Soient $X$ une variété abélienne sur un corps de nombres $E$ et $G$ son groupe de Mumford–Tate. Soit $v$ une valuation de $E$ et pour tout nombre premier $\ell$ tel que $v\left(\ell \right)=0$, soit $F_{\ell }\in G({\mathbf{Q}}_{\ell })$ l’automorphisme de Frobenius (géométrique) de la cohomologie étale $\ell$-adique de $X$. On montre que si $X$ a une bonne réduction ordinaire en $v$, alors il existe $F\in G\left(\mathbf{Q}\right)$ tel que, pour tout $\ell$, $F_{\ell }$ soit conjugué à $F$ dans $G\left({\mathbf{Q}}_{\ell }\right)$. On montre un résultat analogue pour le frobenius de la cohomologie cristalline de la réduction de $X$ modulo $v$.

Let $A$ be a commutative ring with identity and $I$ an ideal of $A$. $A$ is said to be $I$-$clean$ if for every element $a\in A$ there is an idempotent $e=e^2\in A$ such that $a-e$ is a unit and $ae$ belongs to $I$. A filter of ideals, say $ℱ$, of $A$ is Noetherian if for each $I\in ℱ$ there is a finitely generated ideal $J\in ℱ$ such that $J\subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n\left(A\right)$, $J\left(A\right)$, and $A$, in terms of the frame of multiplicative Noetherian filters of ideals of $A$, as well as in terms of more classical ring properties.

Let $R$ be a commutative ring with nonzero identity and $J\left(R\right)$ the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by ${𝔍}_R$, is defined as the graph with vertex set $R\setminus J\left(R\right)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1-xy$ is not a unit of $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_n$. In this paper, we investigate the genus number of the compact Riemann surface in which ${𝔍}_R$ can be embedded and explicitly...

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.The first part begins with the definition and elementary properties of the $\overline{\nu }$ order function associated to an ideal $I$ of a reduced analytic...

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category ${𝒞}_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When ${𝒞}_A$ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module $H_{}^i(M,N)$ is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $To{r}_i^R(N,M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $To{r}_i^R(N,H_I^j\left(M\right))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $To{r}_i^R(N,M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...

We study series of the form ${\displaystyle \sum _M|{Aut}_R{\left(M\right)|}^{-1}{\left|M\right|}^{-u}}$, where $R$ is a commutative local ring, $u$ is a non-negative integer, and the summation extends over all finite $R$-modules $M$, up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If $R$ has additional properties, we will relate the above sum to a limit of zeta functions of the free modules $R^n$, where these zeta functions count $R$-submodules of finite index in $R^n$. In particular we will show that...