Manuel Saorín

University of Murcia | UM · Department of Mathematics

Let P<∞(Λ-mod) be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra Λ. We develop a widely applicable criterion that reduces the test for contravariant finiteness of P<∞(Λ-mod) in Λ-mod to corner algebras eΛe for suitable idempotents e ∈Λ. The reduction substantially facilitates access to the...

Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in B has a pseudocokernel in C that belongs to B. We apply this result and its dual to, among others, preabelian and pretriangulated...

We consider an arbitrary Abelian category \(\mathcal {A}\) and a subcategory \(\mathcal {T}\) closed under extensions and direct summands, and characterize those \(\mathcal {T}\) that are (semi-)special preenveloping in \(\mathcal {A}\); as a byproduct, we generalize to this setting several classical results for categories of modules. For instance,...

Let $\mathcal{P}^{<\infty} (\Lambda$-mod$)$ be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra $\Lambda$. We develop an applicable criterion that reduces the test for contravariant finiteness of $\mathcal{P}^{<\infty} (\Lambda$ -mod$)$ in $\Lambda$-mod to corner algebras $e \Lambda e$ for su...

Given any additive category $\mathcal{C}$ with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory $\mathcal{B}$ is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in $\mathcal{B}$ has a pseudocokernel in $\mathcal{C}$ that belongs to $\mathcal{B}$. We apply this result an...

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is...

We extend to the context of Abelian categories the bijection, classically given in categories of modules, between equivalence classes of tilting (resp., finendo quasi-tilting) modules and special (resp., semi-special) preenveloping torsion classes. We show that, when $\mathcal{A}$ has an epi-generator, the (semi-)special preenveloping torsion class...

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF...

In previous work, based on the work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to the degeneration of modules. In triangulated categories ${\mathcal{T}}$ , it is surprising that the zero object may degenerate. We show that the triangulated subcategory of ${\mathcal{T}}$ generated by th...

We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective...

In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smal\o (HSR) t-structure in the derived category of a Grothendieck category associated to a torsion pair in the latter. We revisit the HRS tilting process deriving from it a lot of i...

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this...

Given a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in a Grothendieck category $\mathcal{G}$, we study when the heart $\mathcal{H}_{\mathbf t}$ of the associated Happel-Reiten-Smalo $t$-structure in the derived category ${\mathbf D}(\mathcal{G})$ is a locally finitely presented or a locally coherent Grothendieck category. Since $\mathcal{H}...

We present a natural extension to functor categories over small preadditive categories of the classical results of Gabriel and Jans classifying, respectively, hereditary torsion pairs and TTF triples in terms of Gabriel topologies and idempotent ideals.

In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We study this systematically. In particular we show that the degeneration of the zero object actually...

We study the lifting problem for recollements of triangulated subcategories of triangulated categories with coproducts and for the associated TTF triples. We prove, under relatively mild assumptions, that, when these latter categories are compactly generated and the subcategories in the recollement contain compact objects the preservation of compac...

Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom\(_\mathrm{A}\)(T,?) is fully faithful and preserves compact objects, in which case nice recollements situations appe...

Let $\mathcal D$ be a triangulated category endowed with a $t$-structure $\mathfrak t=(\mathcal U,\Sigma \mathcal V)$ and denote by $\mathcal H:=\mathcal U\cap \Sigma\mathcal V$ its heart. In this paper we study the following well-known problem: Under what conditions on $\mathcal D$ and $\mathfrak t$ can we say that $\mathcal H$ is a Grothendieck c...

We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

We develop the theory dg algebras with enough idempotents and their dg modules and show their equivalence with that of small dg categories and their dg modules. We introduce the concept of dg adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as...

We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generat...

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion...

Let be a Grothendieck category, let be a torsion pair in and let be the associated Happel–Reiten–Smalø t-structure in the derived category . We prove that the heart of this t-structure is a Grothendieck category if, and only if, the torsionfree class is closed under taking direct limits in .

We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific dis...

We introduce and study the notion of pseudo-Frobenius graded algebra with enough idempotents, showing that it follows the pattern of the classical concept of pseudo-Frobenius (PF) and Quasi-Frobenius (QF) rings, in particular finite dimensional self-injective algebras, as studied by Nakayama, Morita, Faith, Tachikawa, etc. We show that such an alge...

We describe the multiplicative structure of the Hochschild cohomology ring H H ∗(Λ) of the generalized preprojective algebra \(\Lambda =\mathbb {B}_{n}\). This is done by giving the structure of the cohomology groups as modules over the center of Λ and by giving a presentation of H H ∗(Λ), as a bigraded algebra, by means of generators and relations...

Let $R$ be a commutative Noetherian ring and let $\mathcal D(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $\mathcal D(R)$ associated to a left bounded filtration by supports of Spec$(R)$ have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in $\mat...

For a commutative noetherian ring \(R\), we establish a bijection between the resolving subcategories consisting of finitely generated \(R\)-modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R[1]\) in their heart. Under this bijection, the t-structure...

Given associative unital algebras $A$ and $B$ and a complex $T^\bullet$ of $B-A-$bi\-modules, we give necessary and sufficient conditions for the total derived functors, $\Rh_A(T^\bullet,?):\D(A)\longrightarrow\D(B)$ and $?\Lt_BT^\bullet:\D(B)\longrightarrow\D(A)$, to be fully faithful. We also give criteria for these functors to be one of the full...

Given a torsion pair $\mathbf{t} = (\mathcal{T} ;\mathcal{F})$ in a module category $R$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smal\o $\text{ }$ t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\mathbf{t}}$ which is a module category. We also study when such a pair is given by a 2-term complex...

Within the class of finite dimensional mesh algebras, also called m-fold mesh algebras, we identify those which are symmetric and those whose stable module category is weakly Calabi-Yau. We also give, in combinatorial terms, explicit formulas for the period of any such algebra, and for the Calabi-Yau Frobenius and stable Calabi-Yau dimensions, when...

If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard definition of flatness in $Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if any) betw...

The goal of this paper is to give details on categorical constructions related to the categories of Gelfand-Tsetlin modules for associative algebras which first appeared in Yu. A. Drozd, S. A. Futorny and S. A. Ovsienko [NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 79–93 (1994; Zbl 0812.17007)]. We use an alternative approach to show that the categ...

We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of EndH(M) op „ «, then the representation dimension of B 0 Λ = is less or equal to 3 whenever one of the following M H conditions hold: i) H is of finite representation type; ii) H is tame and M is a direct sum of regul...

We study t-structures on D(R) the derived category of modules over a commutative Noetherian ring R generated by complexes in Dfg−(R). We prove that they are exactly the compactly generated t-structures on D(R) and describe them in terms of decreasing filtrations by supports of Spec(R). A decreasing filtration by supports ϕ:Z→Spec(R) satisfies the w...

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coh...

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any...

We study when the stable category A/áTñ{\mathcal A}/\langle{\mathcal T}\rangle of an abelian category A{\mathcal A} modulo a full additive subcategory T{\mathcal T} is balanced and, in case T{\mathcal T} is functorially finite in A{\mathcal A}, we study a weak version of balance for A/áTñ{\mathcal A}/\langle{\mathcal T}\rangle. Precise necessary an...

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 ℵ0-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...

Given a significative class $F$ of commutative rings, we study the precise conditions under which a commutative ring $R$ has an $F$-envelope. A full answer is obtained when $F$ is the class of fields, semisimple commutative rings or integral domains. When $F$ is the class of Noetherian rings, we give a full answer when the Krull dimension of $R$ is...

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in "Parametrizing recollement data", allows us to show that many well-known recollements of right bounded derived categories of algebras are re...

We show that if $\Lambda$ is a $n$-Koszul algebra and $E=E(\Lambda)$ is its Yoneda algebra, then there is a full subcategory $\mathcal{L}_E$ of the category $Gr_E$ of graded $E$-modules, which contains all the graded $E$-modules presented in even degrees, that embeds fully faithfully into the category $C(Gr_\Lambda)$ of cochain complexes of graded...

We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set \phi(i) contains all the inmediate generaliza...

A TTF-triple (C,T,F) in an abelian category is one-sided split in case either (C,T) or (T,F) is a split torsion theory. In this paper we classify one-sided split TTF-triples in module categories, thus completing Jans’ classification of two-sided split TTF-triples and answering a question that has remained open for almost 40 years.

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In our work (9), we complete Jans' classification of TTF-triples (8) by giving a precise description of those two-sided ideals of a ring associated to one-sided split TTF-triples in the corresponding module category.

Killing of supports along subsets U of a group G and regradings along certain maps of groups are studied, in the context of group-graded algebras. We show that, under precise conditions on U and φ, the module theories over the initial and the final algebras are functorially well-connected. Special attention is paid to G=Z, in which case the results...

It is well known that a module M over an arbitrary ring admits an indecomposable decomposition whenever it has the property that every local direct summand of M is a direct summand [28]. Recently, J. L. Gomez Pardo and P. Guil Asensio [18] have shown that requiring this property not only for M but for any direct sum M (ℵ) of copies of M even yields...

The correspondence between the category of modules over a graded algebra and the category of graded modules over its Yoneda algebra was studied by B. Keller [in Representations of algebras. Vol. I, II. Beijing: Beijing Normal University Press. 74-86 (2002; Zbl 1086.18007)] by means of A ∞ -algebras; this relation is very well understood for Koszul...

For every positively graded algebra A, we show that its categories of linear complexes of projectives and almost injectives (see definition below) are both naturally equivalent to the category of graded modules over the quadratic dual algebra A!. In case A = Λ is a graded factor of a path algebra with Yoneda algebra Γ, we show that the category LcΓ...

In the category of finitely generated modules over an artinian ring, we classify all the abelian exact subcategories closed under predecessors or, equivalently, all the split torsion pairs with torsion-free class closed under quotients.

Suppose that $A$ is a semiprimary ring satisfying one of the two conditions: 1) its Yoneda ring is generated in finite degrees; 2) its Loewy length is less or equal than three. We prove that the global dimension of $A$ is finite if, and only if, there is a $m>0$ such that $Ext_A^n(S,S)=0$, for all simple $A$-modules $S$ and all $n\geq m$.

Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension of $HH^1(TA)$ can be derived.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in \Bbb N} M_n$ of finitely generated indecomposable modules $M_n$ is a direct sum of finitely generated objects, th...

This paper is devoted to the study of the endo-structure of infinite direct sums $\bigoplus_{i \in I} M_i$ of indecomposable modules $M_i$ over a ring $R$. It is centered on the following question: If $S = \text{End}_R \bigl( \bigoplus_{i \in I} M_i \bigr)$, how much pressure, in terms of the $S$-structure of $\bigoplus_{i \in I} M_i$, is required...

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A⋊D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ⩽2. In this latter as well as in the hereditary...

The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, gi...

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H 1(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H 1(A) with the rank and p-rank of the fundamental group π1(Δ,I) of (Δ,I). We get explicit formulae for dim H 1(A), when every path...

Let R, S be rings and R MS a faithfully balanced bimodule. It is proved that the groups OutM (R) and OutM (S) of outer automorphisms of R and S which fix (up to isomorphism) the underlying modular structure of M are isomorphic. In case A, B are finite dimensional algebras and A MB is finite dimensional, it is proved that the isomorphism \({\rm Out}...

This is a paper written in 2000, a part of which was published under the title "Automorphism goups of trivial extensions" (J. Pure & Appl. Algebra 166, year 2002). The part on repetitive algebras (Sections 4 and 5) is unpublished.

We fix a finite dimensional vector space and a basis B of V and completely identify the identity component of the stabilizer, under diagonal action by GL(V), of every subspace of a direct sum of tensor or symmetric powers of V that is generated by powers of elements of B.

Given a full subcategory ℱ of a category A, the existence of left ℱ-approximations (or ℱ-preenvelopes) completing diagrams in a unique way is equivalent to the fact that ℱ is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity an...

The primary goal of this work is to develop a strategy to compute PicK(A) and OutK(A) where A is a finite-dimensional algebra over a field K. The basic idea is to put the normal subgroup Inn*(A) of the inner automorphisms of A induced by elements of 1 + J(A), as a common denominator in the fraction AutK(A) / Inn(A). The new numerator AutK(A)/Inn*(A...

A ring R is called left FGF (resp. CF) if every finitely generated (resp. cyclic) left R-module embeds in a free module. In this paper we give new partial affirmative answers to two open problems in the area of Module and Ring Theory, the so-called FGF problem: Is a left FGF ring Quasi-Frobenius? and the CF problem: Is a left CF ring left artinian?...

In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture w...

This paper is divided into four sections. In Section 1 we characterize the morphisms OE : M ! N which admit a decomposition OE = (OE

We characterize two-sided IF-rings such that each of its left modules has a flat envelope which is an essential monomorphism.

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose stru...

Without Abstract

In this paper we study the precise relation between two representations of a given split finite basic dimensional algebra A as a factor of the free path algebra over its quiver (A). After defining the notion of strongly acyclic quiver, we apply the results obtained to develop a method of calculating the group Aut(A)/Inn(A) in the case when (A) is s...

It is proved that if R is a perfect (resp. Artinian) strongly graded ring whose ground subring is, modulo its Jacobson radical, a finite direct product of finite-dimensional simple algebras over (nondenumerable) algebraically closed fields, then the grading group cannot contain an infinite abelian subgroup (resp. must be finite). These results exte...

It is proved that if R is a perfect (resp. Artinian) strongly graded ring whose ground subring is, modulo its Jacobson radical, a finite direct product of finite-dimensional simple algebras over (nondenumerable) algebraically closed fields, then the grading group cannot contain an infinite abelian subgroup (resp. must be finite). These results exte...

A duakity between two chategories C and D is an equivalence of categories between the dual category C^op of C and D. The idea of studying dualities between full subcategories of module categories stems ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=184120&tocid=100000260&page=1-18

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question. It is well-known that a ring for which every left module embeds in a free module is Quasi-Frobenius (QF). However, the...

In this paper we t.ackle the so-called FGF and CF problems, that are still open and, explicitly or implicit,ly, have been aboarded several times in the area of Module and Ring Theory. We give new partial affil.~~~at,ive answers t,o b0t.h problems.