On injectivity of maps between Grothendieck groups induced by completion

We give an example of a local normal domain $R$ such that the map of Grothendieck groups \G(R) \to \G(\hat R) is not injective. We also raise some questions about the kernel of that map

Some observations on local and projective hypersurfaces

Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of \Tor^R(M,N), based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connections and applications to projective hypersurfaces such as intersection dimension of subvarieties and cohomological criterion for splitting of vector bundles are discussed

Asymptotic behavior of Tor over complete intersections and applications

Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that \ell(\Tor_i^R(M,N))<\infty for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of \ell(\Tor_i^R(M,N)) using Eisenbud operators and show that they have well-behaved growth. We define and study a function $\eta^R(M,N)$ which generalizes Serre's intersection multiplicity $\chi^R(M,N)$ over regular local rings and Hochster's function $\theta^R(M,N)$ over local hypersurfaces. We use good properties of $\eta^R(M,N)$ to obtain various results on complexities of \Tor and \Ext, vanishing of \Tor, depth of tensor products, and dimensions of intersecting modules over local complete intersections

generalised q-deformed oscillators and their statistics

We consider a version of generalised $q$-oscillators and some of their applications. The generalisation includes also "quons" of infinite statistics and deformed oscillators of parastatistics. The statistical distributions for different $q$-oscillators are derived for their corresponding Fock space representations. The deformed Virasoro algebra and SU(2) algebra are also treated.Comment: 9 pages,no figure, ([email protected]), ENSLAPP-A-494/9

Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields

In this paper, we are going to compute the average size of 2-Selmer groups of two families of hyperelliptic curves with marked points over function fields. The result will be obtained by a geometric method which could be considered as a generalization of the one that was used previously by Q.P. Ho, V.B. Le Hung, and B.C. Ngo to obtain the average size of 2-Selmer groups of elliptic curves

Global existence for weakly coupled systems of semi-linear structurally damped σ\sigma-evolution models with different power nonlinearities

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped $\sigma$-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m \in [1,2)$, we use $(L^m \cap L^2)- L^2$ and $L^2- L^2$ estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces.Comment: arXiv admin note: text overlap with arXiv:1808.0270

On asymptotic vanishing behavior of local cohomology

Let $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal \fm, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules \{\HH{i}{\fm}{R/I^n}\}_{n\in \NN} for $i<\dim R/I$. We show that, when \chara k= 0, $R/I$ is Cohen-Macaulay, and $I$ is a complete intersection locally on \Spec R \setminus\{\fm\}, the lowest degrees of the modules \{\HH{i}{\fm}{R/I^n}\}_{n\in \NN} are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $I/I^2$. Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field $k$, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.Comment: 13 pages. To appear in Math.

The Type Defect of a Simplicial Complex

Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the type defect of $\Delta$, $\textrm{td}(\Delta)$, is the difference $\dim_k \textrm{Tor}_c^S(S/ I_\Delta,k) - c$, where $c$ is the codimension of $\Delta$ and $S = k[x_1, \ldots x_n]$. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, $\Delta$ is Cohen-Macaulay if $\textrm{td}(\Delta) \leq 0$. On the other hand, if $\Delta$ is a simple graph (viewed as a one-dimensional complex), then $\textrm{td}(\Delta') \geq 0$ for every induced subgraph $\Delta'$ of $\Delta$ if and only if $\Delta$ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call $\textit{treeish}$, and which we generalize to simplicial complexes. We then extend some of our chordality results to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolution.Comment: Several minor changes were made following suggestions by referees. Final versio

Necessary conditions for the depth formula over Cohen-Macaulay local rings

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula \depth(M)+\depth(N)=\depth(R)+\depth(M\otimes_{R}N) to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if \Tor_{i}^{R}(M,N) vanishes for all $i\geq 1$. We also examine the relationship between good depth of $M\otimes_RN$ and the vanishing of \Ext modules, with various applications.Comment: Some results on semi-dualizing modules are added at the en

Representation schemes and rigid maximal Cohen-Macaulay modules

Let k be an algebraically closed field and A be a finitely generated, centrally finite, non- negatively graded (not necessarily commutative) k-algebra. In this note we construct a representation scheme for graded maximal Cohen-Macaulay A modules. Our main application asserts that when A is commutative with an isolated singularity, for a fixed multiplicity, there are only finitely many indecomposable rigid (i.e, with no nontrivial self-extensions) MCM modules up to shifting and isomorphism. We appeal to a result by Keller, Murfet, and Van den Bergh to prove a similar result for rings that are completion of graded rings. Finally, we discuss how finiteness results for rigid MCM modules are related to recent work by Iyama and Wemyss on maximal modifying modules over compound Du Val singularities.Comment: 11 pages -- comments welcome