Homological Systems in Triangulated Categories

We introduce the notion of homological systems Θ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. We prove that, attached to the homological system Θ, there are two standardly stratified algebras A and B, which are derived equivalent. Furthermore, it is proved that the category \(\mathfrak {F}({\Theta }),\) of the Θ-filtered objects in a triangulated category \(\mathcal {T},\) admits in a very natural way a structure of an exact category, and then there are exact equivalences between the exact category \(\mathfrak {F}({\Theta })\) and the exact categories of the Δ-good modules associated to the standardly stratified algebras A and B. Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino (see 6.6 and 6.7 ). We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, t-structures and co-t-structures.     

On Direct Summands of Homological Functors on Length Categories

We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories.     

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.     

相对于半对偶化模的Gorenstein同调维数与Auslander范畴

设$R$是一个局部noether环. 我们在本文中研究了相对于半对偶化模$C$的Gorenstein投射, 内射与平坦模. 给出了$C$-Gorenstein同调维数与$\hat{R}$的Auslander范畴之间的关系.     

On Combinatorial Model Categories

Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke, D. Dugger or J. Lurie. We are contributing to this endeavour by some new results about homotopy equivalences, weak equivalences and cofibrations in combinatorial model categories. Supported by MSM 0021622409 and GAČR 201/06/0664.     

Space-Time Localisation for the Dynamic Model

We prove an a priori bound for solutions of the dynamic equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions. We treat the large- and small-scale behaviour of solutions with completely different arguments. For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure. The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large-scale properties of solutions. They can, for example, be used in a compactness argument to construct solutions on the full space and their invariant measures. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC     

A Model Structure on Categories Related to the Categories of Complexes

Ukrainian Mathematical Journal - We prove a Hinich-type theorem on the existence of a model structure on a category related by adjunction to the category of differential graded modules over a...     

Model Categories in Algebraic Topology

This survey of model categories and their applications in algebraic topology is intended as an introduction for non homotopy theorists, in particular category theorists and categorical topologists. We begin by defining model categories and the homotopy-like equivalence relation on their morphisms. We then explore the question of compatibility between monoidal and model structures on a category. We conclude with a presentation of the Sullivan minimal model of rational homotopy theory, including its application to the study of Lusternik–Schnirelmann category.     

奇点模型范畴的Quillen等价

令R是左Gorenstein环.我们构造了奇点反导出模型范畴和奇点余导出模型范畴(见文[Models for singularity categories,Adv Math.,2014,254:187-232])之间的Quillen等价.作为应用,给出了投射,内射模的正合复形的同伦范畴之间的一个具体的等价■.     

Exact Sequences and Closed Model Categories

For every closed model category with zero object, Quillen gave the construction of Eckman-Hilton and Puppe sequences. In this paper, we remove the hypothesis of the existence of zero object and construct (using the category over the initial object or the category under the final object) these sequences for unpointed model categories. We illustrate the power of this result in abstract homotopy theory given some interesting applications to group cohomology and exterior homotopy groups.     

同调方程

设Ｒ是左、右凝聚环,Ｒ~ωＲ是一个忠实平衡自正交双模．对有限表现左Ｒ－模Ａ和正整数ｎ,本文研究了形如,　的同调方程．给出了模范畴为有限表现右Ｒ-模｝是子模闭的充要条件,并举例说明了该模范畴并非总是子模闭的,     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0.     

Algebras and Modules in Monoidal Model Categories

In recent years the theory of structured ring spectra (formerlyknown as A- and E-ring spectra) has been simplified by the discoveryof categories of spectra with strictly associative and commutativesmash products. Now a ring spectrum can simply be defined asa monoid with respect to the smash product in one of these newcategories of spectra. In this paper we provide a general methodfor constructing model category structures for categories ofring, algebra, and module spectra. This provides the necessaryinput for obtaining model categories of symmetric ring spectra,functors with smash product, Gamma-rings, and diagram ring spectra.Algebraic examples to which our methods apply include the stablemodule category over the group algebra of a finite group andunbounded chain complexes over a differential graded algebra.1991 Mathematics Subject Classification: primary 55U35; secondary18D10.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

Homological projective duality

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.     

半Artin环的同调维数

本主要研究半Artin环和完全环的等价性，以及半Artin环的同调维数．     

Left Determined Model Structures for Locally Presentable Categories

We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are “left determined” in the sense of Rosicky and Tholen.     

Homological algebra of homotopy algebras

We study the varieties that parametrize trigonal curves with assigned Weierstrass points; we prove that they are irreducible and compute their dimensions. To do so, we stratify the moduli space of all trigonal curves with given Maroni invariant.