nLab
internal hom of chain complexes

Idea

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

Definition

Let $R$ be a commutative ring and $𝒜=R$Mod the category of modules over $R$. Write ${\mathrm{Ch}}_{•}(𝒜)$ for the category of chain complexes of $R$-modules.

Definition

For $X,Y\in {\mathrm{Ch}}_{•}(𝒜)$ any two objects, define a chain complex $[X,Y]\in {\mathrm{Ch}}_{•}(𝒜)$ to have components

$$[X,Y{]}_n:=\prod _{i\in \mathbb{Z}}{\mathrm{Hom}}_{R\mathrm{Mod}}(X_i,Y_{i+n})$$[X,Y]_n := \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})

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

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

This defines a functor

$$[-,-]:{\mathrm{Ch}}_{•}(𝒜{)}^{\mathrm{op}}\times {\mathrm{Ch}}_{•}(𝒜)\to {\mathrm{Ch}}_{•}(𝒜).$$[-,-] : 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 $[X,Y]$ in degree 0 coincides with the external hom functor

$$Z_0([X,Y])\simeq {\mathrm{Hom}}_{{\mathrm{Ch}}_{•}}(X,Y).$$Z_0([X,Y]) \simeq Hom_{Ch_\bullet}(X,Y) \,.

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

Proof

By Definition 1 the 0-cycles in $[X,Y]$ are collections of morphisms $\{f_k:X_k\to Y_k\}$ such that

$$f_{k+1}\circ d_X=d_Y\circ f_k.$$f_{k+1} \circ d_X = d_Y \circ f_k \,.

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

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

$${\lambda }_{k+1}\circ d_X+d_Y\circ {\lambda }_k$$\lambda_{k+1} \circ d_X + d_Y \circ \lambda_k

for a collection of maps $\{{\lambda }_k:X_k\to Y_{k+1}\}$. This are precisely the null homotopies.

Related concepts

• tensor product of chain complexes

References

A standard textbook account is

• Charles Weibel, An Introduction to Homological Algebra