Abstract:
We present a general construction of model category structures on the category $\mathbb{C}(\mathfrak{Qco}(X))$ of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme $X$. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of $X$. It does not require closure under direct limits as previous methods. We apply it to describe the derived category $\mathbb D (\mathfrak{Qco}(X))$ via various model structures on $\mathbb{C}(\mathgrak{Qco}(X))$. As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. Finally, we prove that the unrestricted case does not induce a model category structure as above in general.

Abstract:
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".

Abstract:
The process of photoproduction of electron-positron pairs in bent single crystals is considered in this paper. In particular, it is shown that the probability of the process for gamma-quanta with energies from 100 GeV on is significantly higher than the one in an amorphous medium. A possible scenario for the experimental validation of the process is discussed and the positive features of the photoproduction in bent crystals compared to straight ones are underlined from the point of view of possible applications.

Abstract:
Given a positively graded commutative coherent ring A which is finitely generated as an A_0-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y\subseteq Proj A of the form Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all i\in\Omega is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) --> (Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr A.

Abstract:
We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.

Abstract:
Let $A$ be a tubular algebra and let $r$ be a positive irrational. Let ${\mathcal D}_r$ be the definable subcategory of $A$-modules of slope $r$. Then the width of the lattice of pp formulas for ${\mathcal D}_r$ is $\infty$. It follows that if $A$ is countable then there is a superdecomposable pure-injective module of slope $r$.

Abstract:
We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.

Abstract:
Given a commutative ring R (respectively a positively graded commutative ring $A=\ps_{j\geq 0}A_j$ which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y\subset Spec R (respectively Y\subset Proj A) of the form Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\Y_i) quasi-compact and open for all i\in\Omega, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R,O_R)-->(Spec(Mod R),O_{Mod R}) and (Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R}) and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices L_{tor}(Mod R) and L_{tor}(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes perf(R) and the torsion classes of finite type in Mod R is established.