# HH^ZZ Module -- local cohomology of a module

## Synopsis

• Function: cohomology
• Usage:
HH^i(M)
• Inputs:
• i, an integer, which is non negative
• M, , which is graded over its base polynomial ring
• Optional inputs:
• Degree => ..., default value 0,
• Outputs:

## Description

The command computes the local cohomology of the graded module M with respect to the maximal irrelevant ideal (the ideal of variables in the base ring of M).

The package Dmodules has alternative code to compute local cohomology (even in the non homogeneous case)

A very simple example:

 i1 : R = QQ[a,b]; i2 : HH^2 (R^{-3}) o2 = cokernel | b a 0 | | 0 -b a | 2 o2 : R-module, quotient of R i3 : HH^2 (R^{-4}) o3 = cokernel | b a 0 0 | | 0 -b a 0 | | 0 0 -b a | 3 o3 : R-module, quotient of R

Another example, a singular surface in projective fourspace (with one apparent double point):

 i4 : R = ZZ/101[x_0..x_4]; i5 : I = ideal(x_1*x_4-x_2*x_3, x_1^2*x_3+x_1*x_2*x_0-x_2^2*x_0, x_3^3+x_3*x_4*x_0-x_4^2*x_0) 2 2 3 2 o5 = ideal (- x x + x x , x x x - x x + x x , x + x x x - x x ) 2 3 1 4 0 1 2 0 2 1 3 3 0 3 4 0 4 o5 : Ideal of R i6 : M = R^1/module(I) o6 = cokernel | -x_2x_3+x_1x_4 x_0x_1x_2-x_0x_2^2+x_1^2x_3 x_3^3+x_0x_3x_4-x_0x_4^2 | 1 o6 : R-module, quotient of R i7 : HH^1(M) o7 = cokernel | x_4 x_3 x_2 x_1 x_0^3 | 1 o7 : R-module, quotient of R i8 : HH^2(M) o8 = cokernel | x_4 x_3 x_2 x_1 x_0 | 1 o8 : R-module, quotient of R

## Caveat

There is no check made if the given module is graded over the base polynomial ring