Introduction

Injective resolutions are a fundmental homological tool in commutative algebra to understand modules and in algebraic geometry to study sheaves. Local cohomology allows one to study the local geometric features of sheaves. Many invariants such as Castelnuovo-Mumford regularity can be formulated in terms of vanishing of local cohomology in specific degrees.

This project implements the algorithms in [HM05] to compute injective resolutions and local cohomology over monoid algebras (also called semigroup rings in the literature). A key feature of this setting is that monoid algebras also include many non-regular rings. A technical limitation is that for several algorithms only saturated semigroups (i.e. modules over normal monoid algebras) can be considered.

Injective modules over polynomial rings or monoid algebras are generally not finitely generated and hence cannot be algorithmically treated with off-the-shelf tools implemented in OSCAR. However, there exist polyhedral data structures for both injective modules and local cohomlogy that allow to represent them using lattice points and polyhedra. Because resolutions typically are also infinite, we can finitely represent it them only up to a given cohomological degree. This package provides all the necessary data structures and algorithms.

Example

We consider a monoid algebra $k[Q]$, whose monoid $Q$ consists of all the lattice points in a cone over a square. This is also the coordinate ring of the Segre embedding $\mathbb{P}^1\times \mathbb{P}^1 \to \mathbb{P}^3$.

julia> kQ = monoid_algebra([[1, 0, 0], [1, 1, 0], [1, 1, 1], [1, 0, 1]], QQ)
monoid algebra over rational field with cone of dimension 3


julia> a, b, c, d = gens(kQ)
4-element Vector{MonoidAlgebraElem{QQFieldElem, MonoidAlgebra{QQFieldElem, MPolyQuoRing{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}}}:
 x_1
 x_2
 x_3
 x_4

We will compute an injective resolution and local cohomology of $M = k[Q]$ as a $k[Q]$-module. For this we set up $M$ as the quotient of $k[Q]$ by the zero ideal:

julia> I_M = ideal(kQ, [])
ideal over monoid algebra over rational field with cone of dimension 3 generated by 


julia> M = quotient_ring_as_module(I_M)
Graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
represented as subquotient with no relations

Let us take a look at the beginning of an injective resolution until cohomological degree 3.

julia> inj_res = injective_resolution(M, 3)
injective resolution 
  J^0 -> J^1 -> J^2 -> J^3
where 
 J^0 = direct sum of
    k{[0, 0, 0] + F - Q}, where p_F = Ideal ()
 J^1 = direct sum of
    k{[0, 0, -1] + F - Q}, where p_F = Ideal with 11 generators
    k{[-1, -1, -1] + F - Q}, where p_F = Ideal with 11 generators
    k{[-1, -1, 0] + F - Q}, where p_F = Ideal with 11 generators
    k{[-1, 0, 1] + F - Q}, where p_F = Ideal with 11 generators
 J^2 = direct sum of
    k{[-1, -1, -1] + F - Q}, where p_F = Ideal with 13 generators
    k{[-3, -2, -1] + F - Q}, where p_F = Ideal with 13 generators
    k{[-3, -2, -2] + F - Q}, where p_F = Ideal with 13 generators
    k{[-3, -1, -2] + F - Q}, where p_F = Ideal with 13 generators
 J^3 = direct sum of
    k{[-2, -1, -1] + F - Q}, where p_F = Ideal with 14 generators
of Graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
represented as subquotient with no relations
over monoid algebra over rational field with cone of dimension 3

Now we compute local cohomology supported on a face of the cone or respectively, the ideal

julia> I = ideal(kQ, [a, b])
ideal over monoid algebra over rational field with cone of dimension 3 generated by x_1, x_2

The zeroth local cohomology is just the well known module $\Gamma_I (M)$ module. This can be represented with the standard data structures and hence we compute:

julia> H0 = zeroth_local_cohomology(M, I)
Graded submodule of kQ^1 with 0 generators
represented as subquotient with no relations

julia> is_zero(H0)
true

There is a dedicated implementation of is_zero for local cohomology modules and their data structures. The zeroth local cohomology vanishes.

Now let's move to cohomological degree one.

julia> H1 = local_cohomology(M, I, 1)
sector partition of 1-th local cohomology module supported on ideal (x_1, x_2) of 
 Graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
represented as subquotient with no relations


julia> H1_sectors = [h for h in H1.sectors if dim(h.H)>0] #sectors with non-zero local cohomomology
1-element Vector{Oscar.InjectiveResolutions.SectorLC}:
 sector of ZZ^3 with local cohomology of dimension 1

julia> is_zero(H1)
false

julia> length(H1_sectors)
1

This local cohomology is non-zero. There is one sector supporting local cohomology.

The second local cohomology is also non-zero.

julia> H2 = local_cohomology(M, I, 2)
sector partition of 2-th local cohomology module supported on ideal (x_1, x_2) of 
 Graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
represented as subquotient with no relations


julia> H2_sectors = [h for h in H2.sectors if dim(h.H)>0] #sectors with non-zero local cohomology
1-element Vector{Oscar.InjectiveResolutions.SectorLC}:
 sector of ZZ^3 with local cohomology of dimension 1

julia> is_zero(H2)
false

julia> length(H2_sectors)
1

And the third vanishes.

julia> H3 = local_cohomology(M, I, 3)
sector partition of 3-th local cohomology module supported on ideal (x_1, x_2) of 
 Graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
represented as subquotient with no relations


julia> is_zero(H3)
true

Contact

Any questions or comments about injective resolutions and local cohomology in OSCAR can be sent to:

• Anna Hofer
• Thomas Kahle

Acknowledgements

We appreciate extensive support on integrating this code into OSCAR provided by Matthias Zach.