Self-injective algebras and the second Hochschild cohomology group

In this paper we study the second Hochschild cohomology group ${\mathrm{HH}}^2(\Lambda )$ of a finite dimensional algebra Λ. In particular, we determine ${\mathrm{HH}}^2(\Lambda )$ where Λ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Λ; we give a basis for ${\mathrm{HH}}^2(\Lambda )$ in the few cases where it is not zero.     

On a Hölder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on ${?}^p(\Gamma )$. We introduce a modification of mean dimension which is an obstruction for ${?}^p(\Gamma )$ and ${?}^q(\Gamma )$ to be Hölder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The D+E[Γ⁎] construction from Prüfer domains and GCD-domains

Let $D?E$ denote an extension of integral domains, Γ be a nonzero torsion-free grading monoid with $\Gamma \cap ?\Gamma =\{0\}$, ${\Gamma }^{?}=\Gamma ?\{0\}$ and $D+E[{\Gamma }^{?}]=\{f\in E[\Gamma ]|f(0)\in D\}$. In this paper, we give a necessary and sufficient criteria for $D+E[{\Gamma }^{?}]$ to be a Prüfer domain or a GCD-domain.     

Boolean rings and reciprocal eigenvalue properties

Let A be the adjacency matrix of the zero-divisor graph $\mathrm{\Gamma }(R)$ of a finite commutative ring R containing nonzero zero-divisors. In this paper, it is shown that $\mathrm{\Gamma }(R)$ is the zero-divisor graph of a Boolean ring if and only if $\det (A)=-1$. Also, A is similar to plus or minus its inverse whenever R is a Boolean ring. As a consequence, it is proved that $\mathrm{\Gamma }(R)$ is the zero-divisor graph of a Boolean ring if and only if the set of eigenvalues (including multiplicities) of $\mathrm{\Gamma }(R)$ can be partitioned into 2-element subsets of the form $\{\lambda \text{,}\pm 1/\lambda \}$. Furthermore, any finite Boolean ring R is characterized by the degree and coefficients of the characteristic polynomial of A.     

Invariants relatifs : une algèbre extérieure

Let G be a complex reflection group acting on V, M be a finite dimensional G-module and S be the coordinate ring of V. Generalizing results of Orlik and Solomon, and of Shepler, we build an exterior algebra structure on the set of relative invariants (associated to a linear character of G) of the algebra $S?\Lambda (M^1)$. To cite this article: V. Beck, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

On the Erdös–Lax inequality concerning polynomials

Let $P(z)$ be a polynomial of degree n and for any complex number α, let $D_{\alpha }P(z):=nP(z)+(\alpha ?z)P^{\prime }(z)$ denote the polar derivative of $P(z)$ with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.     

Character degree graphs with no complete vertices

Let Γ be a graph in which each vertex is non-adjacent to another different one. We show that, if G is a finite solvable group with abelian Fitting subgroup and with character degree graph $\Gamma (G)=\Gamma$, then G is a direct product of subgroups having a disconnected character degree graph. In particular, Γ is a join of disconnected graphs. We deduce also that solvable groups with abelian Fitting subgroup have a character degree graph with diameter at most 2.     

Sieving and expanders

Let V be an orbit in ${\mathbb{Z}}^n$ of a finitely generated subgroup Λ of ${\mathrm{GL}}_n(\mathbb{Z})$ whose Zariski closure $\mathrm{Zcl}(\Lambda )$ is suitably large (e.g. isomorphic to ${\mathrm{SL}}_2$). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the ‘congruence graphs’ that we associate with V. This expansion property is established when $\mathrm{Zcl}(\Lambda )={\mathrm{SL}}_2$. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of $M_n(F)$ was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where R admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions – satisfied by $M_n(E)$ with the trivial grading – to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.     

Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce ${\mathrm{reg}}_R(M,N)$ by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that ${\mathrm{reg}}_R(M,N)$ is finite in several cases. In the case that the base ring is a field, we show that${\mathrm{reg}}_R(M,N)=\mathrm{reg}(N)?\mathrm{indeg}(M).$ This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen–Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.