LIB "tst.lib"; tst_init(); LIB "sheafcoh.lib"; // Kohomologie der Strukturgarbe von P^7: //---------------------------------------- ring r=0,x(1..8),dp; module M=0; def A=sheafCoh(0,-10,5); def B=sheafCohBGG(0,-10,3); displayCohom(A,-10,5,7); displayCohom(B,-10,3,7); kill r; // Kohomologie der Idealgarbe der Veronese Flaeche in $\P^3$: //------------------------------------------------------------ ring S = 32003, x(0..4), dp; module MI=maxideal(1); attrib(MI,"isHomog",intvec(-1)); resolution kos = nres(MI,0); print(betti(kos),"betti"); matrix alpha0 = random(32002,10,3); module pres = module(alpha0)+kos[3]; attrib(pres,"isHomog",intvec(1,1,1,1,1,1,1,1,1,1)); resolution fcokernel = mres(pres,0); print(betti(fcokernel),"betti"); module dir = transpose(pres); attrib(dir,"isHomog",intvec(-1,-1,-1,-2,-2,-2, -2,-2,-2,-2,-2,-2,-2)); resolution fdir = mres(dir,2); print(betti(fdir),"betti"); ideal I = groebner(flatten(fdir[2])); resolution FI = mres(I,0); print(betti(FI),"betti"); module F=FI[2]; A=sheafCoh(F,-6,6); B=sheafCohBGG(F,-6,6); dimH(3,F,-4); dimH(1,F,1); kill S; // Kohomologie der Idealgarbe einer Flaeche in $\P^4$: //------------------------------------------------------------ ring S = 32003, x(0..4), dp; resolution kos = nres(maxideal(1),0); betti(kos); matrix kos5 = kos[5]; matrix tphi = transpose(dsum(kos5,kos5)); matrix kos3 = kos[3]; matrix psi = dsum(kos3,kos3); matrix beta1 = random(32002,20,2); matrix tbeta1tilde = transpose(psi*beta1); matrix tbeta0 = lift(tphi,tbeta1tilde); matrix kos4 = kos[4]; matrix tkos4pluskos4 = transpose(dsum(kos4,kos4)); matrix tgammamin1 = random(32002,20,1); matrix tgamma0 = tkos4pluskos4*tgammamin1; matrix talpha0 = concat(tbeta0,tgamma0); matrix zero[20][1]; matrix tpsi = transpose(psi); matrix tpresg = concat(tpsi,zero); matrix pres = module(transpose(talpha0)) + module(transpose(tpresg)); module dir = transpose(pres); dir = prune(dir); homog(dir); intvec deg_dir = attrib(dir,"isHomog"); attrib(dir,"isHomog",deg_dir-2); // set degrees resolution fdir = mres(prune(dir),2); print(betti(fdir),"betti"); ideal I = groebner(flatten(fdir[2])); resolution FI = mres(I,0); module F=FI[2]; def A1=sheafCoh(F,-4,6); A1=sheafCoh(F,-3,6,"sres"); def A2=sheafCohBGG(F,-4,6); kill S; // Kohomologie des Morrocks-Mumford-Buendels (Macaulay-Beispiel von Wolfram): // ---------------------------------------------------------------------------- ring R=0,x(0..4),dp; matrix FHM[19][35] = -x(2), 0, -x(1), -x(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(0), 0, 0, 0, 0, 0, -x(4), 0, 0, -x(3), -x(2), 0, 0, 0, x(0), 0, 0, 0, 0, 0, -x(3), 0, 0, 0, 0, 0, -x(1), 0, 0, 0, 0, x(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(3), -x(2), 0, 0, 0, 0, x(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(2), 0, 0, x(1), 0, 0, 0, 0, -x(0), 0, 0, -x(4), 0, 0, -x(3), 0, 0, 0, -x(2), -x(3), 0, -x(3), 0, 0, 0, 0, -x(1), 0, 0, x(0), 0, 0, 0, 0, 0, -x(4), 0, 0, 0, 0, 0, 0, 0, 0, -x(2), 0, 0, 0, 0, 0, x(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(0), 0, 0, 0, 0, 0, 0, -x(3), x(2), 0, 0, 0, 0, 0, 0, 0, 0, -x(1), 0, 0, 0, 0, x(4), 0, x(1), 0, 0, 0, 0, 0, 0, 0, 0, x(2), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(0), 0, 0, 0, x(4), 0, 0, -x(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(0), 0, 0, 0, 0, 0, -x(4), x(2), 0, -x(4), 0, 0, -x(3), 0, 0, 0, -x(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(4), -x(3), 0, -x(2), 0, 0, 0, 0, -x(0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(3), 0, 0, 0, -x(1), 0, 0, 0, 0, 0, 0, 0, -x(4), 0, -x(2), 0, 0, 0, 0, 0, 0, x(4), 0, -x(1), x(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(2), 0, 0, -x(1), 0, 0, -x(1), 0, 0, 0, 0, 0, -x(4), 0, 0, 0, x(3), 0, 0, 0, 0, 0, -x(4), 0, 0, -x(1), 0, 0, 0, 0, 0, 0, 0, -x(1), 0, x(0), 0, 0, 0, 0, 0, 0, -x(4), 0, -x(3), 0, 0, 0, -x(2), 0, 0, 0, 0, 0, 0, 0, x(1), 0, x(3), 0, x(2), 0, 0, 0, 0, 0, 0, 0, x(2), 0, 0, 0, 0, x(0), 0, 0, 0, 0, 0, 0, -x(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(3), 0, 0, 0, x(1), x(4), x(1), 0, x(0), 0, 0, 0, 0, 0, 0, 0, 0, -x(2), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(3), x(2), -x(1), 0, 0, -x(1), 0, 0, 0, -x(0), 0, -x(4), 0, 0, -x(4), 0, x(3), 0, 0, 0, 0, 0, 0, x(2), 0, 0, 0, 0, 0, 0, 0, 0, 0, x(0), 0, -x(2), 0, 0, -x(3), 0, 0, 0, -x(4), 0, 0, 0, 0, 0, 0, 0, 0, -x(2), 0, 0, 0, x(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x(2)*x(4), 0, 0, 0, 0, 0, x(2)*x(3), 0, 0, x(1)*x(3), 0, 0, 0, 0, 0, x(4)^2, 0, 0, 0, x(1)^2, 0, 0, 0, 0, -x(1)*x(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, x(3)*x(4), 0, 0, x(3)^2, 0, 0, x(1)*x(4), 0, 0, 0, 0, 0, x(2)^2, 0, 0, x(1)*x(2), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x(2)*x(3), 0, 0, 0, 0, 0, 0, x(4)^2, 0, 0, 0, x(0)*x(4), 0, x(3)^2, 0, x(2)*x(3), 0, -x(1)*x(4), x(1)*x(3), 0, 0, 0, x(2)^2, 0, 0, x(1)*x(2), -x(0)*x(3), x(0)*x(2), 0, x(3)*x(4), x(1)^2, 0, x(0)*x(1), -x(2)*x(4), x(0)^2, 0, 0, 0, 0, 0, 0, 0, 0, x(4)^2, 0, 0, 0, x(2)*x(4), 0, x(3)^2, 0, x(1)*x(4), x(2)*x(3), -x(0)*x(4), 0, x(0)*x(3), x(1)*x(3), 0, x(2)^2, 0, 0, x(1)*x(2), -x(3)*x(4), x(0)*x(2), 0, 0, x(1)^2, 0, x(0)*x(1), 0, x(0)^2; def M=module(FHM); attrib(M,"isHomog",intvec(4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3)); A1=sheafCohBGG(M,-10,7); A2=sheafCoh(M,-4,3); tst_status(1);$