nLab internal hom of chain complexes

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.

Definition

Let RR be a commutative ring and 𝒜=R\mathcal{A} = RMod the category of modules over RR. Write Ch (𝒜)Ch_\bullet(\mathcal{A}) for the category of chain complexes of RR-modules.

Definition

For X,YCh (𝒜)X,Y \in Ch_\bullet(\mathcal{A}) any two objects, define a chain complex [X,Y]Ch (𝒜)[X,Y] \in Ch_\bullet(\mathcal{A}) to have components

[X,Y] n iHom RMod(X i,Y i+n) [X,Y]_n \,\coloneqq\, \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})

(the collection of degree-nn maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements f[X,Y] nf \in [X,Y]_n by

dfd Yf(1) nfd X. d f \,\coloneqq\, d_Y \circ f - (-1)^{n} f \circ d_X \,.

This defines a functor

[,]:Ch (𝒜) op×Ch (𝒜)Ch (𝒜). [-,-] \;\colon\; Ch_\bullet(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \longrightarrow Ch_\bullet(\mathcal{A}) \,.

Properties

Proposition

The internal hom [,][-,-] (Def. ) together with the tensor product of chain complexes (-)(-)(\text{-})\otimes (\text{-}) endow Ch (𝒜)Ch_\bullet(\mathcal{A}) with the structure of a closed monoidal category.

Proposition

The collection of cycles of the internal hom [X,Y][X,Y] in degree 0 coincides with the external hom functor

Z 0([X,Y])Hom Ch (X,Y). Z_0\big([X,Y]\big) \,\simeq\, Hom_{Ch_\bullet}(X,Y) \,.

The chain homology of the internal hom [X,Y][X,Y] in degree 0 coincides with the homotopy classes of chain maps.

Proof

By Definition the 0-cycles in [X,Y][X,Y] are collections of morphisms {f k:X kY k}\{f_k \colon X_k \to Y_k\} such that

f k+1d X=d Yf k. f_{k+1} \circ d_X \;=\; d_Y \circ f_k \,.

This is precisely the condition for ff to be a chain map.

Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form

λ k+1d X+d Yλ k \lambda_{k+1} \circ d_X + d_Y \circ \lambda_k

for a collection of maps {λ k:X kY k+1}\{\lambda_k : X_k \to Y_{k+1}\}. This are precisely the null homotopies.

From the remark at tensor product of chain complexes we have that the canonical forgetful functor U ChCh (𝒜)(I Ch,):Ch (𝒜)𝒜U_{Ch} \coloneqq Ch_\bullet(\mathcal{A})(I_{Ch},-) \colon Ch_\bullet(\mathcal{A}) \to \mathcal{A} takes a chain complex to its 0-cycles.

Thus the description of the 0-cycles in the above proposition is equivalent to the statement U Ch([,])Ch (𝒜)(,)U_{Ch}([-,-]) \cong Ch_\bullet(\mathcal{A})(-,-), which is true in any closed category.

References

Textbook accounts:

Last revised on August 23, 2023 at 08:45:39. See the history of this page for a list of all contributions to it.