2,159 research outputs found
A compactness result in approach theory with an application to the continuity approach structure
We establish a compactness result in approach theory which we apply to obtain
a generalization of Prokhorov's Theorem for the continuity approach structure.Comment: 10 page
Hochschild cohomology, the characteristic morphism and derived deformations
A notion of Hochschild cohomology of an abelian category was defined by Lowen
and Van den Bergh (2005) and they showed the existence of a characteristic
morphism from the Hochschild cohomology into the graded centre of the (bounded)
derived category. An element in the second Hochschild cohomology group
corresponds to a first order deformation of the abelian category (Lowen and Van
den Bergh, 2006). The problem of deforming single objects of the bounded
derived category was treated by Lowen (2005). In this paper we show that the
image of the Hochschild cohomology element under the characteristic morphism
encodes precisely the obstructions to deforming single objects of the bounded
derived category. Hence this paper provides a missing link between the above
works. Finally we discuss some implications of these facts in the direction of
a ``derived deformation theory''.Comment: 24 page
On compact generation of deformed schemes
We obtain a theorem which allows to prove compact generation of derived
categories of Grothendieck categories, based upon certain coverings by
localizations. This theorem follows from an application of Rouquier's
cocovering theorem in the triangulated context, and it implies Neeman's result
on compact generation of quasi-compact separated schemes. We prove an
application of our theorem to non-commutative deformations of such schemes,
based upon a change from Koszul complexes to Chevalley-Eilenberg complexes.Comment: 21 page
Linear quasi-categories as templicial modules
We introduce a notion of enriched -categories over a suitable
monoidal category, in analogy with quasi-categories over the category of sets.
We make use of certain colax monoidal functors, which we call templicial
objects, as a variant of simplicial objects respecting the monoidal structure.
We relate the resulting enriched quasi-categories to nonassociative Frobenius
monoidal functors, allowing us to prove that the free templicial module over an
ordinary quasi-category is a linear quasi-category. To any dg-category we
associate a linear quasi-category, the linear dg-nerve, which enhances the
classical dg-nerve. Finally, we prove an equivalence between (homologically)
non-negatively graded dg-categories on the one hand and templicial modules with
a Frobenius structure on the other hand, indicating that nonassociative
Frobenius templicial modules and linear quasi-categories can be seen as
relaxations of dg-categories.Comment: 68 pages, no figures; revised introduction, added references, revised
section 4 for readability, results unchange
Abelian and derived deformations in the presence of Z-generating geometric helices
For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in
Z}, is equivalent to the category of "quasi-coherent modules" over an
associated Z-algebra A, we show that under suitable cohomological conditions
"taking quasi-coherent modules" defines an equivalence between linear
deformations of A and abelian deformations of C. If (O(n))_{n in Z} is at the
same time a geometric helix in the derived category, we show that restricting a
(deformed) Z-algebra to a "thread" of objects defines a further equivalence
with linear deformations of the associated matrix algebra.Comment: 21 page
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