Computing coherent cohomology in projective space over a ring

[Jürgen Böhm]

Recently there was a question how to compute the cohomology of a coherent sheaf on the space X = P^n_A where A is a polynomial ring or an affine algebra:

https://mathoverflow.net/questions/476427/is-it-possible-to-use-the-cech-complex-to-compute-coherent-cohomology-in-practic

The OP asked for a way with Cech-cohomology, but I could inform her that it can be done with Macaulay2 using an algorithm I stated in 2018 in MO

https://mathoverflow.net/questions/301923/an-algorithm-to-compute-coherent-sheaf-cohomology-in-projective-space-over-a-rin

Unfortunately, up to now, nobody has taken note of my posting above and so I have no second opinion on its correctness and usefulness. (My own experiments gave plausible results).

I am also not quite sure whether such a calculation is already possible with Macaulay2: When I entered

A = QQ[a]

R = A[x_0..x_3]

m = matrix {{x_0 * x_1, x_2 * x_3}}

F = sheaf coker m

res = apply(0..3, i->HH^i(F))

I get an error in the last line, whereas for A=QQ it works.

Maybe someone could take a look on my posting above and write here about his opinions regarding its correctness and usefulness.

Best regards

Jürgen Böhm

[Jürgen Böhm]

I need to answer myself to post some code which became necessary as the truncate function did not seem to work correctly in many examples:
The following code ran for me successfully on the Web M2 interface. A quick hack for a better truncate (truncate1) and the dirty trick of mapping the result chain complex into a newly made up basering brough success (at least for the example below and some similar examples):

--load "Truncations.m2"

--
-- compute the H^i(proj S, moM~) for S=A[x_1,...,x_n]
--

compCohX = (moM, d) ->
(
S := ring moM;
A := coefficientRing S;

n1 := numgens A;

B := (coefficientRing A)[t_1..t_(n1)];

n := numgens S;
moMt := truncate1(moM, d);
-- the number 10 must be chosen accordingly with n
-- C := presX (moMt, 10);
C := res moMt;
C1 := Hom(C, S^{-n});
Dlis := {};
for i from min(C1) to max (C1) do
(
    Dlis = prepend( transpose lift(matrix basis(0, C1.dd_(i)), A), Dlis);
);
D:= chainComplex(Dlis);
D, C, prune HH(D);

D1 := (map(B,A,gens B))(D);

return D1;
);


truncate1 = (moM, d) ->
(
S := ring moM;
A := coefficientRing S;
n := numgens S;

phi := presentation moM;

cross := gens truncate(d, S);

mat1 := (target(phi)) ** cross;

phi1 := map(moM, source(mat1), mat1);

return image(phi1);

);



--
-- compute a free resolution of moM with S(-d_{i,1}) + .. + S(-d_{i,j_i}) terms
--

presX = (moM, k) ->
(
moN := moM;
philis := {};
moN = prune moN;
i := 0;
phi := presentation moN;
degs := degrees target phi;
print degs;
philis = append(philis, phi);
moN = ker phi;
while (((prune moN) != 0) and (i <= k)) do
(
    phi = presentation moN;
    psi := gens moN;
    philis = append(philis, psi);
    philis = append(philis, phi);
    moN = ker phi;
    i = i + 1;  
);
D := chainComplex(philis);
return D;
);

-*
A=QQ[a,Degrees=>{1:{}}]



S=A[x_0,x_1,x_2, Join => false]

ma = matrix {{(a-1) * x_0, (a-1) * x_1, (a-1) * x_2}}

mo = coker ma

*-



A=QQ[a,Degrees=>{1:{}}];

S=A[x_0..x_4];


ma = matrix {{(a-1) * x_0,(a-1) * x_1, (a-1) * x_2,(a-1) * x_3, (a-1) * x_4, (x_0 * x_1 + x_2 * x_3 + x_4 * x_0 * a)}}

mo1 = coker ma



D1 = compCohX(mo1, 2)

print(D1)


-*

T=A[t,u];

phi = map(T,S, {t^4, t^3*u, a* t^2 * u^2, t * u^3, u^4});

id1 = ker phi;


R=QQ[w_0..w_4]


ma1 = module id1;

na1 = S^1/id1;

phi1 = map(R, S, {w_0, w_1, w_2, w_3, w_4, 0})

ma11 = phi1(ma1)

resol1 = apply(0..4, i->HH^i(sheaf ma11))

print(resol1)

*-