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 — Methodchain_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.
The integer seed indicates the lowest homological degree of a module in the complex.
The function checks whether successive homomorphisms indeed compose to zero.
cochain_complex — Methodcochain_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.
The integer seed indicates the lowest cohomological degree of a module of the complex.
The function checks whether successive homomorphisms indeed compose to zero.
Data Associated to Complexes
Given a complex C,
range(C)refers to the range ofC,obj(C, i)andC[i]to thei-th module ofC, andmap(C, i)to thei-th differential ofC.
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 TReturn 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:-1:6hom — Methodhom(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)
3:5hom_without_reversing_direction — Methodhom_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:-5hom — Methodhom(M::ModuleFP, C::ComplexOfMorphisms{ModuleFP})Return the complex obtained by applying $\text{Hom}($M, $-)$ to C.
tensor_product — Methodtensor_product(C::ComplexOfMorphisms{<:ModuleFP}, M::ModuleFP)Return the complex obtained by applying $\bullet\;\! \otimes$ M to C.
tensor_product — Methodtensor_product(M::ModuleFP, C::ComplexOfMorphisms{ModuleFP})Return the complex obtained by applying M $\otimes\;\! \bullet$ to C.
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