We study the dg-category of twisted complexes on a ringed space and prove that it gives a new dg-enhancement of the derived category of perfect complexes on that space. A twisted complex is a collection of locally defined sheaves together with the homotopic gluing data. We construct a dg-functor from twisted complexes to perfect complexes, which turns out to be a dg-enhancement. This new enhancement has the advantage of being completely geometric and it comes directly from the definition of perfect complex. In addition we will talk about some applications and further topics around twisted complexes.
In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the following two cases: (1) the complexes of sheaves of O-modules on theČech nerve of an open cover of a ringed space (X, O); (2) the complexes of sheaves on the simplicial nerve of a discrete group G acting on a space. The explicit models we obtain in this way are twisted complexes as well as their D-module and G-equivariant versions. As an application we show that there is a stack of twisted perfect complexes.
This paper gives a complete answer of the following question: which
(singular, projective) curves have a categorical resolution of singularities
which admits a full exceptional collection? We prove that such full exceptional
collection exists if and only if the geometric genus of the curve equals to 0.
Moreover we can also prove that a curve with geometric genus equal or greater
than 1 cannot have a categorical resolution of singularities which has a
tilting object. The proofs of both results are given by a careful study of the
Grothendieck group and the Picard group of that curve.Comment: 10 pages, minor change
In this paper we study the descent problem of cohesive modules on compact complex manifolds. For a complex manifold X we could consider the Dolbeault dg-algebra A(X) on it and Block in 2006 introduced a dg-category P A(X) , called cohesive modules, associated with A(X). The same construction works for any open subset U ⊂ X and we obtain a dg-presheaf on X given by U → P A(U) . In this paper we prove that this dg-presheaf satisfies descent for any locally finite open cover of a compact manifold X. This generalizes a result by Ben-Bassat and Block in 2012, which studied the case that X is covered by two open subsets.
In this paper we consider the dg-category of twisted complexes over simplicial spaces. It is clear that a simplicial map f : U → V between simplicial spaces induces a dg-functor f * : Tw(V, R) → Tw(U, R).In this paper we prove that for simplicial homotopic maps f and g, there exists an A∞-natural transformation Φ : f * ⇒ g * between induced dg-functors. Moreover the 0th component of Φ is a weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, we prove that Φ admits an A∞-quasiinverse.
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