Related papers: Krull-Schmidt decompositions for thick subcategori…
In the present article we investigate properties of the category of the integral Grothendieck-Chow motives over a field. We discuss the Krull-Schmidt principle for integral motives, provide a complete list of the generalized Severi-Brauer…
We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Within the stratified context, we investigate connections between big prime…
We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories…
Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct…
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…
Goodwillie's rational isomorphism between relative algebraic K-theory and relative cyclic homology, together with the lambda decomposition of cyclic homology, illustrates the close relationships among algebraic K-theory, cyclic homology,…
We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by…
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…
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem,…
We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-M\'ezard…
In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool…
We introduce a new notion of regularity for rings and exact categories and we show important results in algebraic K-theory. In particular we prove a strong vanishing theorem for Nil groups and give an explicit class of groups, much bigger…
The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the "Lichtenbaum-Quillen dimension" of a variety in terms…
Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor…
Considering the monoidal category $\mathcal{C}$ obtained as modules over a Hopf algebra $H$ in a rigid braided category $\mathcal{B}$, we prove decomposition results for the Hochschild and cyclic homology categories $HH(\mathcal{C})$ and…
We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can…
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of…
For a finite group $G$ and an arbitrary commutative ring $R$, Brou\'e has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the…
We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative…