Functors of the category of abelian sheaves on regular affine scheme

In this short paper, we prove that ifR is a regular local ring of unequal characteristic, then there exists an additive covariant functorG from the category of abelian sheaves on SpecR to the category of abelian groups such that id _R (G(R))>dimG(R). This result shows that the answer to the question 3.8 (ii) in [3] may be negative.     

On the derived category of the classical Godeaux surface

We construct an exceptional sequence of length 11 on the classical Godeaux surface X$X$ which is the Z/5Z$\mathbb{Z}/5\mathbb{Z}$-quotient of the Fermat quintic surface in P³${\mathbb{P}}^3$. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z¹¹⊕Z/5Z${\mathbb{Z}}^{11}\oplus \mathbb{Z}/5\mathbb{Z}$. In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type _E8$E_8$. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.     

The snail lemma in a pointed regular category

In this paper we explore the snail lemma in a pointed regular category. In particular, we show that under the presence of cokernels of kernels, the validity of the snail lemma is equivalent to subtractivity of the category. As a corollary, this gives that in the more restrictive context of a normal category the validity of the snail lemma is equivalent to the validity of the snake lemma.     

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D ^b (X), where X is the blow up of ? ^n?r ×? ^r along a multilinear subspace ? ^n?r?1×? ^r?1 of codimension 2 of ? ^n?r ×? ^r . As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.     

Semi-orthogonal decomposability of the derived category of a curve

We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.     

On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let $\mathsf{X}$ be an Adams geometric stack. We show that $\mathsf{D}({\mathsf{A}}_{\mathsf{qc}}(\mathsf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in [13]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathsf{X}$.     

The generating hypothesis in the derived category of a ring

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author (J. Pure Appl. Algebra 208(2), 2007). We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular and therefore does not satisfy the strong form of the generating hypothesis.     

The bar derived category of a curved dg algebra

Curved A_∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A_∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.     

On the geometry of toric arrangements

Motivated by the counting formulas of integral polytopes, as in Brion and Vergne, and Szenes and Vergne, we start to form the foundations of a theory for toric arrangements, which may be considered as the periodic version of the theory of hyperplane arrangements.     

Irreducible morphisms in the bounded derived category

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ, denoted , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in either f^j is split monomorphism for all j∈Z or split epimorphism, for all j∈Z. Moreover, we prove that all the non-trivial components of the Auslander-Reiten quiver of are of the form ZA_∞.     

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of $\mathrm{Aut}(X)$ contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.     

On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley-Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley-Reisner and affine monoid algebras. We consider (non-pure) shellable fan’s and the Cohen-Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.