-- testing Hom(Module,Module) -- free modules: R = ZZ/101[a,b,c] M = ker vars R N = ideal vars R * M f = map(M, N, gens N//gens M) -- the inclusion of N into M assert isHomogeneous f assert isWellDefined f M' = coker presentation M N' = coker presentation N f' = map(M',M,1) * f * map(N,N',1) assert isHomogeneous f' assert isWellDefined f' assert ( hash (Hom(M,N)).cache.cache === hash (Hom(M,N)).cache.cache ) assert ( hash (M**N).cache.cache === hash (M**N).cache.cache ) assert ( prune coker Hom(M',f') == coker vars R ) assert ( prune coker Hom(M,f') == coker vars R ) assert ( prune coker Hom(M',f) == coker vars R ) assert ( prune coker Hom(M,f) == coker vars R ) assert ( coker Hom(f',M') == 0 ) assert ( coker Hom(f',M) == 0 ) assert ( coker Hom(f,M') == 0 ) assert ( coker Hom(f,M) == 0 ) assert ( prune coker Hom(f',N') == coker vars R ) assert ( prune coker Hom(f,N') == coker vars R ) assert ( prune coker Hom(f',N) == coker vars R ) assert ( prune coker Hom(f,N) == coker vars R ) R = QQ[x] M = x * R^1 H = Hom(M,M) assert isHomogeneous H assert isHomogeneous H h = H_{0} assert isHomogeneous h; g = homomorphism h; assert isHomogeneous g; h' = homomorphism' g assert ( h === h' ) R = QQ[x,y] M = R^{ -1,-2 } N = R^{ -3,-4 } H = Hom(M,N) assert isHomogeneous H assert ( H === Hom(M,N) ) assert isFreeModule H assert( entries homomorphism H_{0} == {{1, 0}, {0, 0}} ) assert( entries homomorphism H_{1} == {{0, 0}, {1, 0}} ) assert( entries homomorphism H_{2} == {{0, 1}, {0, 0}} ) assert( entries homomorphism H_{3} == {{0, 0}, {0, 1}} ) assert( {{1}, {2}} == degrees source homomorphism H_{0} ) assert( 2 == degree source homomorphism H_{0} ) for i from 0 to 3 do ( h = H_{i}; assert isHomogeneous h; g = homomorphism h; assert isHomogeneous g; -- wait for implementation of "adjoint" in general -- assert ( h === homomorphism'_H g ) ) i = homomorphism H_{0} + homomorphism H_{3} assert isHomogeneous i assert ( {2} == degree i ) assert isIsomorphism i j = homomorphism H_{1} + homomorphism H_{2} assert not isHomogeneous j assert isIsomorphism j M = x*M / (y*M) N = x*N / (y*N) H = Hom(M,N) assert isHomogeneous H assert ( H === Hom(M,N) ) -- debugLevel = 1 -- assert( entries homomorphism H_{0} == {{1, 0}, {0, 0}} ) -- assert( entries homomorphism H_{1} == {{0, 0}, {1, 0}} ) -- assert( entries homomorphism H_{2} == {{0, 1}, {0, 0}} ) -- assert( entries homomorphism H_{3} == {{0, 0}, {0, 1}} ) assert( {{2}, {3}} == degrees source homomorphism H_{0} ) assert( 2 == degree source homomorphism H_{0} ) for i from 0 to 3 do ( h = H_{i}; assert (degree h === {0}); assert isHomogeneous h; g = homomorphism h; assert isHomogeneous g; h' = homomorphism' g; assert isHomogeneous h'; -- make h' have the same degree as h h' = map(target h', source h' ** R^{ degree h - degree h' }, h', Degree => degree h); assert isHomogeneous h'; assert ( h === h' ) ) -- S = ZZ/101[a..d] I = monomialCurveIdeal(S, {1,3,4}) R = S/I use R J = module ideal(a,d) K = module ideal(b^2,c^2) JK = Hom(J,K, MinimalGenerators => true) f = JK_{0} g = homomorphism f assert isHomogeneous g assert ( source g === J ) assert ( target g === K ) f' = homomorphism'(g, MinimalGenerators => true) assert (f-f' == 0) assert (degrees f' === {{{1}}, {{0}}}) assert (degrees f === {{{1}}, {{1}}}) assert (degree f == {0}) assert (degree f' == {1}) ----------------------------------------------------------------------------- -- test "compose" R=QQ[x,y] M=image vars R ++ R^2 time f = compose(M,M,M); H = Hom(M,M); assert isHomogeneous H for i to numgens H - 1 do for j to numgens H - 1 do ( g = H_{i}; g' = H_{j}; h = homomorphism g; h' = homomorphism g'; assert ( homomorphism (f * (g ** g')) == h' * h )) H' = trim H assert isHomogeneous H' assert ( ambient H === ambient H' ) toH = inducedMap(H,H') toH' = inducedMap(H',H) assert ( isIsomorphism toH ) assert ( isIsomorphism toH' ) assert ( toH * toH' == 1 ) assert ( toH' * toH == 1 ) for i to numgens H' - 1 do for j to numgens H' - 1 do ( g = H'_{i}; g' = H'_{j}; h = homomorphism (toH * g); h' = homomorphism (toH * g'); assert ( homomorphism (f * ((toH * g) ** (toH * g'))) == h' * h )) S = ZZ/101[a,b,c] A = matrix"a,b,c;b,c,a" B = matrix"a,b;b,c" N = subquotient(A,B) time com = compose(N,N,N) assert isHomogeneous com assert( (minimalPresentation com) === map(cokernel map((S)^1,(S)^{{ -2}},{{b^2-a*c}}), cokernel map((S)^1,(S)^{{ -2}},{{b^2-a*c}}), {{1}})); ----------------------------------------------------------------------------- -- test homomorphism' and homomorphism are inverse R = QQ[x] f = x ++ x^2 assert ( degree f == {0} ) assert isHomogeneous f g = homomorphism' f assert ( degree g == {0} ) assert isHomogeneous g assert ( source g === R^1 ) assert ( target g === Hom(source f,target f) ) f' = homomorphism g assert ( f === f' ) f = x ++ x^2 f = map(target f ** R^{2}, source f, f, Degree => {-2}) assert ( degree f == {-2} ) assert isHomogeneous f g = homomorphism' f assert (degree g == {-2} ) assert isHomogeneous g assert ( source g === R^1 ) assert ( target g === Hom(source f,target f) ) f' = homomorphism g assert ( f === f' ) M = R^{1,2} N = R^{3,5} H = Hom(M,N) assert isHomogeneous H g = H_{0} assert ( source g === R^{2} ) assert ( target g === H ) assert isHomogeneous g f = homomorphism g assert ( source f === M ) assert ( target f === N ) g' = homomorphism' f assert isHomogeneous g' assert ( target g' === H ) assert ( source g' === R^1 ) degree g degree g' assert ( g === map(target g, source g, g', Degree => degree g )) S = QQ[x] M = S^1/x++S^1 presentation M g = M^{1} N = target g Hom(M,N) assert isWellDefined g f = homomorphism' g assert ( target f === Hom(M,N) ) assert ( source f === S^1 ) g' = homomorphism f assert ( g == g' ) S = QQ[a,b] M = S^2/(a,b)++S^2/(b) g = id_M f = homomorphism' g assert isHomogeneous f assert ( g == homomorphism f ) ----------------------------------------------------------------------------- R = QQ[x] M = subquotient( matrix {{x,x^2}} , matrix {{x^3,x^4}} ) p = presentation M assert( target p === cover M ) H = Hom(M,M) peek H.cache f = id_M -- implemented by David: Hom(M,f) Hom(f,M) -- tests for Hom(Matrix,Module) from David: S = ZZ/101[a,b] M = S^1/ideal(a) f = inducedMap(S^1,module ideal(a^3,b^3)) fh = Hom(f,M) assert( (minimalPresentation Hom(f,M) ) === map(cokernel map(S^{{3}},S^{{2}},{{a}}),cokernel map(S^1,S^{{-1}},{{a}}),{{b^3}}) ) assert( target fh === Hom(source f,M) ) assert( source fh === Hom(target f,M) ) S=ZZ/101[a,b,c] I = ideal"a3,b3,c3" f = map(S^1,module I,{{a^3,b^3,c^3}}) fh = Hom(f,S^1) assert( target fh === Hom(source f,S^1) ) assert( source fh === Hom(target f,S^1) ) assert( fh === map(image map(S^{{3},{3},{3}},S^1,{{a^3}, {b^3}, {c^3}}),S^1,{{1}}) ) M = module ((ideal(a,b))^3) M'= module ((ideal(a,b))^2) f=inducedMap(M',M) fh = Hom(M,f) assert (target fh == Hom(M, target f)) assert (source fh == Hom(M, source f)) assert( (minimalPresentation Hom(f,M)) === map(S^1,cokernel(map(S^{{-1}, {-1}},S^{{-2}},{{a}, {-b}})),{{b, a}}) ) assert( (minimalPresentation Hom(f,f)) === map(S^1,cokernel(map(S^{{-1}, {-1}},S^{{-2}},{{a}, {-b}})),{{b, a}}) ) -- bug found by David Treumann: p = 1 q = 1 S = QQ[x,y, Degrees => {{1,0},{0,1}}] E = S^{{ -1,0},{-2,0},{0,-1},{-1,-1},{-1,-1},{-2,-1},{0,-2},{-1,-2}} Xmatrix = transpose matrix{ {0_S,1,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0}, {0,0,0,1,0,0,0,0}, {0,0,0,0,0,p,0,0}, {0,0,0,0,0,1,0,0}, {0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,1}, {0,0,0,0,0,0,0,0} } Ymatrix =transpose matrix{ {0_S,0,0,0,1,0,0,0}, {0,0,0,0,0,1,0,0}, {0,0,0,0,0,0,1,0}, {0,0,0,0,0,0,0,1}, {0,0,0,0,0,0,0,q}, {0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0} } phi = map(E,E**S^{{ -1,0 }}++E**S^{{0,-1}},(x*id_E-Xmatrix) | (y*id_E-Ymatrix)); isHomogeneous phi M = coker phi Ext^1(M,M) -- mismatch here, because trimming of result is ignored ----------------------------------------------------------------------------- -- tests for giving DegreeLimit R = QQ[x,y,z] elapsedTime Hom(R^100, R^100); -- <0.005s elapsedTime Hom(R^{-100}, R^100, DegreeLimit => 0); -- <0.04s elapsedTime assert(numcols basis(0, oo) == 515100) -- <0.02s -- Local Variables: -- compile-command: "make -C $M2BUILDDIR/Macaulay2/packages/Macaulay2Doc/test hom.out" -- End:

Simpan