Related papers: FI-modules and stability for representations of sy…

This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a…

In this paper, we first introduce stable functors with respect to a preenveloping/precovering subcategory and investigate some of their properties. Using that we then introduce and study a relative complete cohomology theory in abelian…

Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and…

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with…

We give explicit, uniform formulas for the graded characters and total ranks of the Lie algebra homology of finite-dimensional representations in all classical types. In many cases, these compute the Tor groups of finite length modules over…

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable…

The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of…

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite…

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a variety $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices…

Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous…

F-bundle is a formal/non-archimedean version of variation of nc-Hodge structures which plays a crucial role in the theory of atoms as birational invariants from Gromov-Witten theory. In this paper, we establish the spectral decomposition…

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant…

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an…

The cohomology of the configuration space of n points in R^3 admits a symmetric group action and has been shown to be isomorphic to the regular representation. One way to prove this is by defining an S^1-action whose fixed point set is the…

We compute the cohomology groups of the automorphism group of the free group $F_n$, with coefficients in arbitrary tensor products of the standard representation $H_1(F_n, \mathbb{Q})$ and its dual, in a range where $n$ is sufficiently…

Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, $M$ a finite $R$--module and $X$ an arbitrary $R$--module. Here, we show that, in the Serre subcategories of the category of $R$--modules, how the…

The IA-automorphism group is the group of automorphisms of the free group $F_n$ that act trivially on the abelianization $F_n^{\mathrm{ab}}$. This group is in many ways analoguous to Torelli groups of surfaces and their higher dimensional…

A Hom-group G is a nonassociative version of a group where associativity, invertibility, and unitality are twisted by a map \alpha: G\longrightarrow G. Introducing the Hom-group algebra KG, we observe that Hom-groups are providing examples…

Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is…

Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the…