Injective Resolutions

Let $k[Q]$ be a monoid algebra for $Q\subset \mathbb{Z}^d$ and, let $M$ be a finitely generated $\mathbb{Z}^d$-graded module over $k[Q]$. An injective resolution is an exact sequence

\[0 \to M \xhookrightarrow{\epsilon} J^0 \xrightarrow{d^0} J^1 \xrightarrow{d^1} \dots \xrightarrow{d^{i-1}} J^i \xrightarrow{d^i} \cdots.\]

The maps $d^j$ are given by monomial matrices. The function injective_resolution computes an injective resolution up to some given cohomological degree. This is an implementation of the algorithms in [HM05].

Note

We require that the monoid algebra $k[Q]$ is normal.

injective_resolution — Method

injective_resolution(M::SubquoModule{<:MonoidAlgebraElem}, i::Int)

Return an injective resolution of $M$ up to cohomological degree i.

Note

The monoid algebra $k[Q]$ must be normal.

Examples

julia> kQ = monoid_algebra([[1,0],[0,1]],QQ)
monoid algebra over rational field with cone of dimension 2

julia> x,y = gens(kQ)
2-element Vector{MonoidAlgebraElem{QQFieldElem, MonoidAlgebra{QQFieldElem, MPolyDecRing{QQFieldElem, QQMPolyRing}}}}:
 x_1
 x_2

julia> I = ideal(kQ,[x^4,x^2*y^2,y^4])
ideal over monoid algebra over rational field with cone of dimension 2 generated by x_1^4, x_1^2*x_2^2, x_2^4

julia> M = quotient_ring_as_module(I)
Graded subquotient of graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
by graded submodule of kQ^1 with 3 generators
  1: x_1^4*e[1]
  2: x_1^2*x_2^2*e[1]
  3: x_2^4*e[1]

julia> injective_resolution(M,2)
injective resolution 
  J^0 -> J^1 -> J^2
where 
 J^0 = direct sum of
    k{[1, 3] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
    k{[3, 1] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
 J^1 = direct sum of
    k{[-1, 3] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
    k{[1, 1] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
    k{[3, -1] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
 J^2 = direct sum of
    k{[-1, -1] + F - Q}, where p_F = Ideal (x_2, x_1, x_1*x_2)
of Graded subquotient of graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
by graded submodule of kQ^1 with 3 generators
  1: x_1^4*e[1]
  2: x_1^2*x_2^2*e[1]
  3: x_2^4*e[1]
over monoid algebra over rational field with cone of dimension 2

julia> kQ = monoid_algebra([[0,1],[1,1],[2,1]],QQ)
monoid algebra over rational field with cone of dimension 2

julia> x,y,z = gens(kQ)
3-element Vector{MonoidAlgebraElem{QQFieldElem, MonoidAlgebra{QQFieldElem, MPolyQuoRing{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}}}:
 x_1
 x_2
 x_3

julia> F = graded_free_module(kQ,2)
Graded free module monoid algebra over rational field with cone of dimension 2^2([0 0]) of rank 2 over monoid algebra over rational field with cone of dimension 2

julia> a = kQ[y y;0 x^2]
[x_2     x_2]
[  0   x_1^2]

julia> b = kQ[x^2*z 0; x^4*y 0; 0 x^5*y; 0 z^3]
[x_1^2*x_3           0]
[x_1^4*x_2           0]
[        0   x_1^5*x_2]
[        0       x_3^3]

julia> M = SubquoModule(F,a,b)
Graded subquotient of graded submodule of F with 2 generators
  1: x_2*e[1] + x_2*e[2]
  2: x_1^2*e[2]
by graded submodule of F with 4 generators
  1: x_1^2*x_3*e[1]
  2: x_1^4*x_2*e[1]
  3: x_1^5*x_2*e[2]
  4: x_3^3*e[2]

julia> injective_resolution(M,2)
injective resolution
  J^0 -> J^1 -> J^2
where
 J^0 = direct sum of
    k{[1, 4] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[5, 7] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[4, 7] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[0, 2] + F - Q}, where p_F = Ideal (x_2, x_3, x_1*x_3)
    k{[1, 2] + F - Q}, where p_F = Ideal (x_1, x_2, x_1*x_3)
 J^1 = direct sum of
    k{[1, 2] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[0, 3] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[-1, 3] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[5, 4] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[0, 5] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[4, 6] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[3, 6] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[-1, -2] + F - Q}, where p_F = Ideal (x_2, x_3, x_1*x_3)
    k{[-4, -2] + F - Q}, where p_F = Ideal (x_1, x_2, x_1*x_3)
 J^2 = direct sum of
    k{[0, 0] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[-1, 1] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[-1, 2] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[-2, 2] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[3, 5] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
    k{[2, 5] + F - Q}, where p_F = Ideal (x_1, x_2, x_3, x_1*x_3)
of Graded subquotient of graded submodule of F with 2 generators
  1: x_2*e[1] + x_2*e[2]
  2: x_1^2*e[2]
by graded submodule of F with 4 generators
  1: x_1^2*x_3*e[1]
  2: x_1^4*x_2*e[1]
  3: x_1^5*x_2*e[2]
  4: x_3^3*e[2]
over monoid algebra over rational field with cone of dimension 2

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

injective_resolution — Method

injective_resolution(I::MonoidAlgebraIdeal,i::Int)

Return an injective resolution of $M = k[Q]/I$ up to cohomological degree i.

Note

The monoid algebra $k[Q]$ must be normal.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Data associated to injective modules

Injective modules in the $\mathbb{Z}^d$-graded setting have the form

\[J = \bigoplus_{i=1}^r k\{a_i + F_i - Q\},\]

where $k\{a_i + F_i - Q\}$ are so-called indecomposable injective modules (see, e.g., Chapter 11 of [MS05]).

Let J be a $\mathbb{Z}^d$-graded injective module

\[J = \bigoplus_{i=1}^r J_i = \bigoplus_{i=1}^r k\{a_i + F_i - Q\}.\]

Then

J.Q_graded_part refers to $J_Q = \bigoplus_{i=1}^r k\{a_i + F_i - Q\}_Q = \bigoplus_{i=1}^r k[Q]/W_i$ for some irreducible ideals $W_1,\dots,W_r$,
J.indec_injectives refers to $J_1,\dots,J_r$, and,
J.monoid_algebra refers to the monoid algebra $k[Q]$.

Data associated to injective resolutions

Let I = injective_resolution(M,i) be an injective resolution (up to cohomological degree i)

\[0 \to M \xhookrightarrow{\epsilon} I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} \cdots \xrightarrow{d^{i-1}} I^i\]

of a finitely generated $Q$-graded module $M$. Then

I.cochain_maps refers to the cochain maps $d^0,d^1,d^2,\dots,d^{i-1}$,
I.inj_mods refers to the injective modules $I^0,I^1,I^2,\dots,I^i$,
I.mod refers to $M$,
I.upto refers to the length of I,
I.shift refers to a degree $\alpha \in \mathbb{Z}^d$, and,
I.Q_graded_part refers to the irreducible resolution of $M$ shifted by $\alpha$.

Irreducible Resolutions

Let $M$ be a finitely generated $\mathbb{Z}^d$-graded module. An irreducible resolution of $M$ is an exact sequence

\[0 \to M \xhookrightarrow{\epsilon} \overline{W}^0 \xrightarrow{d^0} \overline{W}^1 \xrightarrow{d^1} \cdots \xrightarrow{d^{r-1}} \overline{W}^r \rightarrow \cdots,\]

where

\[\overline{W}^i = \bigoplus_{j=1}^{n_i} \overline{W_{i_j}} = \bigoplus_{j=1}^{n_i} k[Q]/W_{i_j}\]

for irreducible ideals $W_{i_j}$. The $k[Q]$-modules $\overline{W}^i$ are called irreducible sums.

Every finitely generated $Q$-graded module has a finite minimal irreducible resolution, i.e., it is finite in length and the components are finite direct sums. It is unique up to isomorphism and obtained as the $Q$-graded part of a minimal injective resolution. For more details see, e.g., Chapter 11 of [MS05].

irreducible_resolution — Function

irreducible_resolution(M::SubquoModule{<:MonoidAlgebraElem}, i::Int = 0)

Return an irreducible resolution of $M$.

Note

The monoid algebra $k[Q]$ must be normal.

Examples

julia> kQ = monoid_algebra([[1,0],[0,1]],QQ)
monoid algebra over rational field with cone of dimension 2

julia> x,y = gens(kQ)
2-element Vector{MonoidAlgebraElem{QQFieldElem, MonoidAlgebra{QQFieldElem, MPolyDecRing{QQFieldElem, QQMPolyRing}}}}:
 x_1
 x_2

julia> I = ideal(kQ,[x^4,x^2*y^2,y^4])
ideal over monoid algebra over rational field with cone of dimension 2 generated by x_1^4, x_1^2*x_2^2, x_2^4

julia> M = quotient_ring_as_module(I)
Graded subquotient of graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
by graded submodule of kQ^1 with 3 generators
  1: x_1^4*e[1]
  2: x_1^2*x_2^2*e[1]
  3: x_2^4*e[1]

julia> irr_res = irreducible_resolution(M)
irreducible resolution 
  W^0 -> W^1
where 
 W^0 = direct sum of
    k{[1, 3] + F - Q}_Q, where p_F = Ideal (x_2, x_1, x_1*x_2)
    k{[3, 1] + F - Q}_Q, where p_F = Ideal (x_2, x_1, x_1*x_2)
 W^1 = direct sum of
    k{[1, 1] + F - Q}_Q, where p_F = Ideal (x_2, x_1, x_1*x_2)
of Graded subquotient of graded submodule of kQ^1 with 1 generator
  1: 1*e[1]
by graded submodule of kQ^1 with 3 generators
  1: x_1^4*e[1]
  2: x_1^2*x_2^2*e[1]
  3: x_2^4*e[1]
over monoid algebra over rational field with cone of dimension 2

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Data associated to irreducible resolutions

Let I = irreducible_resolution(M) be an irreducible resolution

\[0 \to M \xhookrightarrow{\epsilon} \overline{W}^0 \xrightarrow{d^0} \overline{W}^1 \xrightarrow{d^1} \cdots \xrightarrow{d^{r-1}} \overline{W}^r\]

of a $\mathbb{Z}^d$-graded module $M$. Then

I.cochain_maps refers to the cochain maps $d^0,\dots,d^{r-1}$,
I.irr_sums refers to the irreducible sums $\overline{W}^0, \dots, \overline{W}^r$,
I.mod refers to $M$, and,
I.cochain_complex refer to the exact sequence as a ComplexOfMorphisms{ModuleFP}.