Coherent analogues of matrix factorizations and relative singularity categories

Abstract

Abstract. We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.

The Tilting–Cotilting Correspondence

“…To summarize, given a commutative Noetherian ring R of Krull dimension 1 and a multiplicative set of nonzero-divisors in it, we have derived equivalences (25), where ⋆ can be one of b, +, −, ∅, abs+, abs−, ctr, or abs. The ring structure of the topological ring R has an explicit description via (26), and the topology is also the obvious one: S −1 R is discrete, p∈P R p carries the product topology of the adic topologies, and the adelic ring A S = S −1 p∈P R p has a canonical locally pseudocompact topology (in the sense of [35, §4.3]).…”

Flat morphisms of finite presentation are very flat

“…On the other hand, on any quasi-compact semi-separated scheme there are enough flat quasi-coherent sheaves [11,Section 2.4], [4,Lemma A.1]. But arbitrary flat modules over a commutative ring are much more complicated, from the homological point of view, than the projective modules.…”

“…Matrix factorizations are a kind of curved differential modules, so the conventional construction of the derived category does not make sense for them. Instead, one has to use derived categories of the second kind, such as the coderived and the absolute derived categories [14,4]. However, for an exact category of infinite homological dimension, the coderived category may differ from the derived category [17,Section 2.1], [18,Examples 3.3].…”

∞-tilting theory

“…In particular, if A is the category of quasi-coherent sheaves on a quasi-compact semi-separated scheme X, then any quasi-coherent sheaf on X is the quotient of one of the so-called very flat quasi-coherent sheaves [42, Lemma 4.1.1] (see [38,Section 2.4] or [22,Lemma A.1] for the more widely known, but weaker assertion with flat sheaves in place of the very flat ones). If X is covered by n affine open subschemes, then the projective dimension of any very flat quasi-coherent sheaf, as an object of A, does not exceed n, as one can show using aČech resolution for the affine covering, together with the fact that the projective dimension of a very flat module does not exceed 1 (cf.…”