The Stacks project

Lemma 22.5.4. Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ has direct sums and products.

Proof. Omitted. Hint: Just use the direct sums and products as in Lemma 22.4.2. This works because we saw that these functors commute with the forgetful functor to the category of graded $A$-modules and because $\prod $ is an exact functor on the category of families of abelian groups. $\square$

Comments (2)

Comment #8480 by Et on

This proposition isn't hard to prove directly, but I wonder where you use : "and because ∏ is an exact functor on the category of families of abelian groups."?

Comment #9096 by on

To prove that Homs from into a product is the product of the homs (in the homotopy category).

There are also:

  • 3 comment(s) on Section 22.5: The homotopy category

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