Related papers: Hilbert-Kunz functions over rings regular in codim…

Let $G$ be a simple graph on $n$ vertices, and let $J_G$ denotes the corresponding binomial edge ideal in $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, where $\mathbb{K}$ is a field. We show that if a vertex satisfies a certain degree…

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that…

In this paper, we investigate the relationship between the Hilbert functions and the associated properties of the graded modules. To attain this, we construct the graded modules from the sets of points in projective space, $\mathbb{P}_k^n$…

Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let $K$ be its Kahler cone, which is an open, convex subset in the space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with a canonical…

For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite…

Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This…

Thirty years ago, Huneke (for local rings) and Lyubeznik (in general) conjectured that for all regular rings $R$, the local cohomology modules $H^i_I(R)$ have finitely many associated prime ideals. We prove substantial new cases of their…

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the…

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We…

We describe the cone of Hilbert functions of artinian graded modules finitely generated in degree 0 over the polynomial ring R = k[x, y] with the non-standard grading deg(x) = 1 and deg(y) = n, where n is any natural number.

In this article, by using the technique of gluing semigroups, we give infinitely many families of 1-dimensional local rings with non-decreasing Hilbert functions. More significantly, these are local rings whose associated graded rings are…

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real…

Let $M$ be a finitely generated module of dimension $d$ and depth $t$ over a Noetherian local ring ($A, {\mathfrak m}$) and $I$ an ${\mathfrak m}$-primary ideal. In the main result it is shown that the last $t$ Hilbert coefficients…

We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$ f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x) $$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various…

We introduce an analog of the Ziegler spectrum for maximal Cohen-Macaulay modules over a complete Cohen-Macaulay local ring. We define a topology on the space of isomorphism classes of indecomposable maximal Cohen-Macaulay modules and…

In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite…

The ring of symmetric functions can be implemented in the homology of \union_{a,b} Gr(a,a+b), the multiplicative structure being defined from the "direct sum" map. There is a natural circle action (simultaneously on all Grassmannians) under…

Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to…

We prove that if the first tight Hilbert coefficient vanishes then ring is $F$-rational provided it is a Buchsbaum local ring satisfying the $(S_2)$ condition.