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 — Method

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

Given a tuple V of module homorphisms between successive modules over a multivariate polynomial ring, return the chain complex defined by these homomorphisms.

chain_complex(V::Vector{<:ModuleFPHom}; seed::Int = 0)

Given a vector V of module homorphisms between successive modules over a multivariate polynomial ring, return the chain complex defined by these homomorphisms.

Note

The integer seed indicates the lowest homological degree of a module in the complex.

Note

The function checks whether successive homomorphisms indeed compose to zero.

source

cochain_complex — Method

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

Given a tuple V of module homorphisms between successive modules over a multivariate polynomial ring, return the cochain complex defined by these homomorphisms.

cochain_complex(V::Vector{<:ModuleFPHom}; seed::Int = 0)

Given a vector V of module homorphisms between successive modules over a multivariate polynomial ring, return the cochain complex defined by these homomorphisms.

Note

The integer seed indicates the lowest cohomological degree of a module of the complex.

Note

The function checks whether successive homomorphisms indeed compose to zero.

source

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)
Module homomorphism
  from A
  to B

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 — Method

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

Return the complex obtained by applying $\text{Hom}(-,$ M$)$ to C.

If C is a chain complex, return a cochain complex. If C is a cochain complex, return a chain complex.

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 = hom(C, A);

julia> range(D)
3:5

source

hom_without_reversing_direction — Method

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

Return the complex obtained by applying $\text{Hom}(-,$ M$)$ to C.

If C is a chain complex, return a chain complex. If C is a cochain complex, return a cochain complex.

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 = hom_without_reversing_direction(C, A);

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

source

hom — Method

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

Return the complex obtained by applying $\text{Hom}($M, $-)$ to C.

source

tensor_product — Method

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

Return the complex obtained by applying $\bullet\;\! \otimes$ M to C.

source

tensor_product — Method

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

Return the complex obtained by applying M $\otimes\;\! \bullet$ to C.

source

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

Chain and Cochain Complexes

Constructors

Data Associated to Complexes

Examples

Operations on Complexes

Examples

Tests on Complexes

Examples

Maps of Complexes

Types

Constructors