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Paul Balmer

Publications and source records attributed to Paul Balmer.

At least 37 records · Page 2Linked to original sources
arXiv2020

Resolutions by permutation modules

We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

math.RT↗
arXiv2018

Tensor-triangular fields: Ruminations

We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.

math.CT↗
arXiv2018

The frame of smashing tensor-ideals

Given a tensor-triangulated category $T$, we prove that every flat tensor-idempotent in the module category over $T^c$ (the compacts) comes from a unique smashing ideal in $T$. We deduce that the lattice of smashing ideals forms a frame.

math.CT↗
arXiv2018

On the surjectivity of the map of spectra associated to a tensor-triangulated functor

We prove a few results about the map $Spc(F)$ induced on tensor-triangular spectra by a tensor-triangulated functor $F$. First, $F$ is conservative if and only if $Spc(F)$ is surjective on closed points. Second, if $F$ detects tensor-nilpotence of morphisms then $Spc(F)$ is surjective on the whole spectrum. In fact, surjectivity of $Spc(F)$ is equivalent to $F$ detecting the nilpotence of some class of morphisms, namely those morphisms which are nilpotent on their cone.

math.CT↗
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

The spectrum of the equivariant stable homotopy category of a finite group

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

math.AT↗
arXiv2016

Separable extensions in tensor-triangular geometry and generalized Quillen stratification

We study the continuous map induced on spectra by a separable extension of tensor-triangulated categories. We determine the image of this map and relate the cardinality of its fibers to the degree of the extension. We then prove a weak form of descent, "up-to-nilpotence", which allows us to generalize Quillen's Stratification Theorem to equivariant derived categories.

math.CT↗
arXiv2015

Grothendieck-Neeman duality and the Wirthmüller isomorphism

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin-Matlis duality à la Dwyer-Greenlees-Iyengar in the theory of ring spectra, and of Brown-Comenetz duality à la Neeman in stable homotopy theory.

math.CT↗
arXiv2013

Stacks of group representations

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup $H$ can be extended to $G$. We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite $G$-sets (or the orbit category of $G$), with respect to a suitable Grothendieck topology that we call the sipp topology. When $H$ contains a Sylow subgroup of $G$, we use sipp Cech cohomology to describe the kernel and the image of the homomorphism $T(G)\to T(H)$, where $T(-)$ denotes the group of endotrivial representations.

math.RT↗
arXiv2012

Witt groups of Grassmann varieties

We compute the total Witt groups of (split) Grassmann varieties, over any regular base X. The answer is a free module over the total Witt ring of X. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams.

math.AG↗