Related papers: Remarks on the ABG Induction Theorem

We give a geometric realization of cohomologically induced (g,K)-modules. Let (h,L) be a subpair of (g,K). The cohomological induction is an algebraic construction of (g,K)-modules from a (h,L)-module V. For a real semisimple Lie group, the…

Let $G$ be a reductive algebraic group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p$. There are two associated families of finite group schemes, the $r$-th Frobenius kernels, denoted by…

For a positively graded artin algebra $A=\oplus_{n\geq 0}A_n$ we introduce its Beilinson algebra $\mathrm{b}(A)$. We prove that if $A$ is well-graded self-injective, then the category of graded $A$-modules is equivalent to the category of…

For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the category of modules over…

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of…

By results of Rognerud, a source algebra equivalence between two $p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence…

We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider…

We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation…

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of…

Several authors have studied the filtered colimit closure lim(B) of a class B of finitely presented modules. Lenzing called lim(B) the category of modules with support in B, and proved that it is equivalent to the category of flat objects…

We prove that the parabolic induction functor on BGG-category $\mathcal{O}$ associated to a complex reductive Lie algebra is gradable, that is, lifts to graded category $\mathcal{O}$ as constructed by Beilinson-Ginzburg-Soergel. Graded…

Given a ring homomorphism $B \to A$, consider its centralizer $R = A^B$, bimodule endomorphism ring $S = \End {}_BA_B$ and sub-tensor-square ring $T = (A \o_B A)^B$. Nonassociative tensoring by the cyclic modules $R_T$ or ${}_SR$ leads to…

We introduce the notion of the $\infty$-category of (complete) derived $G$-graded modules over a $G$-graded ring $R$ for a torsion-free abelian group $G$, and we study its foundational properties. Moreover, we prove a categorical…

We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine…

We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded…

Let $\mathcal{U}$ be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize $\mathcal{U}$ under the assumption that there exists a commutative algebra $A$ in $\mathcal{U}$ with certain…

Auslander's formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to…

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…

Given a finite ribbon category, which is a particular case of a cyclic algebra over the operad of genus zero surfaces, there are two possibilities for an extension defined on all three-dimensional handlebodies: On the one hand, one can use…

Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let $V$ be an irreducible ${\mathbb F}H$-module and $U$ a quotient of the induced ${\mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$, which is…