complexes.md

CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()

Chain and Cochain Complexes

The general OSCAR type ComplexOfMorphisms{T} allows one to model both chain complexes and cochain complexes (the T refers to the type of the differentials of the complex). In the context of commutative algebra, we handle complexes of modules and module homomorphisms over multivariate polynomial rings. In this section, we first show how to create such complexes. Then we discuss functionality for dealing with the constructed complexes, mainly focusing on chain complexes. Cochain complexes can be handled similarly.

Constructors

chain_complex(V::ModuleFPHom...; seed::Int = 0)

cochain_complex(V::ModuleFPHom...; ssed::Int = 0)

Data Associated to Complexes

Given a complex C,

• range(C) refers to the range of C,
• obj(C, i) and C[i] to the i-th module of C, and
• map(C, i) to the i-th differential of C.

Examples

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> F = free_module(R, 1);

julia> A, _ = quo(F, [x^4*F[1]]);

julia> B, _ = quo(F, [x^3*F[1]]);

julia> a = hom(A, B, [x^2*B[1]]);

julia> b = hom(B, B, [x^2*B[1]]);

julia> C = chain_complex([a, b]; seed = 3)
C_3 <---- C_4 <---- C_5

julia> range(C)
5:-1:3

julia> C[5]
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> x^4*e[1]

julia> delta = map(C, 5)
Map with following data
Domain:
=======
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> x^4*e[1]
Codomain:
=========
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> x^3*e[1]

julia> matrix(delta)
[x^2]

Operations on Complexes

shift(C::ComplexOfMorphisms{T}, d::Int) where T

Return the complex obtained from C by shifting the homological degrees d steps, with maps multiplied by $(-1)^d$.

Examples

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> F = free_module(R, 1);

julia> A, _ = quo(F, [x^4*F[1]]);

julia> B, _ = quo(F, [x^3*F[1]]);

julia> a = hom(A, B, [x^2*B[1]]);

julia> b = hom(B, B, [x^2*B[1]]);

julia> C = chain_complex([a, b]; seed = 3);

julia> range(C)
5:-1:3

julia> D = shift(C, 3);

julia> range(D)
8:-1:6

hom(C::ComplexOfMorphisms{ModuleFP}, M::ModuleFP)

hom_without_reversing_direction(C::ComplexOfMorphisms{ModuleFP}, M::ModuleFP)

hom(M::ModuleFP, C::ComplexOfMorphisms{ModuleFP})

tensor_product(C::ComplexOfMorphisms{ModuleFP}, M::ModuleFP)

tensor_product(M::ModuleFP, C::ComplexOfMorphisms{ModuleFP})

Tests on Complexes

The functions below check properties of complexes:

is_chain_complex(C::ComplexOfMorphisms{ModuleFP})

is_cochain_complex(C::ComplexOfMorphisms{ModuleFP})

is_exact(C::ComplexOfMorphisms{ModuleFP})

Examples

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> F = free_module(R, 1);

julia> A, _ = quo(F, [x^4*F[1]]);

julia> B, _ = quo(F, [x^3*F[1]]);

julia> a = hom(A, B, [x^2*B[1]]);

julia> b = hom(B, B, [x^2*B[1]]);

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> C = chain_complex([a, b]);

julia> is_cochain_complex(C)
false

Maps of Complexes

Types

Constructors