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D.5.18.8 sheafCoh
Procedure from librarysheafcoh.lib (see sheafcoh_lib).
- Usage:
- sheafCoh(M,l,h); M module, l,h int
- Assume:
Mis graded, and it comes assigned with an admissible degree vector as an attribute,h>=l. The baseringShasn+1variables.- 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);
Al 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))'-'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 usesmresto compute the Ext modules. If called with the additional parameter"sres", thesrescommand is used instead.
LIB "sheafcoh.lib"; // // cohomology of structure sheaf on P^4: //------------------------------------------- ring r=0,x(1..5),dp; module M=0; intmat A=sheafCoh(0,-7,2); 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; attrib(M,"isHomog",v); intmat B=sheafCoh(M,-6,2); 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 |