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Juan Omar Gómez

Publications and source records attributed to Juan Omar Gómez.

14 recordsLinked to original sources
arXiv2026

A derived Chouinard's theorem for infinite groups

We establish a Chouinard-type theorem for modules over algebras of cochains on the classifying spaces of a broad class of infinite groups, allowing commutative ring spectra as coefficients. We then use this result to study finiteness properties of the integral cohomology rings of certain infinite groups, as well as stratification of the corresponding module categories. In particular, we complete the (co)stratification of module categories associated to homotopical groups over fields of positive characteristic.

math.AT↗
arXiv2026

The proper stable module category of a group algebra

We introduce the proper stable module category for an arbitrary discrete group over any commutative ring by means of cotorsion pairs and abelian model structures. This category is a well-generated tensor-triangulated category and is compactly generated in case the commutative ring is regular. Our construction resembles the topological approach via proper equivariant stable homotopy theory. Moreover, it agrees with the Mazza-Symonds stable module category, whenever the latter is defined, and with various other stable categories associated to hierarchically defined groups. Along the way, we introduce certain homological dimensions and study them in detail, comparing them with the classical notions.

math.RT↗
arXiv2026

Hopfological algebra, revisited

We propose an $\infty$-categorical approach to Khovanov--Qi's Hopfological algebra that, in particular, refines several foundational aspects of the theory by recasting the previous constructions in terms of $\infty$-categories of modules in monoidal $\infty$-categories. This perspective leads to a more general variant of Hopfological algebra that takes place over an arbitrary rigidly-compactly generated symmetric monoidal stable $\infty$-category, which we also outline in the article. In the appendix, we compare the construction of Hopfological derived categories to that of Holm--Jørgensen's $Q$-shaped derived categories.

math.RT↗
arXiv2026

The classification of integral endotrivial complexes

We describe the group of endotrivial complexes, i.e., the Picard group, of the derived category of permutation modules for a finite group over a commutative Noetherian ring. As a result, we deduce that not every endotrivial with integer coefficients arises from a homotopy representation, i.e., an invertible genuine equivariant spectrum. Along the way, we establish a descent result with respect to subgroups of prime-power order, show that oriented endotrivial complexes are line bundles, that is, locally trivial with respect to an open cover of the Balmer spectrum, and provide a topological construction of forerunner homomorphisms, answering a question of the second-named author.

math.RT↗
arXiv2026

Descent and finite permutation resolutions for discrete groups

Let $G$ be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let $k$ be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective $kG$-modules, the stable module category of $kG$-modules as defined by Mazza-Symonds, and the derived category of permutation $kG$-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any $kG$-module of type $FP_\infty$ is a retract of a module that admits a finite resolution by finitely generated $\natural$-permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer.

math.RT↗
arXiv2026

The importance of being isolated

We give both a sufficient condition for and an obstruction to the derived category of a commutative ring being generated by its residue fields. As an illustration, we exhibit a ring for which Foxby's small support classifies localizing subcategories despite the failure of the local-to-global principle. We also conclude that the residue fields do not generate for a polynomial ring in infinitely many variables. Finally, we give a necessary and sufficient criterion for the derived category to satisfy the local-to-global principle; it turns out to depend solely on the topology of the spectrum.

math.AC↗
arXiv2026

Fiberwise building and stratification in tensor triangular geometry

We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in \mathcal{T}$ is determined by those objects whose image under each $f_i$ lies in the localizing tensor ideal generated by $f_i(x)$ for all $i$. This leads to a fiberwise criterion for stratification in the setting of rigidly-compactly generated tt-categories. As an application, we prove that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is stratified. Moreover, our methods extend to the category of representations of a finite group scheme over a Noetherian base, thereby recovering a recent result from the literature.

math.RT↗
arXiv2026

Conservative geometric functors via purity

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra.

math.AT↗
arXiv2026

Geometric purity and the frame of smashing ideals

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.

math.RT↗
arXiv2025

The Balmer spectrum of integral permutation modules

We extend the analysis of Balmer and Gallauer on the tt-geometry of the small derived category of permutation modules for a finite group over a field to the setting of a commutative Noetherian base. In this general context, we provide a description of the tt-spectrum as a set and reduce the study of its topology to the elementary abelian case. Under certain mild additional assumptions on the ground ring, we further develop their theory of twisted cohomology, which enables us to realize the tt-spectrum as a Dirac scheme when restricted to elementary abelian $p$-groups.

math.RT↗
arXiv2024

On the Picard group of the stable module category for infinite groups

We introduce the stable module $\infty$-category for groups of type $Φ$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $Φ$ which act on a tree, we show that the stable module $\infty$-category decomposes in terms of the associated graph of groups. For groups which admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\infty$-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups.

math.RT↗
arXiv2023

On the commutator length of compact Lie groups

In this short note we show that the path-connected component of the identity of the derived subgroup of a compact Lie group consists just of commutators. We also discuss an application of our main result to the homotopy type of the classifying space for commutativity for a compact Lie group whose path-connected component of the identity is abelian.

math.GR↗
arXiv2022

On the cohomological triviality of the center of the Frattini subgroup

We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order 256. We also show that nonabelian metacyclic p-groups, nonabelian groups of maximal nilpotency class and 2-groups of coclass two satisfy the conjecture.

math.GR↗