nLab
internal hom of chain complexes

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Monoidal categories

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 := \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

df:=d Yf(1) nfd X. d f := d_Y \circ f - (-1)^{n} f \circ d_X \,.

This defines a functor

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

Properties

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([X,Y]) \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 1 the 0-cycles in [X,Y][X,Y] are collections of morphisms {f k:X kY k}\{f_k : 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.

References

A standard textbook account is

Revised on August 26, 2012 23:34:01 by Urs Schreiber (89.204.139.66)