Products in D(A)

Suppose that A is an abelian category with Ab4*, i.e., products exists and are exact. Then a product of quasi-isomorphisms is a quasi-isomorphism and we can define products in D(A) just by taking the product of underlying complexes. If A has just Ab3* (i.e., products exist) then this doesn’t work.

Let A be a Grothendieck abelian category. Then A has Ab3* (this does not follow directly from the definitions, but rather is an example of what Akhil was referring to here). In a nice short paper entitled Resolution of unbounded complexes in Grothendieck categories, C. Serpé shows that the category of unbounded complexes over A has enough K-injectives. There are other references; I like this one because its proof is a modification of Spaltenstein’s argument in his famous paper Resolutions of unbounded complexes. Combining these results we can show products exist in D(A).

In fact, I claim that products exist in D(A) if A has Ab3* and enough K-injective complexes. Namely, suppose that we have a collection of complexes K^*_λ in A parametrized by a set Λ. Choose quasi-isomorphisms K^*_λ —> I^*_λ into K-injective complexes I^*_λ and consider the termwise product

Π_{λ ∈ Λ} I^*_λ

I claim this is a product of the objects K^*_λ in D(A). Namely, it is a result in the Spaltenstein paper that the product of K-injective complexes is K-injective. Hence to check our assertion we need only check this on the level of maps up to homotopy, where it is clear.

OK, now what I want to know is this: Let A be a Grothendieck abelian category and let B ⊂ A be a subcategory such that D_B(A) makes sense. When does D_B(A) have products? Are there some reasonable assumptions we can make to guarantee this?

1 thought on “Products in D(A)

  1. Another way to see that D(A) has products is by using Brown representability. This triangulated category is well generated in the sense of Neeman, so it satisfies Brown’s representability theorem. If for instance D_B(A) is a localizing subcategory generated by a set then it’s also well generated, so it has products.

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