Singular Manual: sheafCoh

  | #### D.5.18.6 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))
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
See also: dimH; sheafCohBGG.

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