Abelian Category

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Omar Jaramillo - One of the best experts on this subject based on the ideXlab platform.

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
    2021
    Co-Authors: Positselski Leonid, Stovicek Jan
    Abstract:

    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 xy-pic; 45 pages, 5 commutative diagram

  • Derived, coderived, and contraderived categories of locally presentable Abelian categories
    2021
    Co-Authors: Positselski Leonid, Stovicek Jan
    Abstract:

    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 xy-pic; 50 pages, 5 commutative diagrams; v.2: Remarks 6.4 and 9.2 inserted, Introduction expanded, many references adde

  • Covers and direct limits: a contramodule-based approach
    'Springer Science and Business Media LLC', 2021
    Co-Authors: Bazzoni Silvana, Positselski Leonid
    Abstract:

    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 $n$-tilting-cotilting 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-$\operatorname{Ext}^1$-self-orthogonal) 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 pb-diagram and xy-pic, 58 pages, 5 commutative diagrams. v.1: This paper is based on Sections 11-15 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 DG-categories
    2021
    Co-Authors: Positselski Leonid
    Abstract:

    This paper is a greatly expanded version of Section 3.2 in arXiv:0905.2621. We construct an "almost involution" assigning a new DG-Category 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 DG-Category of DG-modules over a DG-ring $(R^*,d)$. This provides an appropriate technical background for the definition and discussion of Abelian and exact DG-categories. Examples range from the categories of complexes in Abelian/exact categories to matrix factorization categories, and from curved DG-modules over curved DG-rings to quasi-coherent CDG-modules over quasi-coherent CDG-quasi-algebras over schemes.Comment: LaTeX 2e with xy-pic, 63 pages, 21 commutative diagram

  • Flat ring epimorphisms of countable type
    'Cambridge University Press (CUP)', 2020
    Co-Authors: Positselski Leonid
    Abstract:

    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 Geigle-Lenzing 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 positive-degree Ext-orthogonality to all left $U$-modules and all $\mathbb G$-separated $\mathbb G$-complete left $R$-modules.Comment: LaTeX 2e with pb-diagram and xy-pic, 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.

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
    2021
    Co-Authors: Positselski Leonid, Stovicek Jan
    Abstract:

    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 xy-pic; 45 pages, 5 commutative diagram

  • Derived, coderived, and contraderived categories of locally presentable Abelian categories
    2021
    Co-Authors: Positselski Leonid, Stovicek Jan
    Abstract:

    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 xy-pic; 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
    2020
    Co-Authors: Positselski Leonid, Stovicek Jan
    Abstract:

    Extending the Wedderburn-Artin 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 Iovanov-Mesyan-Reyes. 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 tikz-cd, 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.