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category 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

Contents

Definition

Let π’œ\mathcal{A} be an additive category.

Recall the notion of chain complex, of chain map between chain complexes and of chain homotopy between chain maps in π’œ\mathcal{A}.

Call a chain complex C β€’C_\bullet

  • bounded below if there is kβˆˆβ„•k \in \mathbb{N} such that C n≀k=0C_{n \leq k} = 0;

  • bounded above if there is kβˆˆβ„•k \in \mathbb{N} such that C nβ‰₯k=0C_{n \geq k} = 0;

  • bounded if it is bounded below and bounded above. We have

Definition

Write Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) for the category whose objects are chain complexes in π’œ\mathcal{A} and whose morphisms are chain maps between these.

This is the category of chain complexes in π’œ\mathcal{A}.

Several variants of this category are of relevance.

Definition

Write Ch β€’ +,βˆ’,b(π’œ)β†ͺCh β€’(π’œ)Ch_\bullet^{+,-,b}(\mathcal{A}) \hookrightarrow Ch_\bullet(\mathcal{A}) for the full subcategory on the chain complexes which are, respectively, bounded above, bounded below or bounded.

Definition

Write K(π’œ)K(\mathcal{A}) for the category obtained from Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) by identifying homotopic chain maps.

K(π’œ)(C β€’,D β€’)≔Ch β€’(C β€’,D β€’)/chainβˆ’homotopy. K(\mathcal{A})(C_\bullet, D_\bullet) \coloneqq Ch_\bullet(C_\bullet, D_\bullet)/chain-homotopy \,.

Accordingly K +,βˆ’,b(π’œ)β†ͺK(π’œ)K^{+,-,b}(\mathcal{A}) \hookrightarrow K(\mathcal{A}) denotes the full subcategory on the chain complexes bounded above, bounded below or bounded, respectively.

This is sometimes called the homotopy category of chain complexes. But see the warning on terminology there, as this term is also appropriate for the category in the following remark.

Remark

If π’œ\mathcal{A} is moreover an abelian category, then there is also the derived category D(π’œ)D(\mathcal{A}), obtained from Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) or K(π’œ)K(\mathcal{A}) by universally inverting all quasi-isomorphisms. See at derived category for more on this.

Properties

Abelian category

Theorem

For π’œ\mathcal{A} an abelian category also the category of chain complexes Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) is again an abelian category.

We discuss the ingredients that go into this statement.

(…)

Proposition

For f:C β€’β†’D β€’f : C_\bullet \to D_\bullet a chain map,

  • the complex ker(f)ker(f) of degreewise kernels in π’œ\mathcal{A} is the kernel of ff in Ch β€’(π’œ)Ch_\bullet(\mathcal{A});

  • the complex coker(f)coker(f) of degreewise cokernels in π’œ\mathcal{A} is the cokernel of ff in Ch β€’(π’œ)Ch_\bullet(\mathcal{A}).

Remark

A sequence of chain complexes 0β†’A β€’β†’B β€’β†’C β€’β†’00 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 is a short exact sequence in Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) precisely if each component 0β†’A nβ†’B nβ†’C nβ†’00 \to A_n \to B_n \to C_n \to 0 is a short exact sequence in π’œ\mathcal{A}.

(…)

Closed monoidal structure

Proposition

Equipped with the standard tensor product of chain complexes βŠ—\otimes the category of chain complexes is a monoidal category (Ch β€’(RMod),βŠ—)(Ch_\bullet(R Mod), \otimes). The unit object is the chain complex concentrated in degree 0 on the tensor unit RR of RModR Mod.

Proposition

In fact (Ch β€’,βŠ—)(Ch_\bullet, \otimes) is a closed monoidal category, the internal hom is the standard internal hom of chain complexes.

References

A basic introduction is in chapter 1 of

category: category

Revised on June 20, 2014 14:03:43 by Tobias Fritz (72.0.209.252)