));
+    -- composition of all endomorphisms is the complement
+    comp := product ends;
+    injs  = append(injs,  inducedMap(M, image comp));
+    projs = append(projs, inducedMap(image comp, M, comp));
+    -- assert(0 == intersect apply(ends, ker));
+    -- assert(0 == intersect apply(injs, image));
+    -- assert isIsomorphic(M, directSum apply(projs, target));
+    -- this is the splitting (surjection, inclusion) of M to a module
+    -- whose degree zero endomorphisms have already been computed.
+    (pr0, inc0) := opts#"Splitting" ?? (id_M, id_M);
+    if opts.Verbose then printerr("splitting summands of ranks ",
+	toString apply(injs, i -> rank source i));
+    c := -1; -- component counter
+    comps := for n to #ends list (
+	(pr, inc) := (projs#n, injs#n);
+	(N0, K0) := (target pr, source inc);
+	if (N := prune' N0) == 0 then continue;
+	-- TODO: can we check if M has multiple copies of N quickly?
+	iso := try isomorphism(K0, N0);
+	p := inverse N.cache.pruningMap * pr;
+	i := try inc * iso * N.cache.pruningMap;
+	M.cache#(symbol ^, [c += 1]) = p; -- temporary
+	N.cache.components = summandsFromProjectors(N,
+	    opts, "Splitting" => (p * pr0, try inc0 * i));
+	N);
+    --M.cache.Idempotents = apply(c, i -> M.cache#(symbol ^, [i]));
+    -- Finally, call a helper to add splitting maps
+    splitComponents(M, comps, components))
diff --git a/M2/Macaulay2/packages/DirectSummands/idempotents.m2 b/M2/Macaulay2/packages/DirectSummands/idempotents.m2
new file mode 100644
index 00000000000..c96e34bba2c
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/idempotents.m2
@@ -0,0 +1,275 @@
+--needsPackage "RationalPoints2"
+
+-----------------------------------------------------------------------------
+-- helpers that should probably move to Core
+-----------------------------------------------------------------------------
+
+-- same as flatten(Matrix), but doesn't bother homogenizing the result
+--flatten' = m -> map(R := ring m, rawReshape(m = raw m, raw R^1, raw R^(rawNumberOfColumns m * rawNumberOfRows m)))
+
+leadCoefficient Number := x -> x
+leadMonomial    Number := x -> 0
+
+-- not strictly speaking the "lead" coefficient, but the first nonzero coefficient
+leadCoefficient Matrix := RingElement => m -> if zero m then 0 else (
+    for c to numcols m - 1 do for r to numrows m - 1 do (
+	if not zero m_(r,c) then return leadCoefficient m_(r,c)))
+
+-- not strictly speaking the "lead" monomial, but the first nonzero monomial
+leadMonomial Matrix := RingElement => m -> if zero m then 0 else (
+    for c to numcols m - 1 do for r to numrows m - 1 do (
+	if not zero m_(r,c) then return leadMonomial m_(r,c)))
+
+-- used to be called reduceScalar
+reduceCoefficient = m -> if zero m then m else (
+    map(target m, source m, cover m // leadCoefficient m))
+
+reduceMonomial = m -> if zero m then m else (
+    map(target m, source m, cover m // leadMonomial m))
+
+-- hacky things for CC
+-- TODO: move to Core, also add conjugate Matrix, realPart, imaginaryPart, etc.
+conjugate RingElement := x -> sum(listForm x, (e, c) -> conjugate c * (ring x)_e)
+magnitude = x -> x * conjugate x
+isZero = x -> if not instance(F := ultimate(coefficientRing, ring x), InexactField) then x == 0 else (
+    leadCoefficient magnitude x < 2^(-precision F))
+
+-- borrowed from Varieties as hack to get around
+-- https://github.com/Macaulay2/M2/issues/3407
+flattenMorphism = f -> (
+    g := presentation ring f;
+    S := ring g;
+    -- TODO: sometimes lifting to ring g is enough, how can we detect this?
+    -- TODO: why doesn't lift(f, ring g) do this automatically?
+    map(target f ** S, source f ** S, lift(cover f, S)) ** cokernel g)
+
+-- reduceCoefficient is a kludge to handle the case when h^2 = ah
+isIdempotent = h -> reduceCoefficient(h^2) == reduceCoefficient h
+isWeakIdempotent = h -> all(flatten entries flattenMorphism(reduceCoefficient(h^2) - reduceCoefficient h), isZero)
+--isWeakIdempotent = h -> isZero det cover flattenMorphism(reduceCoefficient(h^2) - reduceCoefficient h)
+
+-----------------------------------------------------------------------------
+
+-- e.g. given a tower such as K[x][y]/I, returns K
+-- TODO: use in localRandom?
+groundField = method()
+groundField Ring := R -> ultimate(K -> if isField K then K else coefficientRing K, R)
+
+potentialExtension = method()
+potentialExtension Module := M -> extField {char generalEndomorphism M}
+potentialExtension CoherentSheaf := M -> potentialExtension module M
+
+-- e.g. given a field isomorphic to GF(p,e), returns e
+fieldExponent = R -> (
+    L := groundField R;
+    (p, e) := (char L, 1);
+    if p == 0 then return e;
+    a := L_0; -- primitive element of L
+    while a^(p^e) != a do (e = e + 1);
+    e)
+
+-- finds the characteristic polynomial of a matrix mod the maximal ideal
+char Matrix := A -> A.cache.char ??= (
+    if numRows A != numColumns A then error "expected a square matrix";
+    b := symbol b;
+    T := (groundField ring A)(monoid[b]);
+    B := sub(cover A, T);
+    I := id_(source B);
+    -- TODO: this is a major step in large examples
+    det(B - T_0 * I, Strategy => Bareiss))
+
+eigenvalues' = A -> (
+    Chi := char A;
+    F := groundField ring A;
+    if instance(F, InexactField) then roots Chi
+    else flatten rationalPoints ideal Chi)
+
+fieldElements = method()
+fieldElements QuotientRing := ZZp -> apply(ZZp.order, i -> i_ZZp)
+fieldElements GaloisField  := GFq -> prepend_(0_GFq) apply(GFq.order - 1, i -> GFq_0^i)
+fieldElements' = memoize fieldElements -- FIXME: don't cache globally
+
+-- dumb search over finite fields ...
+eigenvalues'' = A -> (
+    R := ring A;
+    p := char R;
+    F := groundField R;
+    I := id_(target A);
+    if p == 0 or not F.?order or F.order > 1000 then return eigenvalues' A;
+    select(fieldElements' F, e -> zero det(A - e * I)))
+
+largePower = (p,l,M) -> (
+    if p^l < 2^30 then return M^(p^l);
+    --should have this line check for monomial size of ambient ring also
+    N := M;
+    for i from 1 to l do N = N^p;
+    N)
+
+-- TODO: use BinaryPowerMethod?
+largePower' = (p,l,M) -> (
+    if p^l < 2^30 then return M^(p^l-1);
+    --should have this line check for monomial size of ambient ring also
+    N := M;
+    (largePower(p,l-1,M))^(p-1) * largePower'(p,l-1,M))
+
+lift(CC, CC_*) := opts -> (r, C) -> numeric(precision C, r)
+
+-- adjust as needed LOL
+findErrorMargin = m -> ceiling(log_10 2^(precision ring m))
+
+-- TODO: move to LocalRings?
+residueField = method()
+residueField Ring      := R -> quotient ideal vars R
+residueField LocalRing := R -> target R.residueMap
+
+residueMap' = method()
+residueMap' Ring      := R -> map(quotient ideal vars R, R, vars R % ideal vars R)
+residueMap' LocalRing := R -> map(quotient ideal R.maxIdeal, R, vars baseRing R % R.maxIdeal)
+
+-----------------------------------------------------------------------------
+-- findIdempotents
+-----------------------------------------------------------------------------
+
+-- TODO: findIdem right now will fail if K is not L[a]/f(a);
+-- in general, will need to find a primitive element first
+findIdempotents = method(Options => DirectSummandsOptions)
+findIdempotents CoherentSheaf := opts -> M -> findIdempotents(module M, opts)
+findIdempotents Module        := opts -> M -> (
+    R := ring M;
+    p := char R;
+    F := groundField R;
+    e := fieldExponent R;
+    K := residueMap' R;
+    V := K ** M;
+    inexactFlag := instance(F, InexactField);
+    l := if p == 0 then e else max(e, ceiling log_p numgens M);
+    L := null;
+    -- this is used in generalEndomorphism
+    -- to avoid recomputing the Hom module
+    (pr, inc) := opts#"Splitting" ?? (id_M, id_M);
+    limit := opts.Limit ?? numgens M;
+    tries := opts.Tries ?? defaultNumTries p;
+    for c to tries - 1 do (
+	f := generalEndomorphism(M, pr, inc);
+	fm := sub(K ** cover f, F);
+	if fm == 0 then continue;
+	-- if at most one eigenvalue is found the module is probably indecomposable,
+	-- unless the characteristic polynomial has odd degree, then one is enough.
+	eigen := eigenvalues'' fm;
+	-- we only want eigenvalues in F
+	eigen = nonnull apply(eigen, y -> try lift(y, F));
+	if #eigen <= 1 then (
+	    -- to be used as a suggestion in the error
+	    -- TODO: expand for inexact fields
+	    if L === null and not inexactFlag then L = try extField { char fm };
+	    -- if char fm doesn't factor over F, or if it fully factors
+	    -- but has only one eigenvalue, we can't find an idempotent
+	    if #eigen == 1 and F === L
+	    or #eigen == 0 then continue);
+	-- try to find idempotens from eigenvalues
+	isUsable := gm -> isWeakIdempotent gm and not isSurjective gm and gm != 0;
+	largePow := (j, g) -> largePower'(p, j+1, largePower(p, l, g));
+	-- TODO: use limit here
+        idems := nonnull flatten for y in eigen list (
+	    if p > 0 then for j from 0 to e do (
+		if isUsable largePow(j, fm - y*id_V) then break (j, f - y*id_M))
+	    else if isUsable(fm - y*id_V) then (1, f - y*id_M));
+	idems = select(idems, (j, f) ->
+	    prune' image f != 0 and prune' coker f != 0 and prune' image f != M);
+	if #idems == 0 then continue;
+	return apply(idems, (j, g) -> (
+		idem := if p == 0 then g else largePow(j, g);
+		-- for inexact fields, we compose the idempotent until the determinant is zero
+		if inexactFlag then idem = idem ^ (findErrorMargin idem);
+		idem)));
+    -- TODO: skip the "Try using" line if the field is large enough, e.g. L === K
+    -- TODO: if L is still null, change the error
+    error("no idempotent found after ", tries, " attempts. ",
+	"Try using changeBaseField with ", toString L))
+
+-- for backwards compatibility
+findIdempotent = options findIdempotents >> opts -> M -> first findIdempotents(M, opts)
+
+protect Idempotents
+
+-- only tries to find an idempotent among the generators of End_0(M)
+-- which is in general unlikely to be successful, but it often works!
+-- returns a pair: (idempotent or null, whether M is certified indecomposable)
+findBasicIdempotents = options findIdempotents >> opts -> M -> (
+    M.cache.Idempotents ??= {};
+    if 0 < #M.cache.Idempotents then return (M.cache.Idempotents, false);
+    R := ring M;
+    K := residueMap' R;
+    -- FIXME: this may not be correct
+    if instance(R, LocalRing) then (
+	M = liftUp M;
+	K = residueMap' ring M);
+    B := gensEnd0 M;
+    -- whether all non-identity endomorphisms are zero mod m
+    -- if this remains true till the end, the module is
+    -- certifiably indecomposable.
+    certified := true;
+    -- TODO: searching over 10k generators for F_*(OO_X)
+    -- on Gr(2,4) even over ZZ/3 takes a very long time
+    -- TODO: parallelized this and break on first success
+    idemp := scan(numcols B, c -> (
+	    h := homomorphism B_{c};
+	    if zero h or h == id_M
+	    or zero(hm := K ** cover h) then return;
+	    certified = false;
+	    if isWeakIdempotent hm then break h));
+    if idemp =!= null then (
+	if opts.Verbose then printerr "splitting summands using a basic idempotent";
+	M.cache.Idempotents |= { idemp });
+    (M.cache.Idempotents, certified))
+
+-- this is essentially the Meat-Axe algorithm,
+-- but the process for finding an idempotent for
+-- a module over a polynomial ring makes it distinct.
+summandsFromIdempotents = method(Options => options findIdempotents)
+summandsFromIdempotents Module := opts -> M -> (
+    if opts.Verbose then printerr "splitting summands using idempotents";
+    if rank cover M <= 1 or prune' M == 0 then return {M};
+    idems := try M.cache.Idempotents else {};
+    if 0 == #idems then try
+    idems = findIdempotents(M, opts) else return {M};
+    summandsFromIdempotents(M, idems, opts))
+
+-- keep close to summandsFromProjectors
+summandsFromIdempotents(Module, Matrix) := opts -> (M, h) -> summandsFromIdempotents(M, {h}, opts)
+summandsFromIdempotents(Module, List)   := opts -> (M, ends) -> (
+    checkRecursionDepth();
+    -- in some examples, we use barebones splitComponentsBasic
+    if opts.Strategy & 4 == 4 or not isHomogeneous M
+    then return splitComponentsBasic(M, ends, opts);
+    -- maps from kernel summands and to cokernel summands
+    injs  := apply(ends, h -> inducedMap(M, ker h));
+    projs := apply(ends, h -> inducedMap(coker h, M));
+    -- composition of all endomorphisms is the complement
+    comp := product ends;
+    injs  = append(injs,  inducedMap(M, image comp));
+    projs = append(projs, inducedMap(image comp, M, comp));
+    -- assert(0 == intersect apply(ends, ker));
+    -- assert(0 == intersect apply(injs, image));
+    -- assert isIsomorphic(M, directSum apply(projs, target));
+    -- this is the splitting (surjection, inclusion) of M to a module
+    -- whose degree zero endomorphisms have already been computed.
+    (pr0, inc0) := opts#"Splitting" ?? (id_M, id_M);
+    if opts.Verbose then printerr("splitting summands of ranks ",
+	toString apply(injs, i -> rank source i));
+    c := -1; -- component counter
+    comps := for n to #ends list (
+	(pr, inc) := (projs#n, injs#n);
+	(N0, K0) := (target pr, source inc);
+	if (N := prune' N0) == 0 then continue;
+	-- TODO: can we check if M has multiple copies of N quickly?
+	iso := try isomorphism(K0, N0);
+	p := inverse N.cache.pruningMap * pr;
+	i := try inc * iso * N.cache.pruningMap;
+	M.cache#(symbol ^, [c += 1]) = p; -- temporary
+	N.cache.components = summandsFromIdempotents(N,
+	    opts, "Splitting" => (p * pr0, try inc0 * i));
+	N);
+    --M.cache.Idempotents = apply(c, i -> M.cache#(symbol ^, [i]));
+    -- Finally, call a helper to add splitting maps
+    splitComponents(M, comps, components))
diff --git a/M2/Macaulay2/packages/DirectSummands/large-eigenvalue.m2 b/M2/Macaulay2/packages/DirectSummands/large-eigenvalue.m2
new file mode 100644
index 00000000000..3ac0d42fdd5
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/large-eigenvalue.m2
@@ -0,0 +1,10 @@
+kk = ZZ/5
+R = kk[p_(0,1)..p_(0,2), p_(1,2), p_(0,3), p_(1,3), p_(2,3)]/(p_(1,2)*p_(0,3)-p_(0,2)*p_(1,3)+p_(0,1)*p_(2,3))
+tar = R^{256:{-2}}
+src = R^{256:{-3}}
+mat = 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+-- non-free summand of F_*(OO_X) on X = Gr(1,3) over F_5
+N = coker map(tar, src, mat);
+-- about 3hrs to compute gensEnd0 N
+-- after that, manually find one set of projectors in a few seconds,
+-- split to get the rank 2 summands, then peel those off in a minute
diff --git a/M2/Macaulay2/packages/DirectSummands/large-tests.m2 b/M2/Macaulay2/packages/DirectSummands/large-tests.m2
new file mode 100644
index 00000000000..299775099ca
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/large-tests.m2
@@ -0,0 +1,55 @@
+TEST ///
+  -- testing reduceScalar
+--  restart
+  debug needsPackage "DirectSummands"
+  assert elapsedTime isIdempotent(2 * id_((ZZ/5[x])^500)) -- .03s
+  --
+  R = (ZZ/3)[x,y,z,w]/(x^3*y+y^3*z+x*y*z*w+z^3*w+x*w^3)
+  m = map(subquotient(map(R^36,R^{{0}, {-1}, {-1}, {-1}, {-1}, {-1}, {-1}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}},{{1, 0, 0, 0, 0, 0, 0, 0, y*z, 0, w^2, x*w, y^2, z*w, x*w, x*y+z^2, z^2, y*z, 0, 0, 0, 0, 0, 0, 0, y^2, 0, 0}, {0, 0, z, 0, 0, y, 0, 0, 0, w^2, 0, 0, 0, 0, 0, 0, 0, 0, y^2, 0, x*w, 0, 0, 0, 0, 0, 0, y*w}, {0, w, 0, z, 0, 0, y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, y*w, 0, 0, 0, 0, y^2, 0, x*w, 0, y*z, 0, 0}, {0, y, x, 0, 0, z, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, x*w, 0, y^2, 0, 0, 0}, {0, 0, 0, x, 0, 0, z, x*w, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, 0, 0, 0, 0, 0, 0, y^2, 0}, {0, 0, 0, 0, 0, 0, 0, 0, x*w, 0, y^2, z*w, 0, 0, 0, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, z, x, 0, 0, 0, 0, 0, 0, z^2, x*z, x*w, x*w, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w}, {1, x, 0, 0, z, 0, 0, y*w, y*z, z*w, w^2, x*w, 0, 0, 0, 0, 0, w^2, 0, 0, x*z, 0, z^2, 0, 0, y^2, 0, 0}, {0, z, 0, 0, 0, 0, w, 0, 0, x*z, 0, 0, 0, 0, 0, 0, y*z, x*w, 0, z^2, 0, 0, 0, x*z, 0, x^2, 0, 0}, {0, 0, 0, 0, x, 0, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, x, 0, 0, z, x*w, x*z, y*z, 0, z^2, 0, 0, 0, 0, x^2, x*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, y, x, w, 0, 0, 0, 0, 0, z*w, y*w, x^2, y*z, y*z, y^2+x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w}, {0, w, y, 0, 0, 0, 0, z*w, 0, x*w, 0, 0, 0, 0, 0, 0, y*w, 0, z*w, 0, x^2, 0, 0, 0, 0, 0, 0, 0}, {1, x, 0, y, z, w, 0, x*z, y*z, z*w, w^2, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, x^2, 0, x*w, 0, w^2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, 0, y*w}, {0, 0, 0, 0, x, 0, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0}, {0, y, x, 0, 0, z, 0, y*z, x^2, 0, z*w, y*w, 0, 0, 0, 0, x*w, 0, 0, 0, 0, 0, x*w, 0, 0, 0, 0, z*w}, {0, 0, z, x, w, 0, 0, 0, 0, 0, x*z, 0, x*y, y*z+w^2, z^2, z*w, z*w, y*w, x*w, x*w, 0, 0, z*w, 0, 0, 0, 0, 0}, {0, 0, w, 0, y, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, 0, y*z, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, y*z+w^2, 0, x*y, 0, z^2, 0, y*w, 0, 0}, {1, x, 0, y, z, w, 0, x*z, y*z, z*w, y*z+w^2, x*w, 0, 0, 0, 0, 0, 0, y*w, y*w, 0, 0, 0, 0, x*y, y^2, 0, y*z}, {0, z, 0, 0, 0, 0, w, 0, 0, x*z, x*w, 0, 0, 0, 0, 0, y*z, 0, z^2, 0, 0, 0, 0, 0, 0, 0, x*y, 0}, {0, w, 0, z, 0, 0, y, z*w, z^2, 0, z^2, y*z+w^2, 0, 0, 0, 0, 0, 0, z*w, z*w, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, x, w, 0, 0, 0, 0, 0, 0, 0, x*z, y*w, x*y, 0, y^2, y^2, z*w, 0, w^2, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, 0, y*w, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, w, 0, y, 0, 0, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, x*y, 0, y*w, y*z, 0, 0, z*w, 0, 0}, {0, w, y, 0, 0, 0, 0, z*w, 0, x*w, 0, y*z, 0, 0, 0, 0, y*w, 0, 0, 0, 0, 0, 0, 0, y*w, 0, 0, x*y}, {1, x, 0, 0, z, 0, 0, y*w, y*z, z*w, w^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, 0, 0, y^2, y*w, 0}, {0, 0, z, 0, 0, y, 0, 0, 0, w^2, y*w, z^2, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, x*w, y*z, x*z, x*z, x*y, 0, 0, 0, 0, 0, 0, 0, w^2, 0, 0}, {0, 0, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, w^2, y*z, 0, 0, 0, 0, y*w, 0, 0}, {0, 0, z, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w, y*z, 0, 0, 0, 0}, {0, 0, 0, 0, y, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w, 0, 0}, {0, 0, 0, 0, 0, z, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}),map(R^36,R^{{0}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}},{{1, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 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y, x, 0, 0, z, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, x*w, 0, y^2, 0, 0, 0}, {0, 0, 0, x, 0, 0, z, x*w, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, 0, 0, 0, 0, 0, 0, y^2, 0}, {0, 0, 0, 0, 0, 0, 0, 0, x*w, 0, y^2, z*w, 0, 0, 0, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, z, x, 0, 0, 0, 0, 0, 0, z^2, x*z, x*w, x*w, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w}, {1, x, 0, 0, z, 0, 0, y*w, y*z, z*w, w^2, x*w, 0, 0, 0, 0, 0, w^2, 0, 0, x*z, 0, z^2, 0, 0, y^2, 0, 0}, {0, z, 0, 0, 0, 0, w, 0, 0, x*z, 0, 0, 0, 0, 0, 0, y*z, x*w, 0, z^2, 0, 0, 0, x*z, 0, x^2, 0, 0}, {0, 0, 0, 0, x, 0, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, x, 0, 0, z, x*w, x*z, y*z, 0, z^2, 0, 0, 0, 0, x^2, x*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, y, x, w, 0, 0, 0, 0, 0, z*w, y*w, x^2, y*z, y*z, y^2+x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w}, {0, w, y, 0, 0, 0, 0, z*w, 0, x*w, 0, 0, 0, 0, 0, 0, y*w, 0, z*w, 0, x^2, 0, 0, 0, 0, 0, 0, 0}, {1, x, 0, y, z, w, 0, x*z, y*z, z*w, w^2, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, x^2, 0, x*w, 0, w^2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, 0, y*w}, {0, 0, 0, 0, x, 0, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0}, {0, y, x, 0, 0, z, 0, y*z, x^2, 0, z*w, y*w, 0, 0, 0, 0, x*w, 0, 0, 0, 0, 0, x*w, 0, 0, 0, 0, z*w}, {0, 0, z, x, w, 0, 0, 0, 0, 0, x*z, 0, x*y, y*z+w^2, z^2, z*w, z*w, y*w, x*w, x*w, 0, 0, z*w, 0, 0, 0, 0, 0}, {0, 0, w, 0, y, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, 0, y*z, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, y*z+w^2, 0, x*y, 0, z^2, 0, y*w, 0, 0}, {1, x, 0, y, z, w, 0, x*z, y*z, z*w, y*z+w^2, x*w, 0, 0, 0, 0, 0, 0, y*w, y*w, 0, 0, 0, 0, x*y, y^2, 0, y*z}, {0, z, 0, 0, 0, 0, w, 0, 0, x*z, x*w, 0, 0, 0, 0, 0, y*z, 0, z^2, 0, 0, 0, 0, 0, 0, 0, x*y, 0}, {0, w, 0, z, 0, 0, y, z*w, z^2, 0, z^2, y*z+w^2, 0, 0, 0, 0, 0, 0, z*w, z*w, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, x, w, 0, 0, 0, 0, 0, 0, 0, x*z, y*w, x*y, 0, y^2, y^2, z*w, 0, w^2, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y^2, 0, 0, 0, 0, 0, 0, y*w, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, w, 0, y, 0, 0, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, x*y, 0, y*w, y*z, 0, 0, z*w, 0, 0}, {0, w, y, 0, 0, 0, 0, z*w, 0, x*w, 0, y*z, 0, 0, 0, 0, y*w, 0, 0, 0, 0, 0, 0, 0, y*w, 0, 0, x*y}, {1, x, 0, 0, z, 0, 0, y*w, y*z, z*w, w^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, 0, 0, y^2, y*w, 0}, {0, 0, z, 0, 0, y, 0, 0, 0, w^2, y*w, z^2, 0, 0, 0, 0, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, x*w, y*z, x*z, x*z, x*y, 0, 0, 0, 0, 0, 0, 0, w^2, 0, 0}, {0, 0, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, w^2, y*z, 0, 0, 0, 0, y*w, 0, 0}, {0, 0, z, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w, y*z, 0, 0, 0, 0}, {0, 0, 0, 0, y, x, w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w, 0, 0}, {0, 0, 0, 0, 0, z, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w^2, z*w}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}),map(R^36,R^{{0}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}, {-2}},{{1, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*w, y*z, x*y, x^2+z*w, w^2, x*z}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x*y, 0, z*w, w^2, y*w, y^2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z+w^2, z^2, y*w, x*w, x*y, z*w}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*w, x^2, y^2+x*w, x*y, z*w, y*z}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z^2, x*z, x^2, y*z, 0, x*w}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z*w, x*w, y*z, x*z, y^2, x*y+z^2}})),{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z, 0, 0, 0, w, 0, 0, 0, 0, y, 0, z, z, 0, 0, 0, x, y, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, x, 0, 0, 0, 0, y, z, 0, w, 0, 0, 0, 0, 0, w, y, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, w, z, 0, 0, 0, y, 0, 0, 0, x, z, w, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w, 0, 0, 0, y, x, x, 0, w, z, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, w, x, 0, z, x, 0, 0, 0, y, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x, z, z, 0, 0, 0, 0, x, 0, 0, 0, y, 0, 0, w, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}});
+  assert not elapsedTime isIdempotent m -- 0.002
+  assert elapsedTime isIdempotent(2 * id_(target m))
+  -- FIXME: what were these tests for?
+  --assert try m == quotient(m, map(target m, cover target m, 1), Strategy => "Reflexive") else false
+  --quotient(m, map(target m, cover target m, 1), DegreeLimit => {0})
+  --elapsedTime(m // 1);
+///
+
+///
+  -- from David's email: reaches recursion limit overnight
+  needsPackage "DirectSummands"
+  kk = ZZ/101
+  S = kk[x,y,z]
+  I = monomialIdeal(x^4,x*y,y^4,x*z,y^2*z,z^4)
+  R = S/I
+  F = res(coker vars R, LengthLimit => 5)
+  M = coker F.dd_5;
+  debugLevel = 1
+  elapsedTime L5 = summands M; -- takes ~30min on Fields' server
+  assert(#L5 == 75); -- could be wrong
+  assert isIsomorphic(M, directSum L5)
+///
+
+///
+  -- FIXME: why is this test so slow?
+  n = 3
+  S = (ZZ/2)[x_0..x_(n-1)]
+  R = quotient (ideal vars S)^3
+  F = res coker vars R
+  M = image F.dd_3
+  summands M
+  summands(image F.dd_1, M)
+  -- TODO: have a flag to test for twists of input summands as well
+  summands({image F.dd_1, coker vars R}, M)
+///
+
+///
+  restart
+  debug needsPackage "DirectSummands"
+  K = GF(9)
+  K = ZZ/7
+  R = K[b]
+  f = random(R^95, R^95);
+  elapsedTime select(7, i -> 0 == det(f - i * id_(target f))) -- only 33s for 250x250 over ZZ/7
+  elapsedTime select(K.order, i -> 0 == det(f - K_0^i * id_(target f))) -- only 33s for 250x250 over ZZ/7
+  scan(20, n -> elapsedTime eigenvalues' random(R^(n+5), R^(n+5)));
+///
diff --git a/M2/Macaulay2/packages/DirectSummands/pushforward2.m2 b/M2/Macaulay2/packages/DirectSummands/pushforward2.m2
new file mode 100644
index 00000000000..cb986cac31a
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/pushforward2.m2
@@ -0,0 +1,114 @@
+-----------------------------------------------------------------------------
+-* Pushforwards of coherent sheaves *-
+-----------------------------------------------------------------------------
+
+decomposePushforwardPresentation = method()
+decomposePushforwardPresentation(List, Matrix) := (d, f) -> (
+    if #d == 1 then decomposePushforwardPresentation(d#0, f)
+    else error "not implemented for multigraded maps")
+decomposePushforwardPresentation(ZZ, Matrix) := (d, f) -> (
+    if d <= 0 then error "expected positive degree";
+    if d == 1 then return {f};
+    tardegrees := degrees target f;
+    srcdegrees := degrees source f;
+    -- TODO: hypercube(n, 0, 0) gives an error, but it should return the origin
+    cube := flatten \ entries \ latticePoints hypercube(degreeLength ring f, 0, d - 1);
+    tarclasses := apply(cube, i -> positions(tardegrees, deg -> deg % d == i));
+    srcclasses := apply(cube, i -> positions(srcdegrees, deg -> deg % d == i));
+    -- sorts the degrees of source and column
+    tarclasses = apply(tarclasses, ell -> ell_(last \ sort \\ reverse \ toList pairs tardegrees_ell));
+    srcclasses = apply(srcclasses, ell -> ell_(last \ sort \\ reverse \ toList pairs srcdegrees_ell));
+    apply(tarclasses, srcclasses, (tarclass, srcclass) -> submatrix(f, tarclass, srcclass)))
+
+pushForward' = (f, M, opts) -> (
+    if not isHomogeneous f
+    or not isHomogeneous M
+    then error "expected homogeneous inputs";
+    M = cokernel presentation M;
+    M1 := M / ideal f.matrix;
+    M2 := subquotient(matrix basis M1, relations M);
+    cokernel pushNonLinear(opts, f, M2))
+
+protect Pushforwards
+pushForward(RingMap, SheafOfRings)  := opts -> (f, O) -> (
+    X := variety O;
+    X.cache.Pushforwards   ??= new MutableHashTable;
+    X.cache.Pushforwards#f ??= pushForward(f, O^1, opts))
+pushForward(RingMap, CoherentSheaf) := opts -> (f, G) -> (
+    G.cache.Pushforwards   ??= new MutableHashTable )#f ??= (
+    d := degree f(first gens source f); -- FIXME: this should be the degree of the map
+    if #d == 1 then d = d#0 else error "not implemented for multigraded maps";
+    S := source f;
+    m0 := presentation pushForward'(f, module G, opts);
+    m1 := first decomposePushforwardPresentation(d, m0);
+    m2 := sub(m1, S); -- TODO: do better than sub?
+    (tardegs, srcdegs) := toSequence(-degrees m2 // d);
+    -- TODO: how long does this take? is it worth caching?
+    sheaf prune coker map(S^tardegs, S^srcdegs, m2))
+
+-----------------------------------------------------------------------------
+-- TODO: fix in Core
+-----------------------------------------------------------------------------
+
+ordertab := new HashTable from {
+    Eliminate    => (nR, nS) -> Eliminate nR,
+    ProductOrder => (nR, nS) -> ProductOrder{nR, nS},
+    Lex          => (nR, nS) -> Lex,
+    }
+
+pushNonLinear = (opts, f, M) -> (
+    -- given f: R --> S, and M an S-module, finite over R,
+    -- returns R-presentation matrix for the pushforward of M
+    -- written by Mike Stillman and David Eisenbud
+    (R, S) := (target f, source f);
+    deglen := degreeLength S;
+    n1 := numgens R; -- TODO: what if R is a tower?
+
+    monorder := opts.MonomialOrder;
+    monorder  = if ordertab#?monorder then (ordertab#monorder)(numgens R, numgens S)
+    else error("pushForward: MonomialOrder option expected one of ",
+	demark_", " \\ toString \ keys ordertab);
+
+    J := graphIdeal(f, MonomialOrder => monorder, VariableBaseName => local X);
+    G := ring J;
+    m := presentation M;
+    xvars := map(G, R, submatrix(vars G, toList(0..n1-1)));
+    m1 := presentation (cokernel xvars m  **  cokernel generators J);
+
+    if opts.UseHilbertFunction and all({f, m}, isHomogeneous) then (
+	-- compare with kernel RingMap
+	hf := poincare cokernel m;
+	T := degreesRing G;
+	hf = hf * product(degrees source generators J, d -> 1 - T_d);
+	-- cache poincare
+	poincare cokernel m1 = hf);
+
+    mapbackdeg := d -> take(d, -deglen);
+    -- that choice of degree map was chosen to make the symmetricPower functor homogeneous, but it doesn't have much
+    -- else to recommend it.
+    -- we should really be *lifting* the result to S along the natural map S ---> G
+    S' := newRing(S, Degrees => take(degrees G, n1 - numgens G));
+    mapback := map(S', G, map(S'^1, S'^n1, 0) | vars S', DegreeMap => mapbackdeg );
+
+    -- let's at least check it splits f's degree map:
+    for i from 0 to numgens S - 1 do (
+	e := degree S'_i;
+	if mapbackdeg f.cache.DegreeMap e =!= e
+	then error "not implemented yet: unexpected degree map of ring map");
+
+    g := gb(m1,
+	StopBeforeComputation => opts.StopBeforeComputation,
+	DegreeLimit           => opts.DegreeLimit,
+	PairLimit             => opts.PairLimit);
+    -- MES: check if the monomial order restricts to S.  If so, then do `` forceGB result ''
+    mapback selectInSubring(if numgens target f > 0 then 1 else 0, generators g))
+
+-- addHook((pushForward, RingMap, Module), Strategy => "DefaultFixed",
+--     (opts, f, M) -> (
+-- 	if not isHomogeneous f
+-- 	or not isHomogeneous M
+-- 	then return null;
+-- 	M = cokernel presentation M;
+-- 	M1 := M / ideal f.matrix;
+-- 	M2 := subquotient(matrix basis M1, relations M);
+-- 	cokernel pushNonLinear(opts, f, M2)))
diff --git a/M2/Macaulay2/packages/DirectSummands/tests.m2 b/M2/Macaulay2/packages/DirectSummands/tests.m2
new file mode 100644
index 00000000000..4530508456e
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/tests.m2
@@ -0,0 +1,316 @@
+///
+restart
+debug needsPackage "DirectSummands"
+errorDepth = 2
+///
+
+TEST /// -- basic test
+  -- ~0.2s
+  S = QQ[x,y]
+  M = coker matrix{{1,0},{1,y}}
+  A = summands M
+  B = summands prune M
+  C = summands trim M
+  -- FIXME: this keeps annoyingly breaking
+  assert same(prune \ A, {prune M}, B, prune \ C)
+///
+
+TEST /// -- direct summands of a free module
+  -- ~1.1s
+  R = ZZ/2[x_0..x_5]
+  M = R^{100:-2,100:0,100:2}
+  A = summands M;
+  B = summands changeBaseField(2, M);
+  C = summands changeBaseField(4, M);
+  D = summands changeBaseField(GF 101, M);
+  E = summands changeBaseField(GF(2,2), M);
+  assert same(M, directSum A)
+  assert same apply({A, B, C, D, E}, length)
+///
+
+TEST /// -- direct summands of a multigraded free module
+  debug needsPackage "DirectSummands"
+  -- ~0.05s
+  R = QQ[x,y,z] ** QQ[a,b,c]
+  M = R^{{1, 0}, {1, -1}, {0, 0}, {-1, 0}}
+  assert same(M, directSum summands M)
+  assert same(M, directSum sort summandsFromProjectors M)
+  assert same(M, directSum sort summandsFromIdempotents M)
+///
+
+TEST /// -- direct summands of a ring
+  -- ~0.06s
+  S = ZZ/3[x,y,z]
+  R = ZZ/3[x,y,z,w]/(x^3+y^3+z^3+w^3)
+  f = map(R, S)
+  -- TODO: find a non-F-split example
+  M = pushForward(f, module R)
+  assert(summands M == {S^{0}, S^{-1}, S^{-2}})
+///
+
+TEST /// -- direct summands of a finite dimensional algebra
+  R = ZZ/101[x]/x^3
+  T = R/x
+  f = map(R, T)
+  -- FIXME: pushforward is wrong in this case
+  --assert(3 == #summands pushForward_f R^1)
+  --needsPackage "PushForward"
+  --assert(3 == #summands pushFwd_f R^1)
+  f = map(R, prune T)
+  assert(3 == #summands pushForward_f R^1)
+  f = map(R, ZZ/101)
+  assert(3 == #summands pushForward_f R^1)
+///
+
+TEST /// -- direct summands over field extensions
+  -- ~9s
+  R = (ZZ/7)[x,y,z]/(x^3+y^3+z^3);
+  X = Proj R;
+  M = module frobeniusPushforward(1, OO_X);
+  -* is smartBasis useful? yes!
+  elapsedTime A = End M; -- ~0.65s
+  elapsedTime B = basis({0}, A); -- ~0.23s
+  elapsedTime B = smartBasis({0}, A); -- ~0.03s
+  *-
+  -- if this test fails, check if "try findIdempotents M" if hiding any unexpected errors
+  -- FIXME: this is slow because random homomorphisms shouldn't be over the extended field
+  elapsedTime assert({1, 2, 2, 2} == rank \ summands M) -- 2.28s
+  elapsedTime assert({1, 2, 2, 2} == rank \ summands changeBaseField(GF 7, M)) -- 2.87s -> 2.05
+  elapsedTime assert({1, 2, 2, 2} == rank \ summands changeBaseField(GF(7, 3), M)) -- 3.77s -> 2.6
+  elapsedTime assert(toList(7:1)  == rank \ summands changeBaseField(GF(7, 2), M)) -- 2.18s -> 0.47
+///
+
+TEST ///
+  debug needsPackage "DirectSummands"
+  K = GF(7, 2)
+  R = K[x..z]/(x^3+y^3+z^3)
+  M = coker map(R^{6:{-1}}, R^{6:{-2}}, {
+	  {(-a-2)*z, -2*y, (-a+1)*y, 0, x, (-2*a+1)*z},
+	  {(2*a+3)*y, 0, x, (a+2)*z, (3*a-3)*z, y},
+	  {x, (3*a-2)*z, (-a+2)*z, (-2*a+1)*y, (2*a-3)*y, 0},
+	  {(2*a-1)*z, (a+1)*y, (-3*a+1)*y, x, 0, (-2*a-1)*z},
+	  {(-a+3)*y, x, 0, (-2*a-2)*z, z, -3*a*y},
+	  {0, a*z, -2*z, (a-3)*y, (2*a-1)*y, x}})
+  assert({1,1} == rank \ summandsFromProjectors M)
+  assert({1,1} == rank \ summandsFromIdempotents M)
+///
+
+TEST /// -- testing the local case
+  -- the structure is significantly altered by homogenizing modules
+  -- simpler example: nodal cubic in affine vs projective plane
+  debug needsPackage "DirectSummands"
+  k = ZZ/2
+  -- D_4^1 singularity
+  R = k[x,y,z]/(x^2*y + x*y^2 + x*y*z + z^2)
+  M = frobeniusPushforward(1, R)
+  -- uses a basic idem
+  elapsedTime assert(toList(4:1) == rank \ summands M) -- ~2s
+  elapsedTime assert(toList(4:1) == rank \ summandsFromIdempotents M) -- ~0s
+  --
+  k = ZZ/2
+  R = k[x,y,z,h]/(x^2*y + x*y^2 + x*y*z + z^2*h)
+  M = frobeniusPushforward(1, R)
+  elapsedTime assert(toList(8:1) == rank \ summands M) -- <2s
+  --elapsedTime assert(toList(8:1) == rank \ summandsFromProjectors M)  -- 6s
+  --elapsedTime assert(toList(8:1) == rank \ summandsFromIdempotents M) -- 10s
+  --
+  k = ZZ/2
+  R = k[x,y]/(x^2-y^3-y^2)
+  M = frobeniusPushforward(1, R)
+  elapsedTime assert({1,1} == rank \ summands M) -- 3s
+  --
+  R = k[x,y,z]/(x^2*z-y^3-y^2*z)
+  M = frobeniusPushforward(1, R)
+  elapsedTime assert(toList(4:1) == rank \ summands M) -- <2s
+///
+
+TEST /// -- Grassmannian example
+  X = Proj quotient Grassmannian(1, 3, CoefficientRing => ZZ/3);
+  elapsedTime F = frobeniusPushforward(1, OO_X); -- <1s in char 2 & 3
+  elapsedTime assert(splice{65:1, 8:2} == rank \ summands F) -- ~8s
+///
+
+TEST ///
+  -- ~1.1s
+  R = ZZ/32003[x,y,z]/(x^3, x^2*y, x*y^2, y^4, y^3*z)
+  C = res(coker vars R, LengthLimit => 3)
+  D = res(coker transpose C.dd_3, LengthLimit => 3)
+  M = coker D.dd_3
+  elapsedTime L = summands M
+  assert(8 == #L)
+  assert all(L, isHomogeneous)
+  assert isIsomorphic(M, directSum L)
+  assert all(8, i -> same { M, target M_[i], source M^[i] }
+      and same { L#i, target M^[i], source M_[i] })
+  --elapsedTime profile summands M;
+  --profileSummary "DirectSum"
+///
+
+TEST ///
+  -- ~1.7s
+  n = 4
+  S = ZZ/32003[x_0..x_(n-1)]
+  I = trim minors_2 matrix { S_*_{0..n-2}, S_*_{1..n-1}}
+  R = quotient I
+  C = res coker vars R
+  M = prune image C.dd_3
+  elapsedTime L = summands M
+  assert(6 == #L)
+  all(6, i -> isWellDefined M^[i] and isWellDefined M_[i]
+      and M^[i] * M_[i] == id_(L#i))
+///
+
+TEST /// -- testing in char 0
+  -- FIXME:
+  --S = ZZ[x,y];
+  --assert(2 == #summands coker matrix "x,y;y,x")
+  S = QQ[x,y];
+  assert(2 == #summands coker matrix "x,y; y,x")
+  assert(1 == #summands coker matrix "x,y;-y,x")
+  debug needsPackage "DirectSummands"
+  S = QQ[a,b,c,d];
+  assert(3 == #summands coker matrix "a,b,c,d;d,a,b,c;c,d,a,b;b,c,d,a")
+  K = toField(QQ[i]/(i^2+1));
+  S = K[x,y];
+  assert(2 == #summands coker matrix "x,y; y,x")
+  assert(2 == #summands coker matrix "x,y;-y,x")
+  S = K[a,b,c,d];
+  assert(4 == #summands coker matrix "a,b,c,d;d,a,b,c;c,d,a,b;b,c,d,a")
+  S = CC[x,y];
+  -- FIXME scan(20, i -> assert(set summands coker matrix {{x,y},{-y,x}} == set {cokernel matrix {{x-ii*y}}, cokernel matrix {{x+ii*y}}}))
+///
+
+TEST /// -- testing inhomogeneous examples
+  debug needsPackage "DirectSummands"
+  S = GF(2,2)[x,y,z];
+  -- homogeneous baseline, used as control
+  M = coker matrix matrix"x,y,z;y,z,x;z,x,y"
+  assert(3 == #summands M)
+  assert(3 == #summandsFromIdempotents M)
+  assert(3 == #summandsFromProjectors M)
+  assert isIsomorphic(directSum summands M, M, Tries => 10)
+  --
+  -- TODO: this is locally zero, but can we diagonalize it?
+  M = coker matrix matrix"1,y,z;y,1,x;z,x,1"
+  assert(summands M == {M})
+  assert(summandsFromIdempotents M == {M})
+  R = S_(ideal vars S)
+  M = coker matrix matrix"1,y,z;y,1,x;z,x,1"
+  assert(summands M == {M})
+  assert(summandsFromIdempotents M == {M})
+  --
+  S = QQ[x,y,z];
+  M = coker matrix matrix"x,y,z;y,z,x;z,x,y"
+  assert(2 == #summands M)
+  assert(2 == #summandsFromProjectors M)
+  assert isIsomorphic(directSum summands M, M)
+  -- TODO: this is locally zero, but can we diagonalize it?
+  M = coker matrix matrix"1,y,z;y,1,x;z,x,1"
+  assert(summands M == {M})
+  assert(summandsFromIdempotents M == {M})
+///
+
+TEST ///
+  kk = ZZ/101
+  S = kk[x,y,z]
+  P = Proj S
+  T = tangentSheaf P
+  R = S/(x*y-z^2)
+  M = module T ** R
+  -- the module doesn't split, but the sheaf does
+  assert(1 == length summands M)
+  assert(2 == length summands sheaf M)
+///
+
+TEST ///
+  debug needsPackage "DirectSummands"
+  K = ZZ/7
+  R = K[x,y,z]/(x^3+y^3+z^3)
+  X = Proj R
+  --
+  F1 = frobeniusPushforward(1, OO_X);
+  elapsedTime assert({1, 2, 2, 2} == rank \ summands F1) -- 2s
+  elapsedTime L1 = summands changeBaseField(2, F1); -- 5s
+  assert(toList(7:1) == rank \ L1)
+  --
+  F2 = frobeniusPushforward(1, L1#1);
+  elapsedTime assert({7} == rank \ summands F2) -- 2s
+  L = potentialExtension F2
+  elapsedTime L2 = summands changeBaseField(L, F2); -- projectors 14s, idempotents 85s->45s
+  assert(toList(7:1) == rank \ L2)
+  -- tests largepowers, but is very slow:
+  -- findIdempotents changeBaseField(L, F2)
+///
+
+TEST ///
+  debug needsPackage "DirectSummands"
+  kk = ZZ/13
+  S = kk[x,y,z]
+  R = S/(x*z-y^2)
+  M = module frobeniusPushforward(1, OO_(Proj R));
+  elapsedTime L = summands(M, Verbose => true);
+  elapsedTime assert({1,12} == last \ isomorphismTally L);
+  elapsedTime L = summands(S^3000, Verbose => true);
+  elapsedTime assert({3000} == last \ isomorphismTally L);
+  -- nonstandard graded case
+  kk = ZZ/11
+  S = kk[x,y,z, Degrees => {5,1,5}]
+  R = S/(x*z-y^10)
+  M = module frobeniusPushforward(1, OO_(Proj R));
+  elapsedTime L = summands(M, Verbose => true);
+  elapsedTime assert({1,2,2,2,2,2} == last \ isomorphismTally L);
+///
+
+TEST ///
+  kk = ZZ/13
+  R = kk[x,y]
+  m = matrix {{x, 2*y}, {-y, x}}
+  assert(1 == # summands coker m)
+  assert(2 == # summands coker(m ++ m))
+  assert(2 == # summands changeBaseField_2 coker m)
+  assert(4 == # summands changeBaseField_2 coker(m ++ m))
+  --
+  kk = ZZ/32003
+  R = kk[x,y]
+  m = matrix {{x, y}, {-y, x}}
+  assert(1 == # summands coker m)
+  -- FIXME: this seems to almost always fail!!!
+  --assert(2 == # summands coker(m ++ m))
+  assert(2 == # summands changeBaseField_2 coker m)
+  assert(4 == # summands changeBaseField_2 coker(m ++ m))
+  end--
+  findProjectors(coker(m ++ m), Tries => 100)
+  factor char generalEndomorphism(coker m ++ coker m)
+  eigenvalues'' generalEndomorphism(coker m ++ coker m)
+///
+
+///
+  restart
+  errorDepth=2
+  debug needsPackage "DirectSummands"
+  -- TODO: ARRGGAGGGHHHH GF is fucking up 'a'
+  R = ZZ/101[a,b, Degrees => {6,2}]/(a^2+b^6)
+  assert(2 == #summands coker matrix {{a, b^3}, {-b^3, a}})
+  R = ZZ/32003[a,b, Degrees => {6,2}]/(a^2+b^6)
+  assert(1 == #summands coker matrix {{a, b^3}, {-b^3, a}})
+  assert(2 == #summands changeBaseField(2, coker matrix {{a, b^3}, {-b^3, a}}))
+  R = ZZ/32003[a,b]/(a^2+b^6)
+  assert(1 == #summands coker matrix {{a, b^3}, {-b^3, a}})
+  assert(2 == #summands changeBaseField(2, coker matrix {{a, b^3}, {-b^3, a}}))
+  R = GF(32003, 2)[a,b, Degrees => {6,2}]/(a^2+b^6)
+  assert(2 == #summands coker matrix {{a, b^3}, {-b^3, a}})
+
+  R = GF(32003, 2)[a,b]/(a^2+b^6)
+  assert(2 == #summands coker matrix {{a, b^3}, {-b^3, a}})
+
+  M = coker matrix {{a, b^3}, {-b^3, a}}
+  findIdempotents M
+  summands changeBaseField(2, M)
+///
+
+load "./large-tests.m2"
+
+end--
+
+restart
+elapsedTime check "DirectSummands" -- ~48s
diff --git a/M2/Macaulay2/packages/DirectSummands/trivial-algorithm.m2 b/M2/Macaulay2/packages/DirectSummands/trivial-algorithm.m2
new file mode 100644
index 00000000000..e5544693682
--- /dev/null
+++ b/M2/Macaulay2/packages/DirectSummands/trivial-algorithm.m2
@@ -0,0 +1,158 @@
+needsPackage "RationalPoints2"
+debug needsPackage "DirectSummands"
+
+List * Set := List => (x,y) -> select(x, i -> y#?i)
+
+idempotentsFromElimination = M -> (
+    R := ring M;
+    F := groundField R;
+    B := gensEnd0 M;
+    -- TODO: can we reduce all to the residue field and solve there?
+    H := apply(numcols B, i -> homomorphism B_{i}) - set { id_M, 0 * id_M };
+    T := R(monoid[Variables => #H]);
+    f := map(T, R);
+    A := sum(gens T, H, (g, h) -> g * (f ** h));
+    I := trim ideal last coefficients(cover(A^2 - A),
+	Variables => first entries f(vars R));
+    -- decompose I
+    T' := F(monoid T);
+    f' := map(T', T);
+    pts := rationalPoints f'(I);
+    -- if #pts == 2 then module is indecomposable
+    idems := apply(pts, pt -> sum(H, pt, times)) - set { id_M, 0 * id_M };
+    G := mingens sum(idems, image @@ homomorphism');
+    idems * set apply(numcols G, i -> homomorphism inducedMap(target B, , G_{i}))
+)
+
+summandsFromElimination = M -> prune \ image \ idempotentsFromElimination M
+
+end--
+restart
+needs "DirectSummands/elimination.m2"
+
+S = (ZZ/2)[x]
+M = S^1 ++ S^1/(x^2)
+elapsedTime summandsFromElimination M
+elapsedTime summands M
+idempotentsFromElimination M
+
+S = (ZZ/2)[x,y,z]
+I = ideal(x^3+y^3+z^3)
+R = S/I
+X = Proj R
+M = module frobeniusPushforward(1, R)
+elapsedTime summandsFromElimination M -- faster
+elapsedTime summands M
+
+S = (ZZ/5)[x,y,z,w]
+I = ideal(x^3+y^3+z^3+w^3)
+R = S/I
+X = Proj R
+M = module frobeniusPushforward(1, OO_X)
+elapsedTime summandsFromElimination M -- doesn't finish
+elapsedTime summands M
+
+K = QQ
+R = K[a,b,c,d];
+M = coker matrix"a,b,c,d;d,a,b,c;c,d,a,b;b,c,d,a"
+elapsedTime summandsFromElimination M -- doesn't work
+elapsedTime summands M
+
+-- FIXME: rationalPoints doesn't work over this field
+K = toField(QQ[i]/(i^2+1));
+S = K[x,y,z];
+M = coker matrix matrix"1,y,z;y,1,x;z,x,1"
+elapsedTime summandsFromElimination M
+elapsedTime summands M
+
+
+--- initial attempt
+-- TODO: delete
+
+T := R(monoid[]);
+use source psi
+A=sub(cover sum for i to numcols B - 1 list a_i * homomorphism B_(i-1), Ra)
+J=trim ideal cover(A^2-A)
+degree eliminate(apply(gens R,i->sub(i,ambient Ra)),  sub(J,ambient Ra)+ideal(Ra))
+decompose J
+
+---
+ungraded case:
+d = numgens End M0
+(Ra,psi) = flattenRing( R[a_1..a_d])
+M=M0**source psi
+E=End M
+use source psi
+A=sub(cover sum for i from 1 to d list a_i * homomorphism E_(i-1),Ra)
+J=trim ideal cover(A^2-A)
+eliminate(apply(gens R,i->sub(i,ambient Ra)),  sub(J,ambient Ra)+ideal(Ra))
+decompose (J+ideal(x,y,z))
+
+----
+S=(ZZ/2)[x,y,z]
+I=ideal(x^3+y^3+z^3)
+R=S/I
+X=Proj R
+M0 = module frobeniusPushforward(1,R)
+R=ring M0
+d = rank End sheaf M0
+(Ra,psi) = flattenRing( R[a_1..a_d])
+M=M0**source psi
+B=smartBasis({0,0},End( M, DegreeLimit=>0))
+use source psi
+A=sub(cover sum for i from 1 to rank source B list a_i * homomorphism B_(i-1),Ra)
+J=trim ideal cover(A^2-A)
+degree eliminate(apply(gens R,i->sub(i,ambient Ra)),  sub(J,ambient Ra)+ideal(Ra))
+netList decompose J
+
+restart
+debug needsPackage "DirectSummands"
+needsPackage "RationalPoints2"
+
+S = (ZZ/2)[x]
+M0 = S^1 ++ S^1/(x^2)
+d = numcols basis(0, End M0)
+(S', psi) = flattenRing( S[a_0..a_(d-1)])
+M = M0 ** source psi
+B = smartBasis({0}, End( M0, DegreeLimit => 0))
+
+A = sum(numcols B, i -> a_i * map(M ** S', M ** S', psi cover homomorphism B_i))
+J = trim ideal cover(A^2 - A)
+netList decompose J
+I = first decompose J
+syz gens I
+
+
+
+S=(ZZ/5)[x,y,z,w]
+I=ideal(x^3+y^3+z^3+w^3)
+R=S/I
+X=Proj R
+M0 = module frobeniusPushforward(1,OO_X)
+d = rank End sheaf M0
+(Ra,psi) = flattenRing( R[a_1..a_d])
+M=M0**source psi
+B=smartBasis({0,0}, End(M, DegreeLimit=>0))
+use source psi
+A=sub(cover sum for i from 1 to rank source B list a_i * homomorphism B_(i-1),Ra)
+J=trim ideal cover(A^2-A)
+use ambient Ra
+degree eliminate(apply(gens R,i->sub(i,ambient Ra)),  sub(J,ambient Ra)+ideal(Ra))
+
+summands M0
+
+K = QQ
+R = K[a,b,c,d];
+M0=coker matrix"a,b,c,d;d,a,b,c;c,d,a,b;b,c,d,a"
+r = rank source basis(0, End M0)
+(Ra,psi) = flattenRing( R[aa_1..aa_r])
+M=M0**source psi
+B=smartBasis({0,0},End( M, DegreeLimit=>0))
+use source psi
+A=sub(cover sum for i from 1 to rank source B list aa_i * homomorphism B_(i-1),Ra)
+J=trim ideal cover(A^2-A)
+use ambient Ra
+rationalPoints(sub(J,ambient Ra)+ideal Ra+ideal( apply(gens R,i->sub(i,ambient Ra))))
+decompose(J+ideal(x,y,z,w))
+
+summands M0