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.     

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.     

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.     

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.     

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 properties of Sklyanin algebras

Summary. To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (“Sklyanin algebras”), having the Hilbert series of a polynomial algebra. In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other “Sklyanin-type” algebras. Oblatum 25-XI-1993     

Homological characterization of the unknot

Given a knot K in the 3-sphere, let Q_K be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H₂(Q_K)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let Q_K^π be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.     

Homological Properties of Quantum Polynomials

We study the endomorphim semigroup of a general quantum polynomial ring, its finite groups of automorphisms, and homological properties, as a module over the skew group ring of a finite group of automorphisms. Moreover, properties of the division ring of fractions are considered.     

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd_R over a commutative ring R satisfies the inequality if Λ embeds uniformly into Γ and holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry. Received: November 2004 Revision: April 2004 Accepted: April 2004     

Subfactor Categories of Triangulated Categories

Let 𝒳 ? 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4 Beligiannis , A. , Reiten , I. ( 2007 ). Homological and Homotopical Aspects of Torsion Theories . Memoirs of the AMS 883 : 426 – 454 . [Google Scholar]]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions.     

Restricted Homological Dimensions of Complexes

We define and study the notions of restricted Tor-dimension and Ext-dimension for unbounded complexes of left modules over associative rings. We show that, for a right (respectively, left) homologically bounded complex, our definition agrees with the small restricted flat (respectively, injective) dimension defined by Christensen et al. Furthermore, we show that the restricted Tor-dimension defined in this paper is a refinement of the Gorenstein flat dimension of an unbounded complex in some sense. In addition, we give some results concerning restricted homological dimensions under a base change over commutative Noetherian rings.