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D.5.10.6 sheafCoh
Procedure from library sheafcoh.lib (see sheafcoh_lib).
- Usage:
- sheafCoh(M,l,h); M module, l,h int
- Assume:
M is graded, and it comes assigned with an admissible degree
vector as an attribute, 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 is displayed in a diagram of the following form:
| | 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))
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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;
def A=sheafCoh(0,-7,2);
==> -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);
def B=sheafCoh(M,-6,2);
==> -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
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See also:
dimH;
sheafCohBGG.
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