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R = ZZ[x,y,z] modules = { image matrix {{x^2,x,y}}, coker matrix {{x^2,y^2,0},{0,y,z}}, image matrix {{x,y}} ++ coker matrix {{y,z}}, R^{-1,-2,-3} } scan(modules, M -> assert( cover exteriorPower(2, M) == exteriorPower(2, cover M) )) scan(modules, M -> assert( cover cokernel M_{1} == cover M )) scan(modules, M -> assert( cover M == target presentation M )) table(modules, modules, (P, Q) -> assert( cover P ** cover Q == cover(P ** Q) )); -- see https://github.com/Macaulay2/M2/issues/2550 (t, M) = toSequence timing symmetricPower(2, R^(splice{3:2, 16:1, 3:0})) assert(flatten degrees M == splice{3:-4, 16:-3, 3:-2, 2:-4, 16:-3, 3:-2, -4..-3, 15:-3, 19:-2, 3:-1, 15:-2, 3:-1, 14:-2, 3:-1, 13:-2, 3:-1, 12:-2, 3:-1, 11:-2, 3:-1, 10:-2, 3:-1, 9:-2, 3:-1, 8:-2, 3:-1, 7:-2, 3:-1, 6:-2, 3:-1, 5:-2, 3:-1, 4:-2, 3:-1, 3:-2, 3:-1, 2:-2, 3:-1, -2..-1, 2:-1, 6:0}) assert(t < 1) -- R = ZZ/101 exteriorPower(3,R^5) R = ZZ/101[a..d] I = monomialCurveIdeal(R,{1,3,4}) M = Ext^2(coker generators I, R) prune exteriorPower(3,M) exteriorPower(0,R^3) exteriorPower(0,M) prune exteriorPower(1,M) exteriorPower(2,M) exteriorPower(-1,M) exteriorPower(-2,M) M = subquotient(matrix{{a,b,c}}, matrix{{a^2,b^2,c^2,d^2}}) N = subquotient(matrix{{a^2,b^2,c^2}}, matrix{{a^3,b^3,c^3,d^3}}) m = map(N,M,matrix(R,{{1,0,0},{0,1,0},{0,0,1}})) source m target m trim ker m M1 = coker presentation M N1 = coker presentation N m1 = map(N1,M1,matrix m) M2 = trim exteriorPower(2,M) N2 = trim exteriorPower(2,N) -- R = ZZ/101[a .. i] m = genericMatrix(R,a,3,3) assert( exteriorPower(1,m) == m ) assert( minors(1,m) == image vars R ) assert( exteriorPower(2,m*m) == exteriorPower(2,m)*exteriorPower(2,m) ) assert( exteriorPower(2,m) == matrix { {-b*d+a*e, -b*g+a*h, -e*g+d*h}, {-c*d+a*f, -c*g+a*i, -f*g+d*i}, {-c*e+b*f, -c*h+b*i, -f*h+e*i}} ) assert( exteriorPower(3,m) == matrix {{-c*e*g+b*f*g+c*d*h-a*f*h-b*d*i+a*e*i}} ) -- R = ZZ/101[a..d] I = monomialCurveIdeal(R,{1,3,4}) A = R/I jacobian A singA = minors(codim ideal presentation A, jacobian A) generators gb singA -- fitting ideal R = ZZ/101[x]; k = coker vars R; M = R^3 ++ k^5; assert( fittingIdeal(0,M) == ideal 0_R ) assert( fittingIdeal(1,M) == ideal 0_R ) assert( fittingIdeal(2,M) == ideal 0_R ) assert( fittingIdeal(3,M) == ideal x^5 ) assert( fittingIdeal(4,M) == ideal x^4 ) assert( fittingIdeal(5,M) == ideal x^3 ) assert( fittingIdeal(6,M) == ideal x^2 ) assert( fittingIdeal(7,M) == ideal x ) assert( fittingIdeal(8,M) == ideal 1_R ) assert( fittingIdeal(9,M) == ideal 1_R )