arXiv subjects

Alina Iacob

Publications and source records attributed to Alina Iacob.

At least 19 recordsLinked to original sources

arXiv2026

Balanced pairs of Cartan-Eilenberg complexes over virtually Gorenstein rings

Let R be a ring and let Ch(R) be the category of complexes of left R-modules. We consider the C-E (Cartan-Eilenberg) exact structure on Ch(R) and show that it is an efficient exact category. We prove that if R is a left virtually Gorenstein ring, the pair (C-E(GProj),C-E(GInj)) is a C-E-admissible balanced pair. Under suitable additional hypotheses, the converse holds. We also consider the C-E version of Tate cohomology and establish C-E analogues of the Avramov-Martsinkovsky exact sequences and balance results for both Ext and Tor.

math.RA↗

arXiv2026

Weakly Ding injective complexes

Working over a (left) coherent ring, we consider the class of weakly Ding injective complexes. These are the cycles of the exact complexes of FP-injective complexes that stay exact when applying $\Hom(A,-)$ for any FP-injective complex $A$. We study the cotorsion pair generated by the class of all such complexes, and exhibit it as part of an abelian model structure. As an application we show that when $R$ is a Ding-Chen ring, its stable chain complex category is compactly generated and triangle equivalent to the stable category of four Frobenius categories. They are the categories of all (i) complexes of Gorenstein injective modules, (ii) complexes of Gorenstein projective modules, (iii) complexes of Gorenstein flat-cotorsion modules, and (iv) complexes of Gorenstein FP-pro-injective modules.

math.AC↗

arXiv2026

Gorenstein flat preenvelopes and weakly Ding injective covers

We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping

math.KT↗

arXiv2025

Homological dimensions of complexes over coherent regular rings

We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of R-modules agrees with its graded-injective (resp. graded-projective) dimension. The same is shown for the analogous dimensions based on FP-injective R-modules, and on flat R-modules.

math.AC↗

arXiv2024

The class of Gorenstein injective modules is covering if and only if it is closed under direct limits

We prove that the class of Gorenstein injective modules, $\mathcal{GI}$, is special precovering if and only if it is covering if and only if it is closed under direct limits. This adds to the list of examples that support Enochs' conjecture:\\ "Every covering class of modules is closed under direct limits".\\ We also give a characterization of the rings for which $\mathcal{GI}$ is covering: the class of Gorenstein injective left $R$-modules is covering if and only if $R$ is left noetherian, and such that character modules of Gorenstein injective left $R$ modules are Gorenstein flat.

math.AC↗

arXiv2023

Direct limits of Gorenstein injective modules

One of the open problems in Gorenstein homological algebra is: when is the class of Gorenstein injective modules closed under arbitrary direct limits? It is known that if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under direct limits, then the ring is noetherian. The open problem is whether or not the converse holds. We give equivalent characterizations of $\mathcal{GI}$ being closed under direct limits. More precisely, we show that the following statements are equivalent:\\ (1) The class of Gorenstein injective left $R$-modules is closed under direct limits.\\ (2) The ring $R$ is left noetherian and the character module of every Gorenstein injective left $R$-module is Gorenstein flat.\\ (3) The class of Gorenstein injective modules is covering and it is closed under pure quotients.\\ (4) $\mathcal{GI}$ is closed under pure submodules.

math.AC↗

arXiv2023

Gorenstein projective precovers and finitely presented modules

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a "reduction" property - we show that, over any ring, it suffices to consider finitely presented modules: if there exists a nonnegative integer $n$ such that every finitely presented module has Gorenstein projective dimension $\le n$, then the class of Gorenstein projective modules is special precovering.

math.AC↗

arXiv2022

Model structures and relative Gorenstein flat modules and chain complexes

A recent result by J. Šaroch and J. Šťov\'ıček asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by Šaroch and Šťov\'ıček, plus some other arguments coming from model theory, we determine some conditions for $\mathcal{B}$ so that the class of Gorenstein $\mathcal{B}$-modules is closed under extensions. This will allow us to show approximation properties concerning these modules, and also to obtain a relative version of the model structure described before. Moreover, we also present and prove our results in the category of complexes of left $R$-modules, study other model structures on complexes constructed from relative Gorenstein flat modules, and compare these models via computing their homotopy categories.

math.CT↗

arXiv2021

Ding injective envelopes in the category of complexes

A complex $X$ is called Ding injective if there exists an exact sequence of injective complexes $\ldots \rightarrow E_1 \rightarrow E_0 \rightarrow E_{-1} \rightarrow \ldots$ such that $X = Ker(E_0 \rightarrow E_{-1})$, and the sequence remains exact when the functor $Hom(A,-)$ is applied to it, for any $FP$-injective complex $A$. We prove that, over any ring $R$, a complex is Ding injective if and only if it is a complex of Ding injective modules. We use this to show that the class of Ding injective complexes is enveloping over any ring.

math.AC↗

arXiv2021

Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jorgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair $\mathfrak{D} = (\mathcal{L},\mathcal{A})$. This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo-Gillespie-Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfect) cotorsion pair, over any ring. Completeness of the Gorenstein flat cotorsion pair over any ring arises from the same duality pair.

math.KT↗

arXiv2020

Projectively coresolved Gorenstein flat and Ding projective modules

We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, $\mathcal{PGF}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, $\mathcal{PGF}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{DP})$. We show that $\mathcal{PGF} = \mathcal{DP}$ if and only if every Ding projective module is Gorenstein flat. This is the case if the ring $R$ is coherent for example. We include an example to show that the coherence is a sufficient, but not a necessary condition in order to have $\mathcal{PGF} = \mathcal{DP}$. We also show that $\mathcal{PGF} = \mathcal{DP}$ over any ring $R$ of finite weak Gorenstein global dimension (this condition is also sufficient, but not necessary). We prove that if the class of Ding projective modules, $\mathcal{DP}$, is covering then the ring $R$ is perfect. And we show that, over a coherent ring $R$, the converse also holds. We also give necessary and sufficient conditions in order to have $\mathcal{PGF} = \mathcal{GP}$, where $\mathcal{GP}$ is the class of Gorenstein projective modules.

math.RA↗

arXiv2020

Acyclic complexes and Gorenstein rings

For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $\mathcal{A}$. We use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $\mathcal{DI}$ for Ding injective modules, and $\mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are equivalent over any ring $R$: (1) Every exact complex of injective modules is totally acyclic. (2) Every exact complex of Gorenstein injective modules is in $\widetilde{\mathcal{GI}}$. (3) Every complex in $dw(\mathcal{GI})$ is dg-Gorenstein injective. We show that the analogue result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. if moreover, the ring is $n$-perfect for some integer $n \ge 0$, then the three equivalent statements for flat and Gorenstein flat modules are also equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings: Let $R$ be a commutative coherent ring. The following statements are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules. (2) every exact complex of injectives has all its cycles Ding injective modules and every $R$-module M such that $M^+$ is Gorenstein flat is Ding injective. If moreover the ring $R$ has finite Krull dimension then statements (1), (2) above are also equivalent to (3) $R$ is a Gorenstein ring (in the sense of Iwanaga).

math.RA↗

arXiv2018

Tate-Betti and Tate-Bass numbers

We define Tate-Betti and Tate-Bass invariants for modules over a commutative noetherian local ring R. Then we show the periodicity of these invariants provided that R is a hypersurface. In case R is also Gorenstein, we show that a finitely generated R-module M and its Matlis dual have the same Tate-Betti and Tate-Bass numbers.

math.KT↗

arXiv2017

$\rm{FP}_{n}$-injective and $\rm{FP}_{n}$-flat covers and preenvelopes, and Gorenstein AC-flat covers

We prove that, for any $n \geq 2$, the classes of $\rm{FP}_{n}$-injective modules and of $\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\rm{FP}_{\infty}$-injective and $\rm{FP}_{\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example we show that this class is precovering over any ring $R$. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When $R$ is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. We also prove that if the class of Gorenstein AC-flat modules is closed under extensions, then this class of modules is covering.

math.AC↗

arXiv2016

Totally acyclic complexes

For a given class of modules $\A$, we denote by $\widetilde{\A}$ the class of exact complexes $X$ having all cycles in $\A$, and by $dw(\A)$ the class of complexes $Y$ with all components $Y_j$ in $\A$. We consider a two sided noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (flat, projective respectively) $R$-modules. We prove (Theorem 1) that the following are equivalent: 1. Every exact complex of injective modules is totally acyclic. 2. Every exact complex of Gorenstein injective modules is in $\widetilde{\mathcal{GI}}$. 3. Every complex in $dw(\mathcal{GI})$ is dg-Gorenstein injective. Theorem 2 shows that the analogue result for complexes of flat and Gorenstein flat modules also holds. We prove (Corollary 1) that, over a commutative noetherian ring $R$, the equivalent statements in Theorem 1 (as well as their counterparts from Theorem 2) hold if and only if the ring is Gorenstein. Thus we improve on a result of Iyengar's and Krause's; in [18] they proved that for a commutative noetherian ring $R$ with a dualizing complex, the class of exact complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if $R$ is Gorenstein. We are able to remove the dualizing complex hypothesis. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 6) that for such a ring $R$ that also has finite finitistic flat dimension, every complex of injective (left and respectively right) $R$-modules is totally acyclic if and only if $R$ is a Gorenstein ring.

math.AC↗

arXiv2016

A Zariski-local notion of F-total acyclicity for complexes of sheaves

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local - i.e. it can be verified on any open affine covering of the scheme - and it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent theorems on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint to the inclusion of these totally acyclic complexes into the homotopy category of complexes of flat sheaves.

math.AC↗

arXiv2016

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.

math.AC↗

arXiv2016

Gorenstein projective precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In section 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.

math.AC↗