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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., Comment: 7 pages

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 ${\mathbb Z}/3^r\times\Sigma_3$ ($r> 0$) 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 $G$., Comment: 11 pages

We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex., Comment: 5 pages

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 $\alpha:k[t]/(t^p) \to kG$ 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., Comment: 20 pages

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., Comment: 21 pages

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., Comment: 15 pages, The Version of Record of this article is published in Beitr\"age zur Algebra und Geometrie and is available on line at https://doi.org/10.1007/s13366-021-00595-y

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.

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., Comment: 7 pages

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.

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., Comment: 36 pages

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., Comment: 19 pages, minor modifications following refereeing

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., Comment: 14 pages

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., Comment: Final version, 11 pages, to appear in Canadian Mathematical Bulletin

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.

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.

We initiate the investigation of the projective varieties $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These varieties $\mathbb E(r,\mathfrak g)$ are the natural ambient varieties for generalized support varieties for restricted representations of $\mathfrak g$. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional $\mathfrak u(\mathfrak g)$-module $M$, the local $(r,j)$-radical rank and local $(r,j)$-socle rank, functions which are lower/upper semicontinuous on $\mathbb E(r,\mathfrak g)$. Examples are given of $\mathfrak u(\mathfrak g)$-modules for which some of these rank functions are constant., Comment: Replaces 1st half of arXiv:1207.5898 (with same title). To appear in the Journal of Algebra

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak g$. These are sheaves on locally closed subvarieties of the projective variety $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of $\mathfrak g$ of dimension $r$. We show that representations of constant radical or socle rank studied in \cite{CFP3} which generalize modules of constant Jordan type lead to algebraic vector bundles on $\mathbb E(r,\mathfrak g)$. For $\mathfrak g = Lie(G)$, the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $\mathbb E(r, \mathfrak g)$., Comment: Replaces second half of arXiv:1207.5898 . To appear in Crelle

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.

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.

The theory of support varieties gives a rich supply of examples of thick subcategories of the stable module category of a finite group algebra. We study direct sum decompositions of such categories. We give examples where there are finer decompositions than one might originally expect, and relate this to Linckelmann's theory of block varieties.

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.

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 $E(r, g)$., Comment: The paper has been split into two at the request of the referee, the first part has the same title as the older combined version

A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results., Comment: 18 pages

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the simple rational $G$-module of highest weight $\lambda$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $\lambda$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(\lambda))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(\lambda))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type., Comment: 29 pages, GAP code included as an ancillary file. Rewritten to include the adjoint representation in types An, B2, and Cn. Corrections made to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional minor corrections and improvements also implemented

We initiate the study of representations of elementary abelian $p$-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by $r$ nilpotent elements, $k[t_1,..., t_r]/(t^p_1,..., t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we prove that for an arbitrary finite group scheme G, and for any fixed integer n > 0, there are only finitely many isomorphism classes of endotrivial modules of dimension n. This provides evidence to support the speculation that the group of endotrivial modules for a finite group scheme is always finitely generated. The result also has some applications to questions about lifting and twisting the structure of endotrivial modules in the case that G is an infinitesimal group scheme associated to an algebraic group.

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight., Comment: 24 pages, 5 figures, 6 tables. Typos corrected and some proofs streamlined over previous version

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we investigate endotrivial modules over arbitrary finite group schemes. Our results can be applied to computing the endotrivial group for several classes of infinitesimal group schemes which include the Frobenius kernels of parabolic subgroups, and their unipotent radicals(for reductive algebraic groups). For G reductive, we also present a classification of simple, induced/Weyl and tilting modules (G-modules) which are endotrivial over the Frobenius kernel G_r of G.