Homological Epimorphisms Research Articles

WHEN IS THE SILTING-DISCRETENESS INHERITED?

Abstract We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

Triangle equivalences of Gorenstein defect categories induced by homological epimorphisms

We give a sufficient condition that a certain homological epimorphism between two finite dimensional algebras induces a triangle equivalence between their Gorenstein defect categories as well as between their stable categories of Gorenstein projective modules.

Rank functions on triangulated categories

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms

Our aim is to give an explicit description of the Arens-Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens-Michael envelope belongs to a class of completions introduced by R.~Goodman in 70s. To find a precise form of this algebra we preliminary characterize the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions. As a corollary, it is shown that the Arens-Michael envelope considered above is a homological epimorphism. So we get a positive answer to a question investigated earlier by Dosi and Pirkovskii.

Homological epimorphisms, homotopy epimorphisms and acyclic maps

Abstract We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

Matlis category equivalences for a ring epimorphism

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u:R⟶U. Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U-modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required.

Fabric idempotents and higher Auslander-Reiten theory

Over a finite-dimensional algebra A, simple A-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings and they have also appeared in work concerning the finitistic dimension conjecture. If we however work in a d-cluster-tilting subcategory, then not all simples are contained in the subcategory. In this context, a replacement might be to work with idempotents instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localisable objects studied by Chen-Krause, and show that they provide rich combinatorial properties to illustrate the theory. In particular, we use fabric idempotents to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.

Weakly Stable Torsion Classes

Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring $A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by studying torsion and completion with respect to a single regular and normal element.

Homological epimorphisms, compactly generated t-structures and Gorenstein-projective modules

The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i*, i!, j!, j*, j*), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i* preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i*, i!, j!, j*, j*) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, A B has finite projective dimension and j! restricts to D b (C-mod), then this recollement induces an unbounded ladder (B-Gproj,A-Gproj, C-Gproj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.

Recollements and stratifying ideals

Surjective homological epimorphisms with stratifying kernel can be used to construct recollements of derived module categories. These ‘stratifying’ recollements are derived from recollements of module categories. Can every recollement be put in this form, up to equivalence? A negative answer will be given after providing a characterisation of recollements equivalent to stratifying ones. Moreover, criteria for a ring epimorphism to be ‘stratifying’ will be presented as well as constructions of such epimorphisms.

Smashing localizations of rings of weak global dimension at most one

We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Prüfer domains which are strongly discrete.

Homological epimorphisms, recollements and Hochschild cohomology – with a conjecture by Snashall–Solberg in view

We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig–Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate and formulate a variation of the finite generation conjecture by Snashall–Solberg.

Singular equivalences induced by homological epimorphisms

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.

The telescope conjecture for hereditary rings via Ext-orthogonal pairs

For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2.

Recollements and tilting objects

We study connections between recollements of the derived category D (Mod R ) of a ring R and tilting theory. We first provide constructions of tilting objects from given recollements, recovering several different results from the literature. Secondly, we show how to construct a recollement from a tilting module of projective dimension one. By Nicolás and Saorín (2009) [31], every recollement of D (Mod R ) is associated to a differential graded homological epimorphism λ : R → S . We will focus on the case where λ is a homological ring epimorphism or even a universal localization. Our results will be employed in a forthcoming paper in order to investigate stratifications of D (Mod R ).

Homological Epimorphisms of Differential Graded Algebras

Let R and S be differential graded algebras. In this article, we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories In particular, this characterisation generalises the theory of Geigle and Lenzing's homological epimorphisms of rings, described in [3]. Furthermore, there is an application of the main result to Dwyer and Greenlees's Morita theory.

Parametrizing recollement data for triangulated categories

We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a ℵ 0 -perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of H. Krause's work [Henning Krause, Cohomological quotients and smashing localizations, Amer. J. Math. 127 (2005) 1191–1246]. This bijection implies the following weak version of the generalized smashing conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects.

Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

Hochschild cohomology of algebras with homological ideals

Let $\varphi: A \to B$ be a homological epimorphism of $k$-algebras. We investigate the relationship of the Hochschild cohomologies $H^{i}(A)$ and $H^{i}(B)$ of $A$ and $B$, and show that they can be connected by a long exact sequence. In particular, if $A$ is a quasihereditary algebra and $B$ is the quotient of $A$ by a minimal heredity ideal, then the long exact sequence provides information on $H^{i}(A)$, $H^{i}(B)$ and the extension groups between costandard modules and standard modules, thus one can actually compute $H^{i}(A)$ inductively. As a consequence, we obtain the Hochschild cohomology of all nonsemisimple Temperley-Lieb algebras and representation-finite Schur algebras.

On homomorphisms from a fixed representation to a general representation of a quiver

We study the dimension of the space of homomorphisms from a given representation X of a quiver to a general representation of dimension vector 3. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as 3 increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring. In this paper we prove a number of results about hom(X, 3), the dimension of the space of homomorphisms from a fixed representation X of a quiver to a general representation of dimension vector 3. Our basic result relates this number to the dimension of a subset of a Grassmannian of submodules of X. This result is in the spirit of A. Schofield's paper on general representations of quivers [S2], and generalizes one of his results. We derive two corollaries concerning the asymptotic behaviour of hom(X, r/3), of interest in themselves, and also because of their connection with another theorem of Schofield, that the homological epimorphisms from a path algebra to a simple artinian ring are in 1-1 correspondence with indivisible Schur roots. (A homological epimorphism is a ring homomorphism R -* S with S OR S -S and Tori(S, S) = 0 for i > 0. This correspondence was announced by Schofield in a lecture in March 1995 in Krippen, Germany, but actually dates from 1991.) In fact our second corollary is already known, proved by Schofield and used in the proof of the correspondence, but his proof of the corollary involves difficult results about semi-invariants of quivers, whereas the proof here is quite elementary. Let K be an algebraically closed field, Q a finite quiver with vertex set I, and let (-,-) be the Ringel form on Z'. If 3 E N' we write RePKQW() = 17 Hom(K8('), K:(j)) a:i-j for the configuration space of representations of Q of dimension vector 3, and if y E RePKQ(/) we write Ky for the corresponding left KQ-module. If X is a finitely generated left KQ-module and y E RePKQ (/3) then Hom(X, Ky) is a finite dimensional vector space. Moreover the function y F dim Hom(X, Ky) is an upper semicontinuous function on RePKQ(/3), and it follows that the minimum value of this function, denoted hom(X,/3), is also its general value. The rank of a homomorphism X -* Ky is the dimension vector of its image. For any Ol E N' the set of homomorphisms of rank at most Ol is a closed subset of Hom(X, Ky). It follows that there is a unique maximal rank Yx,y of homomorphisms from X to Ky, Received by the editors July 21, 1995. 1991 Mathematics Subject Classification. Primary 16G20; Secondary 14M15. ?1996 American Mathematical Society