-- Copyright 1995 by Daniel R. Grayson importFrom_Core { "generatorSymbols", } Ext(ZZ, Module, Module) := Module => opts -> (i,M,N) -> ( R := ring M; if not isCommutative R then error "'Ext' not implemented yet for noncommutative rings."; if R =!= ring N then error "expected modules over the same ring"; if i < 0 then R^0 else if i === 0 then Hom(M, N, opts) else ( C := resolution(M,LengthLimit=>i+1); b := C.dd; complete b; prune' := if opts.MinimalGenerators then prune else identity; prune' if b#?i then ( if b#?(i+1) then homology(Hom(b_(i+1), N, opts), Hom(b_i, N, opts)) else cokernel Hom(b_i, N, opts)) else ( if b#?(i+1) then kernel Hom(b_(i+1), N, opts) else Hom(C_i, N, opts)))) Ext(ZZ, Matrix, Module) := Matrix => opts -> (i,f,N) -> ( R := ring f; if not isCommutative R then error "'Ext' not implemented yet for noncommutative rings."; if R =!= ring N then error "expected modules over the same ring"; prune' := if opts.MinimalGenerators then prune else identity; if i < 0 then map(R^0, R^0, {}) else if i === 0 then Hom(f, N, opts) else prune'( g := resolution(f,LengthLimit=>i+1); Es := Ext^i(source f, N, opts); Et := Ext^i(target f, N, opts); psi := if Es.cache.?pruningMap then Es.cache.pruningMap else id_Es; phi := if Et.cache.?pruningMap then Et.cache.pruningMap else id_Et; psi^-1 * inducedMap(target psi, target phi, Hom(g_i, N, opts)) * phi)) -- TODO: is this correct? -- c.f. https://github.com/Macaulay2/M2/issues/246 Ext(ZZ, Module, Matrix) := Matrix => opts -> (i,N,f) -> ( R := ring f; if not isCommutative R then error "'Ext' not implemented yet for noncommutative rings."; if R =!= ring N then error "expected modules over the same ring"; prune' := if opts.MinimalGenerators then prune else identity; if i < 0 then map(R^0, R^0, {}) else if i === 0 then Hom(N, f, opts) else prune'( C := resolution(N,LengthLimit=>i+1); Es := Ext^i(N, source f, opts); Et := Ext^i(N, target f, opts); psi := if Es.cache.?pruningMap then Es.cache.pruningMap else id_Es; phi := if Et.cache.?pruningMap then Et.cache.pruningMap else id_Et; phi^-1 * inducedMap(target phi, target psi, Hom(C_i, f, opts)) * psi)) -- total ext over complete intersections -- change 2009/1/12: -- some modifications contributed 23 Mar 2007 by "Paul S. Aspinwall" <psa@cgtp.duke.edu>, -- but we also get rid of the fudge factor entirely, depending instead on automatic -- computation of the heft vector -- also see Complexes/Ext.m2 Ext(Module, Module) := Module => opts -> (M, N) -> ( -- c.f. caching in Hom(Module,Module) cacheModule := youngest(M.cache.cache, N.cache.cache); cacheKey := (Ext,M,N); if cacheModule#?cacheKey then return cacheModule#cacheKey; B := ring M; if B =!= ring N then error "expected modules over the same ring"; if not isCommutative B then error "'Ext' not implemented yet for noncommutative rings."; if not isHomogeneous B then error "'Ext' received modules over an inhomogeneous ring"; if not isHomogeneous N or not isHomogeneous M then error "'Ext' received an inhomogeneous module"; if N == 0 or M == 0 then return cacheModule#cacheKey = B^0; p := presentation B; A := ring p; I := ideal mingens ideal p; n := numgens A; c := numgens I; if c =!= codim B then error "total Ext available only for complete intersections"; f := apply(c, i -> I_i); pM := lift(presentation M,A); pN := lift(presentation N,A); M' := cokernel ( pM | p ** id_(target pM) ); N' := cokernel ( pN | p ** id_(target pN) ); assert isHomogeneous M'; assert isHomogeneous N'; C := complete resolution M'; X := getSymbol "X"; K := coefficientRing A; S := K(monoid [X_1 .. X_c, toSequence A.generatorSymbols, Degrees => { apply(0 .. c-1, i -> prepend(-2, - degree f_i)), apply(0 .. n-1, j -> prepend( 0, degree A_j)) }]); -- make a monoid whose monomials can be used as indices Rmon := monoid [X_1 .. X_c,Degrees=>{c:{2}}]; -- make group ring, so 'basis' can enumerate the monomials R := K Rmon; -- make a hash table to store the blocks of the matrix blks := new MutableHashTable; blks#(exponents 1_Rmon) = C.dd; scan(0 .. c-1, i -> blks#(exponents Rmon_i) = nullhomotopy (- f_i*id_C)); -- a helper function to list the factorizations of a monomial factorizations := (gamma) -> ( -- Input: gamma is the list of exponents for a monomial -- Return a list of pairs of lists of exponents showing the -- possible factorizations of gamma. if gamma === {} then { ({}, {}) } else ( i := gamma#-1; splice apply(factorizations drop(gamma,-1), (alpha,beta) -> apply (0..i, j -> (append(alpha,j), append(beta,i-j)))))); scan(4 .. length C + 1, d -> if even d then ( scan( flatten \ exponents \ leadMonomial \ first entries basis(d,R), gamma -> ( s := - sum(factorizations gamma, (alpha,beta) -> ( if blks#?alpha and blks#?beta then blks#alpha * blks#beta else 0)); -- compute and save the nonzero nullhomotopies if s != 0 then blks#gamma = nullhomotopy s; )))); -- make a free module whose basis elements have the right degrees spots := C -> sort select(keys C, i -> class i === ZZ); Cstar := S^(apply(spots C, i -> toSequence apply(degrees C_i, d -> prepend(i,d)))); -- assemble the matrix from its blocks. -- We omit the sign (-1)^(n+1) which would ordinarily be used, -- which does not affect the homology. toS := map(S,A,apply(toList(c .. c+n-1), i -> S_i), DegreeMap => prepend_0); Delta := map(Cstar, Cstar, transpose sum(keys blks, m -> S_m * toS sum blks#m), Degree => { -1, degreeLength A:0 }); DeltaBar := Delta ** (toS ** N'); if debugLevel > 10 then ( assert isHomogeneous DeltaBar; assert(DeltaBar * DeltaBar == 0); stderr << describe ring DeltaBar <<endl; stderr << toExternalString DeltaBar << endl; ); -- now compute the total Ext as a single homology module prune' := if opts.MinimalGenerators then prune else identity; cacheModule#cacheKey = prune' homology(DeltaBar,DeltaBar))

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