h=method() -- computes the generators of H^1(Amitsur complex) h(ZZ,Ideal,Ideal):=(n,J,JJ)-> ( sbs1=basis(n,S2); mn=degree source sbs1; q=module S3/JJ; ff=S3^{mn:-n}; mat=fxy(sbs1)+fyz(sbs1)-fxz(sbs1); ke=kernel map(q,ff,mat); BB=gens ke; BB=compress retS BB; mp=map(fxy(sbs1))*BB; sbt=basis(n,S1); Dn=ideal(fx1(sbt)-fy1(sbt))+J; zz=mingens ideal(fss1(mp)%Dn); zz ) k = 2 kk = ZZ S1 = kk[x_1..x_k] S2 = kk[x_1..x_k,y_1..y_k] S3 = kk[x_1..x_k,y_1..y_k,z_1..z_k] use S3 fx = map(S3,S1,{x_1..x_k}) fy = map(S3,S1,{y_1..y_k}) fz = map(S3,S1,{z_1..z_k}) use S2 fx1 = map(S2,S1,{x_1..x_k}) fy1 = map(S2,S1,{y_1..y_k}) fss1 = map(S2,S3,{x_1..x_k,y_1..y_k,k:0}) retS1 = map(kk,S2,{2*k:0}) use S3 fxy = map(S3,S2,{x_1..x_k,y_1..y_k}) fxz = map(S3,S2,{x_1..x_k,z_1..z_k}) fyz = map(S3,S2,{y_1..y_k,z_1..z_k}) retS = map(kk,S3,{3*k:0}) use S1 -- S1 = C = ZZ[x_1,x_2] i = matrix{{x_1^2,x_1*x_2-x_2^2,x_2^3}} -- f1, f2, f3; B = ZZ[f1,f2,f3] J = ideal (fx1(i)-fy1(i)); -- S2/J = C\otimes_B C JJ = ideal (fxy(gens J),fxz(gens J)); -- S3/JJ = C\otimes_B C\otimes_B C n=5 -- the degree where the nontrivial 1-cocycle lives In = super basis(n,J); -- basis for degree n of J bs1 = basis(n,S1); -- basis of the space of forms of degree 5 in C Dn = (fx1(bs1)-fy1(bs1))|In; -- In + coboundaries bs2 = basis(n,S2); -- basis of the space of forms of degree 5 in S2 = C\otimes C lbs2 = numgens source bs2 lDn = numgens source Dn q = module S3/JJ; -- C\otimes_B C\otimes_B C ff = S3^{lbs2:-n}; mat = fxy(bs2)+fyz(bs2)-fxz(bs2); -- the differential d_1 in the Amitsur complex BB = compress retS(gens kernel map(q,ff,mat)); -- cocycles of degree 5 mat = mutableMatrix(kk,lbs2,lDn) for idn from 0 to lDn-1 do ( po = Dn_idn_0; for isbs from 0 to lbs2-1 do mat_(isbs,idn) = retS1(po//bs2_isbs_0); ) mat = matrix(mat); -- the differential d_0 in the Amitsur complex end restart load "noneffective.m2" prune cokernel BB -- The ZZ-module of 1-cocycles of degree 5 is a direct summand in C_5 prune cokernel mat -- The ZZ-module of 1-coboundaries of degree 5 is a direct summand in C_5 minimalPresentation subquotient(BB,mat) -- The first Amitsur cohomology group is free of rank 1 (h(5,J,JJ))_0_0 -- the cocycle f factor oo -- its representation as a product