Related papers: Finiteness of minimal modular symbols for SL(n)

Let G be a linear algebraic group defined over a finite field F_q. We present several connections between the isogenies of G and the finite groups of rational points G(F_q^n). We show that an isogeny from G' to G over F_q gives rise to a…

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower…

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we prove that for an arbitrary finite group scheme G, and for any fixed integer n > 0, there are only finitely many…

By recent work on some conjectures of Pillay, each definably compact group $G$ in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' $G^{00}$ such that the quotient $G/G^{00}$, equipped with the…

Modular symbols for the congruence subgroup $\Gamma_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to…

Let $R$ be a commutative ring and $\Gamma$ be an infinite discrete group. The algebraic $K$-theory of the group ring $R[\Gamma]$ is an important object of computation in geometric topology and number theory. When the group ring is…

Let ${\rm GK}(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that…

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are…

Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local…

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…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a…

Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the…

We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient…

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation…

We define non-commutative schemes by using prime ideals of non-commutative rings, and discuss the \'etale cohomology, the Betti cohomology, and the fundamental groups of non-commutative schemes. For non-commutative schemes which are finite…

For an arbitrary infinite field k of characteristic p > 0, we describe the structure of a block of the algebraic monoid M_n(k) (all n x n matrices over k), or, equivalently, a block of the Schur algebra S(n,p), whose simple modules are…

The (co)homological dimension of homomorphism $\phi:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $\phi:G\to H$ of a…

We obtain the formula computing the number of isomorphic classes of element systems with characters over finite commutative group $G$.