Comparison of spectral sequences involving bifunctors

Suppose given functors A x A' -F-> B -G-> C between abelian categories, an object X in A and an object X' in A' such that certain conditions hold. We show that, E_1-terms exempt, the Grothendieck spectral sequence of the composition of F(X,-) and G evaluated at X' is isomorphic to the Grothendieck spectral sequence of the composition of F(-,X') and G evaluated at X. So instead of "resolving X' twice", we may just as well "resolve X twice".Comment: Typos in Secs. 8.1, 8.2 corrected. Appeared in Documenta Math. 13, 677-737, 200

Nonisomorphic Verdier octahedra on the same base

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.Comment: typos correcte

On lifting stable diagrams in Frobenius categories

Suppose given a Frobenius category E, i.e. an exact category with a big enough subcategory B of bijectives. Let_E_ := E/B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in an additive category A, in which case_E_ is the homotopy category of complexes K(A). Suppose given a finite poset D that satisfies the combinatorial condition of being ind-flat. Then, given a diagram of shape D with values in_E_ (i.e. commutative up to homotopy), there exists a diagram consisting of pure monomorphisms with values in E (i.e. commutative) that is isomorphic, as a diagram with values in_E_, to the given diagram.Comment: Added discussion of the connection to related work of Cooke, Dwyer-Kan-Smith and Mitchell. To appear in Homology, Homotopy and Application