Singular Manual: sheafCoh

D.5.18.9 sheafCoh_w

Procedure from library sheafcoh.lib (see sheafcoh_lib).

Usage:

sheafCoh(M,l,h); M module, l,h int, w intvec

Assume:

M is graded by w, h>=l. The basering S has n+1 variables.

Return:

intmat, cohomology of twists of the coherent sheaf F on P^n associated to coker(M). The range of twists is determined by l, h.

Display:

The intmat can be displayed in a diagram of the following form: with displayCohom(A,l,h,nvars(r)-1);

                l            l+1                      h
  ----------------------------------------------------------
      n:     h^n(F(l))    h^n(F(l+1))   ......    h^n(F(h))
           ...............................................
      1:     h^1(F(l))    h^1(F(l+1))   ......    h^1(F(h))
      0:     h^0(F(l))    h^0(F(l+1))   ......    h^0(F(h))
  ----------------------------------------------------------
    chi:     chi(F(l))    chi(F(l+1))   ......    chi(F(h))

A '-' in the diagram refers to a zero entry.

Note:

The procedure is based on local duality as described in [Eisenbud: Computing cohomology. In Vasconcelos: Computational methods in commutative algebra and algebraic geometry. Springer (1998)].
By default, the procedure uses mres to compute the Ext modules. If called with the additional parameter "sres", the sres command is used instead.

Example:

LIB "sheafcoh.lib";
//
// cohomology of structure sheaf on P^4:
//-------------------------------------------
ring r=0,x(1..5),dp;
module M=0;
intmat A=sheafCoh_w(0,-7,2,intvec(0));
A;
==> 15,5,1,0,0,0,0,0,0,0,
==> 0,0,0,0,0,0,0,0,0,0,
==> 0,0,0,0,0,0,0,0,0,0,
==> 0,0,0,0,0,0,0,0,0,0,
==> 0,0,0,0,0,0,0,1,5,15 
displayCohom(A,-7,2,nvars(r)-1);
==>       -7  -6  -5  -4  -3  -2  -1   0   1   2
==> --------------------------------------------
==>   4:  15   5   1   -   -   -   -   -   -   -
==>   3:   -   -   -   -   -   -   -   -   -   -
==>   2:   -   -   -   -   -   -   -   -   -   -
==>   1:   -   -   -   -   -   -   -   -   -   -
==>   0:   -   -   -   -   -   -   -   1   5  15
==> --------------------------------------------
==> chi:  15   5   1   0   0   0   0   1   5  15
//
// cohomology of cotangential bundle on P^3:
//-------------------------------------------
ring R=0,(x,y,z,u),dp;
resolution T1=mres(maxideal(1),0);
module M=T1[3];
intvec v=2,2,2,2,2,2;
intmat B=sheafCoh_w(M,-6,2,v);
displayCohom(B,-6,2,nvars(R)-1);
==>       -6  -5  -4  -3  -2  -1   0   1   2
==> ----------------------------------------
==>   3:  70  36  15   4   -   -   -   -   -
==>   2:   -   -   -   -   -   -   -   -   -
==>   1:   -   -   -   -   -   -   1   -   -
==>   0:   -   -   -   -   -   -   -   -   6
==> ----------------------------------------
==> chi: -70 -36 -15  -4   0   0  -1   0   6