Triangulated categories of extensions and the Second Isomorphism Theorem for triangulated categories

Details Triangulated categories of extensions and the Second Isomorphism Theorem for triangulated categories Triangulated categories of extensions and the Second Isomorphism Theorem for triangulated categories

2012-01-12

arxiv.org/abs/1201.2499v1

Mathematics

Representation Theory

8 pages

Scientific paper

Let T be a triangulated category with triangulated subcategories X and Y. We show that the subcategory of extensions X * Y is triangulated if and only if Y * X is contained in X * Y. In this situation, we show the following analogue of the Second Isomorphism Theorem: (X * Y) / X is equivalent to Y / (X \cap Y) and (X * Y) / Y is equivalent to X / (X \cap Y). This follows from the existence of a stable t-structure (X / (X \cap Y), Y / (X \cap Y)) in (X * Y) / (X \cap Y). We use the machinery to give a recipe for constructing triangles of recollements and recover some triangles of recollements from the literature.

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