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Jon F. Carlson

Publications and source records attributed to Jon F. Carlson.

At least 19 recordsLinked to original sources
arXiv2024

Locally dualizable modules abound

It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the derived category of $A$ than those that are obtained as retracts of images of perfect $A$-complexes. An analogous result is established dealing with the stable module category of the group algebra, over a countable field of positive characteristic $p$, of an elementary abelian $p$-group of rank at least 3.

math.AC↗
arXiv2023

Modules with finitely generated cohomology

Let $G$ be a finite group and $\mathsf{k}$ a field of characteristic $p$. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod$(\mathsf{k}G)$ consisting of modules whose cohomology is finitely generated over $\mathsf{H}^*(G,\mathsf{k})$ is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order $p$ in $G$ is $p$-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non $p$-nilpotent centralisers. The groups involved are ${\mathbb Z}/3^r\timesΣ_3$ ($r> 0$) in characteristic three and ${\mathbb Z}/2\times A_4$ in characteristic two. As a consequence, in these cases the bounded derived category of $C^*BG$ (cochains on $BG$ with coefficients in $\mathsf{k}$) is generated by $C^*BS$, where $S$ is a Sylow $p$-subgroup of $G$.

math.RT↗
arXiv2022

Idempotent modules, locus of compactness and local supports

Let $kG$ be the group algebra of a finite group scheme defined over a field $k$ of characteristic $p>0$. Associated to any closed subset $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$ is a thick tensor ideal subcategory of the stable category of finitely generated $kG$-module, whose closure under arbitrary direct sums is a localizing tensor ideal in the stable category of all $kG$-modules. The colocalizing functor from the big stable category to this localizing subcategory is given by tensoring with an idempotent module $\mathcal{E}$. A property of the idempotent module is that its restriction along any flat map $α:k[t]/(t^p) \to kG$ is a compact object. For any $kG$-module $M$, we define its locus of compactness in terms of such restrictions. With some added hypothesis, in the case that $V$ is a closed point, for a $kG$-module $M$, we show that in the stable category $\operatorname{Hom}(\mathcal{E}, M)$ is finitely generated over the endomorphism ring of $\mathcal{E}$, provided the restriction along an associated flat map is a compact object. This leads to a notion of local supports. We prove some of its properties and give a realization theorem.

math.RT↗
arXiv2022

The endomorphism ring of the trivial module in a localized category

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$. Let $\mathcal{C}$ denote the Verdier localization of the stable module category $\operatorname{stmod}(kG)$ at $\mathcal{M}$. We show that if $V$ is a finite collection of closed points and if the $p$-rank every maximal elementary abelian $p$-subgroups of $G$ is at least 3, then the endomorphism ring of the trivial module in $\mathcal{C}$ is a local ring whose unique maximal ideal is infinitely generated and nilpotent. In addition, we show an example where the endomorphism ring in $\mathcal{C}$ of a compact object is not finitely presented as a module over the endomorphism ring of the trivial module.

math.RT↗
arXiv2022

Nilpotence and Duality in the Complete Cohomology of a Module

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. We show that the ring has two distinguished ideals $I^* \subseteq J^* \subseteq \mathcal{E}_M^*$ such that $I^*$ is bounded above in degrees, $\mathcal{E}_M^*/J^*$ is bounded below in degree and $J^*/I^*$ is eventually periodic with terms of bounded dimension. We prove that if $M$ is neither projective nor periodic, then the subring of all elements in negative degrees in $\mathcal{E}_M^*$ is a nilpotent algebra.

math.RT↗
arXiv2021

Torsion Free Endotrivial Modules for Finite Groups of Lie Type

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\ell$-subgroups in any finite group of Lie type, for any prime $\ell$, which may be of independent interest.

math.GR↗
arXiv2020

Negative cohomology and the endomorphism ring of the trivial module

Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or group scheme. We show that the negative Tate cohomology ring $\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of $kG$-modules for $G$ an extension of $H$. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero component of a localization of the cohomology ring of the group.

math.RT↗
arXiv2020

Bounded complexes of permutation modules

Let $k$ be a field of characteristic $p > 0$. For $G$ an elementary abelian $p$-group, there exist collections of permutation module such that if $C^*$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^*$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^*$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^*$ is contractible.

math.GR↗
arXiv2020

Relatively Projectivity and the Green correspondence for complexes

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for a subgroup $H$, often the normalizer of a $p$-subgroup of $G$. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.

math.RT↗
arXiv2017

Separable commutative rings in the stable module category of cyclic groups

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group.

math.RT↗
arXiv2016

Hopf algebra structures and tensor products for group algebras

The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the cohomology ring generated by the Bocksteins of the degree one elements are independent of the choice of these coalgebra structures.

math.RT↗
arXiv2016

Ghosts and Strong Ghosts in the Stable Module Category

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study some variations of ghosts maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghost can be detected on a finite range of Tate cohomology. We also consider maps which mimic ghosts in high degrees.

math.RT↗
arXiv2015

The Graded Center of a Triangulated Category

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor We show that such natural transformations which have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

math.RT↗
arXiv2015

Endotrivial Modules for Finite Groups of Lie Type A in Nondefining Characteristic

Let $G$ be a finite group such that $\text{SL}(n,q)\subseteq G \subseteq \text{GL}(n,q)$ and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial $k(G/Z)$-modules where $k$ is an algebraically closed field of characteristic $p$ such that $p$ does not divide $q$. The results in this paper complete the classification of endotrivial modules for all finite groups of Lie Type $A$, initiated earlier by the authors.

math.GR↗
arXiv2015

The torsion group of endotrivial modules

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in the case when S is abelian. This provides a classification of all torsion endotrivial kG-modules in that case.

math.GR↗
arXiv2014

Elementary subalgebras of Lie algebras

We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the representations of $g$. For various choices of $g$ and $r$, we identify the geometric structure of $E(r,g)$. We show that special classes of (restricted) representations of $g$ lead to algebraic vector bundles on $E(r,g)$. For $g = Lie(G)$ the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $E(r, g)$.

math.RA↗
arXiv2014

On the structure of cohomology rings of p-nilpotent Lie algebras

In this paper the authors investigate the structure the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose $E_{2}$-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen-Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as ring with the $E_{2}$-term. We present criteria for the collapsing of this spectral sequence and provide many examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.

math.GR↗