Opened 6 years ago

Last modified 5 years ago

#803 new bug

Can have a module have two different Hilbert series ??

Reported by: wbruns@… Owned by: hannes
Priority: major Milestone: 4-2-0 and higher
Component: singular-kernel Version: 4-0-3
Keywords: Hilbert series, minbase Cc: conca@…

Description

The computation below is part of an ongoing project with Aldo Conca.

We want to test the question whether the module K3 is Cohen-Macauly and self-dual. We use the fact that K3 is a module over K0 which is a Gorenstein ring, namely the coordinate ring of the Grassmannian G(5,2) of dim 3 vector spaces in the 5-dimensional vector space.

LIB "homolog.lib"; loaded /usr/bin/../share/singular/LIB/homolog.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/primdec.lib (4.0.2.0,Apr_2015) loaded /usr/bin/../share/singular/LIB/ring.lib (4.0.2.2,Jan_2016) loaded /usr/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014) loaded /usr/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/deform.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/sing.lib (4.0.0.0,Jun_2013) loaded /usr/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014) int m=2; int n=5; int r=2; ring R=0,(x(1..m*n),t),dp; matrix M[m][n]=x(1..m*n); ideal I=minor(M,r); ideal J=x(1..m*n),t*I; ideal Z=0; bigint NN=binomial(m,r)*binomial(n,r)+m*n; int nn=0; for(int i=1;i<=NN;i=i+1){nn=nn+1;} ring S=0,(y(1..m*n),z(1..(nn-m*n))),dp; setring R; map f=S,J; setring S; ideal K=preimage(R,f,Z); K=std(K); qring REES=K; ideal KK=y(1..m*n); module MM=0; module K0=KoszulHomology?(KK,MM,0); module K3=KoszulHomology?(KK,MM,3); K3=std(K3); hilb(K3); 4 t0 40 t1 -880 t2 5980 t3 -23140 t4 59488 t5 -108680 t6 145860 t7 -145860 t8 108680 t9 -59488 t10 23140 t11 -5980 t12 880 t13 -40 t14 -4 t15

4 t0 92 t1 4 t2 dimension (proj.) = 6 degree (proj.) = 100

Comment: At this point we are happy since the result coincides with an independent computation of the Hilbert series. Moreover, the h-vector (4,92,4) fulfills 2 necessary conditions: (i) it has nonnegative enties, (ii) it is palindromic. Now we want to test our conjecture.

module K3D=Hom(K3,K0); dimension of Hom: 7

K3D=std(K3D); hilb(K3D); 2 t0 62 t1 -980 t2 6190 t3 -23140 t4 58214 t5 -104676 t6 138710 t7 -137280 t8 101530 t9 -55484 t10 21866 t11 -5980 t12 1090 t13 -140 t14 18 t15 -2 t16

2 t0 88 t1 8 t2 2 t3 dimension (proj.) = 6 degree (proj.) = 100

Comment: Our conjecture is dstroyed ... K3D has adifferent h-vector ... But now a miracle happens...

K3D=minbase(K3D); K3D=std(K3D); hilb(K3D); 4 t0 40 t1 -880 t2 5980 t3 -23140 t4 59488 t5 -108680 t6 145860 t7 -145860 t8 108680 t9 -59488 t10 23140 t11 -5980 t12 880 t13 -40 t14 -4 t15

4 t0 92 t1 4 t2 dimension (proj.) = 6 degree (proj.) = 100

Comment: Whow !!!!!!!!!! It has the right h-vecor -- or not? Conjecture rescued. How can minbase change the h-vector ??????????????

quit;

Change History (1)

comment:1 Changed 5 years ago by hannes

Owner: changed from somebody to hannes
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