The Experts below are selected from a list of 9018 Experts worldwide ranked by ideXlab platform
Omar Jaramillo  One of the best experts on this subject based on the ideXlab platform.

auslander reiten sequences and t structures on the homotopy Category of an Abelian Category
Journal of Algebra, 2011CoAuthors: Erik Backelin, Omar JaramilloAbstract:We use tstructures on the homotopy Category Kb(Rmod) for an artin algebra R and Wattsʼ representability theorem to give an existence proof for Auslander–Reiten sequences of Rmodules. This framework naturally leads to a notion of generalized (or higher) Auslander–Reiten sequences.

auslander reiten sequences and t structures on the homotopy Category of an Abelian Category
arXiv: Representation Theory, 2009CoAuthors: Erik Backelin, Omar JaramilloAbstract:We use $t$structures on the homotopy Category $K^b(Rmod)$ for an artin algebra $R$ and Watts' representability theorem to give an existence proof for AuslanderReiten sequences of $R$modules.
Positselski Leonid  One of the best experts on this subject based on the ideXlab platform.

Derived, coderived, and contraderived categories of locally presentable Abelian categories
2021CoAuthors: Positselski Leonid, Stovicek JanAbstract:For a locally presentable Abelian Category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the Category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived Category $\mathsf D(\mathsf B)$ is generated, as a triangulated Category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck Abelian Category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived Category $\mathsf D(\mathsf A)$ is generated, as a triangulated Category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact Category with an object size function and prove that the derived Category of any such exact Category with exact $\kappa$directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived Category of any locally presentable Abelian Category has Hom sets.Comment: LaTeX 2e with xypic; 45 pages, 5 commutative diagram

Derived, coderived, and contraderived categories of locally presentable Abelian categories
2021CoAuthors: Positselski Leonid, Stovicek JanAbstract:For a locally presentable Abelian Category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the Category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived Category $\mathsf D(\mathsf B)$ is generated, as a triangulated Category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck Abelian Category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived Category $\mathsf D(\mathsf A)$ is generated, as a triangulated Category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact Category with an object size function and prove that the derived Category of any such exact Category with exact $\kappa$directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived Category of any locally presentable Abelian Category has Hom sets.Comment: LaTeX 2e with xypic; 50 pages, 5 commutative diagrams; v.2: Remarks 6.4 and 9.2 inserted, Introduction expanded, many references adde

Covers and direct limits: a contramodulebased approach
'Springer Science and Business Media LLC', 2021CoAuthors: Bazzoni Silvana, Positselski LeonidAbstract:We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the $n$tiltingcotilting correspondence situation, if $\mathsf A$ is a Grothendieck Abelian Category and the related Abelian Category $\mathsf B$ is equivalent to the Category of contramodules over a topological ring $\mathfrak R$ belonging to one of certain four classes of topological rings (e.g., $\mathfrak R$ is commutative), then the left tilting class is covering in $\mathsf A$ if and only if it is closed under direct limits in $\mathsf A$, and if and only if all the discrete quotient rings of the topological ring $\mathfrak R$ are perfect. More generally, if $M$ is a module satisfying a certain telescope Hom exactness condition (e.g., $M$ is $\Sigma$pure$\operatorname{Ext}^1$selforthogonal) and the topological ring $\mathfrak R$ of endomorphisms of $M$ belongs to one of certain seven classes of topological rings, then the class $\mathsf{Add}(M)$ is closed under direct limits if and only if every countable direct limit of copies of $M$ has an $\mathsf{Add}(M)$cover, and if and only if $M$ has perfect decomposition. In full generality, for an additive Category $\mathsf A$ with (co)kernels and a precovering class $\mathsf L\subset\mathsf A$ closed under summands, an object $N\in\mathsf A$ has an $\mathsf L$cover if and only if a certain object $\Psi(N)$ in an Abelian Category $\mathsf B$ with enough projectives has a projective cover. The $1$tilting modules and objects arising from injective ring epimorphisms of projective dimension $1$ form a class of examples which we discuss.Comment: LaTeX 2e with pbdiagram and xypic, 58 pages, 5 commutative diagrams. v.1: This paper is based on Sections 1115 and 19 of the long preprint arXiv:1807.10671v1, which was divided into three parts. v.2: Many important improvements and additions based on new results in arXiv:1909.12203 and particularly in arXiv:1911.11720; new Section 5 inserted. v.4: Final versio

Exact DGcategories
2021CoAuthors: Positselski LeonidAbstract:This paper is a greatly expanded version of Section 3.2 in arXiv:0905.2621. We construct an "almost involution" assigning a new DGCategory to a given one, and use this construction in order to recover, say, the Abelian Category of graded modules over the graded ring $R^*$ from the DGCategory of DGmodules over a DGring $(R^*,d)$. This provides an appropriate technical background for the definition and discussion of Abelian and exact DGcategories. Examples range from the categories of complexes in Abelian/exact categories to matrix factorization categories, and from curved DGmodules over curved DGrings to quasicoherent CDGmodules over quasicoherent CDGquasialgebras over schemes.Comment: LaTeX 2e with xypic, 63 pages, 21 commutative diagram

Flat ring epimorphisms of countable type
'Cambridge University Press (CUP)', 2020CoAuthors: Positselski LeonidAbstract:Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$module $U$ has projective dimension at most $1$. Furthermore, the Abelian Category of left contramodules over the completion of $R$ at $\mathbb G$ fully faithfully embeds into the GeigleLenzing right perpendicular subCategory to $U$ in the Category of left $R$modules, and every object of the latter Abelian Category is an extension of two objects of the former one. We discuss conditions under which the two Abelian categories are equivalent. Given a right linear topology on an assocative ring $R$, we consider the induced topology on every left $R$module, and for a perfect Gabriel topology $\mathbb G$ compare the completion of a module with an appropriate Ext module. Finally, we characterize the $U$strongly flat left $R$modules by the two conditions of left positivedegree Extorthogonality to all left $U$modules and all $\mathbb G$separated $\mathbb G$complete left $R$modules.Comment: LaTeX 2e with pbdiagram and xypic, 64 pages, 6 commutative diagrams + Corrigenda, LaTeX 2e with ulem.sty, 8 pages; v.6: main file unchanged, corrigenda added (two mistakes, one in Remark 3.3 and the other one in some proofs in Section 5, discussed and corrected, main results unaffected); v.7: main file unchanged, third section added to corrigenda (clarifying confusion in the last Remark 11.3
Erik Backelin  One of the best experts on this subject based on the ideXlab platform.

auslander reiten sequences and t structures on the homotopy Category of an Abelian Category
Journal of Algebra, 2011CoAuthors: Erik Backelin, Omar JaramilloAbstract:We use tstructures on the homotopy Category Kb(Rmod) for an artin algebra R and Wattsʼ representability theorem to give an existence proof for Auslander–Reiten sequences of Rmodules. This framework naturally leads to a notion of generalized (or higher) Auslander–Reiten sequences.

auslander reiten sequences and t structures on the homotopy Category of an Abelian Category
arXiv: Representation Theory, 2009CoAuthors: Erik Backelin, Omar JaramilloAbstract:We use $t$structures on the homotopy Category $K^b(Rmod)$ for an artin algebra $R$ and Watts' representability theorem to give an existence proof for AuslanderReiten sequences of $R$modules.
Stovicek Jan  One of the best experts on this subject based on the ideXlab platform.

Derived, coderived, and contraderived categories of locally presentable Abelian categories
2021CoAuthors: Positselski Leonid, Stovicek JanAbstract:For a locally presentable Abelian Category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the Category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived Category $\mathsf D(\mathsf B)$ is generated, as a triangulated Category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck Abelian Category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived Category $\mathsf D(\mathsf A)$ is generated, as a triangulated Category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact Category with an object size function and prove that the derived Category of any such exact Category with exact $\kappa$directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived Category of any locally presentable Abelian Category has Hom sets.Comment: LaTeX 2e with xypic; 45 pages, 5 commutative diagram

Derived, coderived, and contraderived categories of locally presentable Abelian categories
2021CoAuthors: Positselski Leonid, Stovicek JanAbstract:For a locally presentable Abelian Category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the Category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived Category $\mathsf D(\mathsf B)$ is generated, as a triangulated Category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck Abelian Category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived Category $\mathsf D(\mathsf A)$ is generated, as a triangulated Category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact Category with an object size function and prove that the derived Category of any such exact Category with exact $\kappa$directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived Category of any locally presentable Abelian Category has Hom sets.Comment: LaTeX 2e with xypic; 50 pages, 5 commutative diagrams; v.2: Remarks 6.4 and 9.2 inserted, Introduction expanded, many references adde

Topologically semisimple and topologically perfect topological rings
2020CoAuthors: Positselski Leonid, Stovicek JanAbstract:Extending the WedderburnArtin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the Abelian Category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the Abelian Category of discrete right modules over the same ring is split (equivalently, semisimple). Our results in this direction complement those of IovanovMesyanReyes. An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equivalences and implications between which we prove. In particular, all the conditions are equivalent for topological rings with a countable base of neighborhoods of zero and for topologically right coherent topological rings. Considering the rings of endomorphisms of modules as topological rings with the finite topology, we establish a close connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. We show that any topologically agreeable split Abelian Category is Grothendieck and semisimple. We also prove that a module $\Sigma$coperfect over its endomorphism ring has a perfect decomposition provided that either the endomorphism ring is commutative or the module is countably generated, partially answering a question of Angeleri Hugel and Saorin.Comment: LaTeX 2e with tikzcd, 66 pages; new Section 13 (on topologically coherent rings) inserted, final Section 14 (former 13) rewritten with much more complete picture obtained using new results of arXiv:1911.11720, some proofs improved and simplified in Sections 10 and 12, Lemma 7.2 deleted as no longer relevant, Examples 3.7(2) and 9.4 inserted, references added, abstract and introduction update
Zhang Haicheng  One of the best experts on this subject based on the ideXlab platform.

Drinfeld doubles via derived Hall algebras and Bridgeland's Hall algebras
'Annals of Mathematics', 2021CoAuthors: Xu Fan, Zhang HaichengAbstract:summary:Let ${\cal A}$ be a finitary hereditary Abelian Category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of ${\cal A}$ via its derived Hall algebra and Bridgeland's Hall algebra of $m$cyclic complexes