Resolutions in factorization categories.(English)Zbl 1353.13016

Matrix factorizations were introduced by Eisenbud in 1980 as a tool for studying free resolutions of modules over local hypersurface rings. Since then, matrix factorizations have proven useful in such diverse fields as algebraic geometry, mathematical physics, and knot theory.
In the article under review, the authors introduce a new framework for the study of matrix factorizations, which they call a factorization category. We give the definition here. Fix an abelian category $$\mathcal{A}$$, an autoequivalence $$\Phi$$ of $$\mathcal{A}$$, and a natural transformation $$w: \text{id}_{\mathcal{A}} \to \Phi$$ such that $$w_{\Phi(A)} = \Phi(w_A)$$ for all objects $$A \in \mathcal{A}$$. A factorization of the triple $$(\mathcal{A}, \Phi, w)$$ is a pair $$E^{-1}, E^0$$ of objects in $$\mathcal{A}$$ along with a pair of morphisms $\phi_E^{-1}: \Phi^{-1}(E^0) \to E^{-1}$
$\phi_E^{0}: E^{-1} \to E^{0}$ such that $$\phi_E^0 \phi_E^{-1} = \Phi^{-1}(w_{E^0})$$ and $$\Phi(\phi_E^{-1}) \phi_E^0 = w_{E^{-1}}$$. $$E^0$$ and $$E^{-1}$$ are called the components of the factorization. Morphisms of factorizations are defined component-wise in the evident way, so that factorizations form an abelian category. Applying formalism introduced by L. Positselski in [Mem. Am. Math. Soc. 996, 133 p. (2011; Zbl 1275.18002)], the authors introduce the notions of absolute derived, co-derived, and contra-derived categories of factorizations. The goal of the article is to study derived functors between such categories.
In Section 2, basics on factorization categories and their various derived categories are discussed. In Section 3, the authors discuss specific methods for “resolving” factorizations; that is, replacing them, up to various notions of derived isomorphism, with factorizations whose components are injective/projective. Section 4 contains a discussion of derived functors between the various derived categories of factorizations, and, in Section 5, the authors apply their results to some concrete examples arising in algebraic geometry.

MSC:

 13D09 Derived categories and commutative rings 18E30 Derived categories, triangulated categories (MSC2010)

Zbl 1275.18002
Full Text:

References:

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