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Dolors Herbera

Publications and source records attributed to Dolors Herbera.

At least 19 recordsLinked to original sources
arXiv2026

Trace ideals and uniserial modules

We thoroughly investigate the trace ideals of projective modules over the endomorphism ring of a uniserial module. After the work of Dubrovin and Puninski, it is known that this class of rings provides examples of trace ideals of projective right modules that are not trace ideals of projective left modules. In this paper we further investigate when this happens, giving an intrinsic description of such trace ideals and their properties. We also use the theory associated to lifting projective modules modulo a trace ideal to give an alternative approach to Puninski's construction of a direct summand of a serial module that is not serial.

math.RA↗
arXiv2026

Semi-Bousfield classes and nonmonotone perversities

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).

math.CT↗
arXiv2025

A family of examples of generalized perfect rings

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A.~Amini, B.~Amini, M.~Ershad and H.~Sharif (2007).

math.RA↗
arXiv2025

Relatively big projective modules and their applications to direct sum decompositions

Countably generated projective modules that are relatively big with respect to a trace ideal were introduced by P. Příhoda, as an extension of Bass' uniformly big projectives. It has already been proved that there are a number of interesting examples of rings whose countably generated projective modules are always relatively big. In this paper, we increase the list of such examples, showing that it includes all right noetherian rings satisfying a polynomial identity. We also show that countably generated projective modules over locally semiperfect torsion-free algebras over $h$-local domains are always relatively big. This last result applies to endomorphism rings of finitely generated torsion-free modules over $h$-local domains. As a consequence, we can give a complete characterization of those $h$-local domains of Krull dimension $1$ for which every direct summand of a direct sum of copies of a single finitely generated torsion-free module is again a direct sum of finitely generated modules.

math.AC↗
arXiv2025

A poisonous example to explicit resolutions of unbounded complexes

We show that various methods for explicitly building resolutions of unbounded complexes in fact fail when applied to a rather simple and explicit complex. We show that one way to rescue these methods is to assume Roos (Ab.4$^*$)-$k$ axiom, which we adapt to encompass also resolutions in the framework of relative homological algebra. In the end we discuss the existence of model structures for relative homological algebra for unbounded complex under the relative (Ab.4$^*$)-$k$ condition, and present a variety of examples where our results apply.

math.RA↗
arXiv2025

Torsion-free Modules over Commutative Domains of Krull Dimension One

Let $R$ be a domain of Krull dimension one, we study when the class $\mathcal{F}$ of modules over $R$ that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If $R$ is local, we show that $\mathcal{F}$ is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, $R$ is noetherian this is equivalent to say that the normalization of $R$ is a local ring. If $R$ is an $h$-local domain of Krull dimension $1$ and $\mathcal{F}_R$ is closed under direct summands, then the property is inherited by the localizations of $R$ at maximal ideals. Moreover, any localizations of $R$ at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is $2$-generated. The converse is true when the domain $R$ is, in addition, integrally closed, or noetherian semilocal or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countable generated modules in $\mathcal{F}$ are determined by their genus.

math.AC↗
arXiv2024

The Mittag-Leffler condition descents via pure monomorphisms

This notes aims to clarify the proof given by Raynaud and Gruson that the Mittag-Leffler property descents via pure rings monomorphism of commutative rings. A consequence of that is that projectivity dencents via such ring homomorphisms, a revision of the proof also allows to prove that the property of being pure-projective also descents via pure monomorphisms between commutative rings}.

math.AC↗
arXiv2024

Cotorsion pairs and Tor-pairs over commutative noetherian rings

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.

math.AC↗
arXiv2023

Big pure projective modules over commutative noetherian rings: comparison with the completion

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module $M$, we consider $\mathrm{Add}\, (M)$, which consists of direct summands of direct sums of copies of $M$. We are primarily interested in the case where $R$ is a one-dimensional, local domain, and in torsion-free (or Cohen-Macaulay) modules. We show that, even in this case, $\mathrm{Add}\, (M)$ can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of $M$ and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behavior of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid $V^*(M)$, of isomorphism classes of countably generated modules in $\mathrm{Add}\, (M)$ with the addition induced by the direct sum. We show that $V^*(M)$ is a submonoid of $V^*(M\otimes _R \widehat R)$, this allows us to make computations with examples and to prove some realization results.

math.AC↗
arXiv2013

Cotilting modules over commutative noetherian rings

Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.

math.AC↗
arXiv2013

Reconstructing projective modules from its trace ideal

We make a detailed study of idempotent ideals that are traces of countably generated projective right modules. We associate to such ideals an ascending chain of finitely generated left ideals and, dually, a descending chain of cofinitely generated right ideals. The study of the first sequence allows us to characterize trace ideals of projective modules and to show that projective modules can always be lifted modulo the trace ideal of a projective module. As a consequence we give some new classification results of (countably generated) projective modules over particular classes of semilocal rings. The study of the second sequence leads us to consider projective modules over noetherian FCR-algebras; we make some constructions of non-trivial projective modules showing that over such rings the behavior of countably generated projective modules that are not direct sum of finitely generated ones is, in general, quite complex.

math.RA↗
arXiv2013

The Inversion Height of the Free Field is Infinite

Let X be a finite set with at least two elements, and let k be any commutative field. We prove that the inversion height of the embedding k ---> D, where D denotes the universal (skew) field of fractions of the free algebra k , is infinite. Therefore, if H denotes the free group on X, the inversion height of the embedding of the group algebra k[H] into the Malcev-Neumann series ring is also infinite. This answer in the affirmative a question posed by Neumann in 1949 [27, p. 215]. We also give an infinite family of examples of non-isomorphic fields of fractions of k with infinite inversion height. We show that the universal field of fractions of a crossed product of a commutative field by the universal enveloping algebra of a free Lie algebra is a field of fractions constructed by Cohn (and later by Lichtman). This extends a result by A. Lichtman.

math.RA↗
arXiv2011

Infinitely generated projective modules over pullbacks of rings

We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup\{\infty\})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters \cite{FS}. We also provide a rich variety of examples of semilocal rings having non finitely generated projective modules that are finitely generated modulo the Jacobson radical.

math.RA↗
arXiv2009

Almost free modules and Mittag--Leffler conditions

Drinfeld recently suggested to replace projective modules by the flat Mittag--Leffler ones in the definition of an infinite dimensional vector bundle on a scheme $X$. Two questions arise: (1) What is the structure of the class $\mathcal D$ of all flat Mittag--Leffler modules over a general ring? (2) Can flat Mittag--Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi--coherent sheaves on $X$? We answer (1) by showing that a module $M$ is flat Mittag--Leffler, if and only if $M$ is $\aleph_1$--projective in the sense of Eklof and Mekler. We use this to characterize the rings such that $\mathcal D$ is closed under products, and relate the classes of all Mittag--Leffler, strict Mittag--Leffler, and separable modules. Then we prove that the class $\mathcal D$ is not deconstructible for any non--right perfect ring. So unlike the classes of all projective and flat modules, the class $\mathcal D$ does not admit the homotopy theory tools developed recently by Hovey . This gives a negative answer to (2).

math.RA↗
arXiv2009

Big projective modules over noetherian semilocal rings

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as $V^*(R)$ for a noetherian semilocal ring such that $R/J(R)\cong M_{n_1}(D_1)\times ... \times M_{n_k}(D_k)$ for suitable division rings $D_1,..., D_k$.

math.RA↗
arXiv2007

Cotorsion pairs generated by modules of bounded projective dimension

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.

math.RA↗
arXiv2007

Mittag-Leffler conditions on modules

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.

math.RA↗