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Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of…

Holm (H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543-560) considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his…

Let $R$ be a local or positively graded ring with a regular presentation $R \cong Q/I$ where $I$ is a monomial ideal generated by $n$ elements on a regular sequence. In Briggs-Grifo-Pollitz (2025), the authors classify the cohomological…

Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^{\phi}\!M$ be the abelian group $M$ with…

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the…

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of…

The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When…

Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be…

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth…

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coeffiecient ring $R$), which is assumed to carry an involution of the form $X^*=Y$, $R^*\subseteq R$. We prove…

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…

Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the…

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category,…

Let X be a smooth real algebraic variety. Let $\xi$ be a distribution on it. One can define the singular support of $\xi$ to be the singular support of the $D_X$-module generated by $\xi$ (some times it is also called the characteristic…

We consider a finite dimensional strongly $G$-graded algebra $A$ with { self-injective} $1$-component $B$, and in our main result we prove that the induction from $B$ to $A$ of a basic support $\tau$-tilting pair of $B$-modules is a support…

Graded Hecke algebras can be constructed in terms of equivariant cohomology and constructible sheaves on nilpotent cones. In earlier work, their standard modules and their irreducible modules where realized with such geometric methods. We…

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let…

We characterize the indecomposable transjective modules over an arbitrary cluster-tilted algebra that do not lie on a local slice, and we provide a sharp upper bound for the number of (isoclasses of) these modules.

The aim of this note is a combinatorial description of a category of $D$-modules over an affine space, smooth along the stratification defined by an arrangement of hyperplanes. These $D$-modules are assumed to satisfy certain non-resonance…

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base…