Mathematics – Representation Theory
Scientific paper
2012-04-01
Mathematics
Representation Theory
24 pages
Scientific paper
The Grothendieck construction of a diagram X of categories can be seen as a process to construct a single category Gr(X) by gluing categories in the diagram together. Here we formulate diagrams of categories as oplax functors from a small category I to the 2-category k-Cat of small k-categories for a fixed commutative ring k. In our previous paper we defined derived equivalences of those oplax functors. Roughly speaking two oplax functors X, X' : I \to k-Cat are derived equivalent if there exist a family of triangle equivalences F(i) from the derived categories D(Mod X(i)) to D(Mod X'(i)) indexed by the objects i in I such that for each morphism a : i \to j in I F(j)D(Mod X)(a) is naturally isomorphic to D(Mod X')(a)F(i), where D(Mod X)(a) : D(Mod X(i)) \to D(Mod X'(i)) is given by the left derived tensor product with an X(i)-X'(i)-bimodule \bar{X(a)} naturally defined by the functor X(a) : X(i) \to X(j). In this paper we glue the derived equivalences F(i) together to obtain a derived equivalence between Grothendieck constructions Gr(X) and Gr(X'), which shows that if oplax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two k-categories with a G-action for a group G are "G-equivariantly" derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two k-algebras A and A' are derived equivalent, then so are the path categories AQ and A'Q for any quiver Q; so are the incidence algebras AS and A'S for any poset S; and so are the semigroup algebras AG and A'G for any semigroup G. Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence.
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