-- Currently, the definition of Tor is provided by the Core. -- We anticipate migrating that code and def to this file (June 2021). -- duplicated from OldChainComplexes/Tor.m2 -- TODO: documentation is still in Macaulay2Doc Tor(ZZ, Module, Module) := Module => opts -> (i, M, N) -> ( R := ring M; if not isCommutative R then error "'Tor' 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 M ** N else ( C := freeResolution(M, LengthLimit => i+1); b := C.dd.map; if b#?i then ( if b#?(i+1) then homology(b#i ** N, b#(i+1) ** N) else kernel(b#i ** N)) else ( if b#?(i+1) then error "internal error" else C_i ** N) ) ) Tor(ZZ, Matrix, Module) := Matrix => opts -> (i,f,N) -> ( R := ring f; if not isCommutative R then error "'Tor' not implemented yet for noncommutative rings."; if R =!= ring N then error "expected modules over the same ring"; if i < 0 then map(R^0, R^0, {}) else if i === 0 then f ** N else ( g := freeResolution(f, LengthLimit => i+1); Tsrc := Tor_i(source f, N); Ttar := Tor_i(target f, N); inducedMap(Ttar, Tsrc, g_i ** N) ) ) Tor(ZZ, Module, Matrix) := Matrix => opts -> (i,M,f) -> ( R := ring f; if not isCommutative R then error "'Tor' not implemented yet for noncommutative rings."; if R =!= ring M then error "expected modules over the same ring"; if i < 0 then map(R^0, R^0, {}) else if i === 0 then M ** f else ( F := freeResolution(M, LengthLimit => i+1); g := F ** f; HH_i g ) ) torSymmetry = method(Options => {LengthLimit => infinity}) torSymmetry(ZZ, Module, Module) := Matrix => opts -> (i, M, N) -> ( FM := freeResolution(M, LengthLimit => opts.LengthLimit); FN := freeResolution(N, LengthLimit => opts.LengthLimit); TorMN := Tor_i(M,N); TorNM := Tor_i(N,M); alpha := gens TorMN; -- The minus sign below is introduced to be compatible with -- signs arising in the tensor product of complexes. for j from 0 to i-1 do ( beta := ((dd^FM_(i-j) ** FN_j) * alpha) // ((-1)^(i-j) * (FM_(i-j-1) ** dd^FN_(j+1))); alpha = beta); delta := (tensorCommutativity(M, FN_i) * (map(M, FM_0, 1) ** FN_i) * alpha) // inducedMap(FN_i ** M, ker (dd^FN_i ** M)); map(TorNM, TorMN, delta) ) connectingTorMap = method(Options => {Concentration => null, LengthLimit => infinity}) connectingTorMap(Module, Matrix, Matrix) := ComplexMap => opts -> (M, g, f) -> ( F := freeResolution(M, LengthLimit => opts.LengthLimit); connectingMap(F ** g, F ** f, Concentration => opts.Concentration) ) connectingTorMap(Matrix, Matrix, Module) := ComplexMap => opts -> (g, f, M) -> ( (g', f') := horseshoeResolution(g, f, LengthLimit => opts.LengthLimit); connectingMap(g' ** M, f' ** M, Concentration => opts.Concentration) ) TEST /// -* restart needsPackage "Complexes" *- S = ZZ/101[a..f]; m = genericSymmetricMatrix(S, 3) I = trim minors(2, m) M = S^1/I N = coker vars S f1 = torSymmetry(2,M,N) f2 = torSymmetry(2,N,M) assert(f1*f2 == 1) assert(f2*f1 == 1) -- Now we try to relate this to maps of complexes. FM = freeResolution M FN = freeResolution N (lo, hi) = concentration FM E = directSum for i from lo to hi list complex(cover HH_i(FM ** N), Base => i) FMN = FM ** N f = map(FMN, E, i -> map(FMN_i, E_i, gens HH_i(FMN))) g = map(FMN, FM ** FN, FM ** augmentationMap FN) assert isWellDefined f assert isWellDefined g assert isQuasiIsomorphism g assert(target f === target g) h = liftMapAlongQuasiIsomorphism(f, g) assert isWellDefined h assert isComplexMorphism h assert(g * (f // g) == f) MFN = M ** FN; g' = map(MFN, FM ** FN, augmentationMap FM ** FN); assert isWellDefined g' assert isComplexMorphism g' FNM = FN ** M; h' = map(FNM, source h, tensorCommutativity(complex M, FN) * g' * h); assert isWellDefined h' K = ker dd^FNM; p = canonicalMap(FNM, K); q = inducedMap(HH FNM, K); assert(source p === source q) r = q * liftMapAlongQuasiIsomorphism(h',p); assert(p * (h' // p) == h') assert all(lo..hi, i -> matrix r_i == matrix torSymmetry(i, M, N)) /// TEST /// -* restart needsPackage "Complexes" *- S = QQ[a]; C1 = complex {matrix{{1_S,1},{1,1},{1,1}}} C2 = complex {matrix{{a,a,a,a},{a,a,a,a},{a,a,a,a},{a,a,a,a},{a,a,a,a}}} C12 = C1 ** C2 C21 = C2 ** C1 f1 = tensorCommutativity(C1, C2) assert isWellDefined f1 assert isComplexMorphism f1 /// TEST /// S = ZZ/101[a..d]; C = koszulComplex{a,b,c,d} D = koszulComplex{a^2,b^3,c^4} f1 = tensorCommutativity(C,D) assert isWellDefined f1 assert isComplexMorphism f1 /// TEST /// -- Example: where M is not a quotient of S^1. S = ZZ/101[a..f] m = genericSymmetricMatrix(S, 3) I = trim minors(2, m) M = prune subquotient(vars S, gens I) -- basically `M = prune truncate(1, S^1/I)` elapsedTime Tor_2(M,M); elapsedTime f1 = torSymmetry(2,M,M); elapsedTime f2 = torSymmetry(2,M,M); assert(f1*f2 == 1) assert(f2*f1 == 1) /// TEST /// S = ZZ/101[a..d]; I = ideal(c^3-b*d^2, b*c-a*d) J = ideal(a*c^2-b^2*d, b^3-a^2*c) g = map(S^1/(I+J), S^1/I ++ S^1/J, {{1,1}}) f = map(S^1/I ++ S^1/J, S^1/intersect(I,J), {{1},{-1}}) assert isShortExactSequence(g,f) kk = coker vars S delta = connectingTorMap(kk, g, f) delta' = connectingTorMap(g, f, kk) assert isWellDefined delta assert isWellDefined delta' F = freeResolution kk LES = longExactSequence(F ** g, F ** f); assert all(3, i -> dd^LES_(3*(i+1)) == delta_(i+1)) assert(HH LES == 0) -- another way (g',f') = horseshoeResolution(g,f); assert isShortExactSequence(g',f') LES' = longExactSequence(g' ** kk, f' ** kk); assert(HH LES' == 0) assert all(3, i -> dd^LES'_(3*(i+1)) == delta'_(i+1)) -- now we show commutativity of some squares -- which show that delta, delta' are isomorphic. h2 = torSymmetry(1, kk, source f) * delta_2 ts = torSymmetry(2, kk, target g) ts' = map(source delta'_2, source ts, ts) h2' = delta'_2 * ts' assert(h2 == h2') -- now for delta_3 h3 = torSymmetry(2, kk, source f) * delta_3 ts3 = torSymmetry(3, kk, target g) ts3' = map(source delta'_3, source ts3, ts3) h3' = delta'_3 * ts3' assert(h3 == h3') /// TEST /// -* restart needsPackage "Complexes" *- -- for quotient rings, some methods require LengthLimit. S = ZZ/101[a..e]/(e^2); I = ideal(c^3-b*d^2, b*c-a*d) J = ideal(a*c^2-b^2*d, b^3-a^2*c) g = map(S^1/(I+J), S^1/I ++ S^1/J, {{1,1}}) f = map(S^1/I ++ S^1/J, S^1/intersect(I,J), {{1},{-1}}) assert isShortExactSequence(g,f) kk = coker vars S delta = connectingTorMap(kk, g, f, LengthLimit => 5); delta' = connectingTorMap(g, f, kk, LengthLimit => 5); assert isWellDefined delta assert isWellDefined delta' F = freeResolution(kk, LengthLimit => 5) LES = longExactSequence(F ** g, F ** f); assert all(3, i -> dd^LES_(3*(i+1)) == delta_(i+1)) assert(HH LES == 0) -- another way (g',f') = horseshoeResolution(g,f, LengthLimit => 5); assert isShortExactSequence(g',f') LES' = longExactSequence(g' ** kk, f' ** kk); assert(HH LES' == 0) assert all(3, i -> dd^LES'_(3*(i+1)) == delta'_(i+1)) -- now we show commutativity of some squares -- which show that delta, delta' are isomorphic. h2 = torSymmetry(1, kk, source f, LengthLimit => 5) * delta_2 ts = torSymmetry(2, kk, target g, LengthLimit => 5) ts' = map(source delta'_2, source ts, ts) h2' = delta'_2 * ts' assert(h2 == h2') -- now for delta_3 h3 = torSymmetry(2, kk, source f, LengthLimit => 5) * delta_3 ts3 = torSymmetry(3, kk, target g, LengthLimit => 5) ts3' = map(source delta'_3, source ts3, ts3) h3' = delta'_3 * ts3' assert(h3 == h3') ///

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