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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…

In this paper we study the category of discrete G-spectra for a profinite group G. We consider an embedding of module objects in spectra into a category of module objects in discrete G-spectra, and study the relationship between the…

We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with…

We equip the type $A$ diagrammatic Hecke category with a special derivation, so that after specialization to characteristic $p$ it becomes a $p$-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the…

For a profinite group $G$, we define an $S[[G]]$-module to be a certain type of $G$-spectrum $X$ built from an inverse system $\{X_i\}_i$ of $G$-spectra, with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal subgroup…

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant…

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…

We investigate the group of normalized units of the group algebra $\mathbb{Z}_{p^e}G$ of a finite abelian $p$-group $G$ over the ring $\mathbb{Z}_{p^e}$ of residues modulo $p^e$ with $e\geq 1$.

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T:= Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let…

The noncommutative stable homotopy category $\mathtt{NSH}$ is a triangulated category that is the universal receptacle for triangulated homology theories on separable $C^*$-algebras. We show that the triangulated category $\mathtt{NSH}$ is…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set…

We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the…

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth…

For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is…

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\mathbb{F}$, let $W_k(\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\mathbb{F}$. We show that the group…

In this paper we discuss different properties of noncommutative schemes over a field. We define a noncommutative scheme as a differential graded category of a special type. We study regularity, smoothness and properness for noncommutative…

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to…