arXiv subjects

Peter Schenzel

Publications and source records attributed to Peter Schenzel.

32 records · Page 2Linked to original sources

arXiv2013

A note on the Matlis dual of a certain injective hull

Let $(R,\mathfrak{m})$ denote a local ring with $E = E_R(R/\mathfrak{m})$ the injective hull of the residue field. Let $\mathfrak{p} \in \Spec R$ denote a prime ideal with $\dim R/\mathfrak{p} = 1$, and let $E_R(R/\mathfrak{p})$ be the injective hull of $R/\mathfrak{p}$. As the main result we prove that the Matlis dual $\Hom_R(E_R(R/\mathfrak{p}), E)$ is isomorphic to $\hat{R_{\mathfrak{p}}}$, the completion of $R_{\mathfrak{p}}$, if and only if $R/\mathfrak{p}$ is complete. In the case of $R$ a one dimensional domain there is a complete description of $Q \otimes_R \hat{R}$ in terms of the completion $\hat{R}$.

math.AC↗

arXiv2013

Projective surfaces of maximal sectional regularity

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve $C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C \subset \mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\widetilde{X} \subset \mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of these surfaces satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate various of their cohomological invariants as well as their Betti numbers. We study the the extremal variety $\mathbb{F}(X)$ of these surfaces $X$, that is the closed union of the extremal secant lines of all smooth hyperplane section curves of $X$. We show that $\mathbb{F}(X)$ is either a plane or that otherwise $r =5$ and $\mathbb{F}(X)$ is a rational smooth threefold scroll $S(1,1,1) \subset \mathbb{P}^5$.

math.AG↗

arXiv2013

Algebraic properties of the binomial edge ideal of complete bipartite graph

Let $J_G$ denote the binomial edge ideal of a connected undirected graph on $n$ vertices. This is the ideal generated by the binomials $x_iy_j - x_jy_i, 1\leq i < j \leq n,$ in the polynomial ring $S= K[x_1,...,x_n,y_1,...,y_n]$ where $\{i,j\}$ is an edge of $G$. We study the arithmetic properties of $S/J_G$ for $G$, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of $S/J_G$.

math.AC↗

arXiv2012

On an endomorphism ring of local cohomology

Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is $H^d_I(R) \not= 0,$ we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case $R$ is complete we prove - as a technical tool - that $H^d_I(R) \simeq H^d_{\mathfrak m}(R/J)$ for a certain ideal $J \subset R.$ Thus, properties of $H^d_I(R)$ and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.

math.AC↗

arXiv2009

Projective Curves with maximal regularity and applications to syzygies and surfaces

We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves ${\mathcal C} \subset \mathbb P^r_K$ of maximal regularity with $°{\mathcal C} \leq 2r -3.$ In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces $X \subset \mathbb P^r_K$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for "an early decent of the Hartshorne-Rao function" of such surfaces. We use this criterion to give a lower bound on the degree for a class of these surfaces. Then, we study surfaces $X \subset \mathbb P^r_K$ for which $h^1(\mathbb P^r_K, {\mathcal I}_X(1))$ takes a value close to the possible maximum $°X - r +1.$ We give a lower bound on the degree of such surfaces. We illustrate our results by a number of examples, computed by means of {\sc Singular}, which show a rich variety of occuring phenomena.

math.AG↗

arXiv2009

On Lyubeznik's invariants and endomorphisms of local cohomology modules

Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case of $(R,\mathfrak m)$ a regular local ring containing a field $B$ is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number $l = \dim_k \Ext_R^d(k,H^c_I(R)).$ In particular $R \simeq B$ if and only if $l = 1.$ Moreover, we show that the natural homomorphism $\Ext_R^d(k, H^c_I(R)) \to k$ is non-zero.

math.AC↗

arXiv2008

On connectedness and indecomposibility of local cohomology modules

Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes the intersection of the highest dimensional primary components of $I.$ This is a partial extension of a result shown by Hochster and Huneke in the case $I$ the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of $H^c_I(R)$ is a local Noetherian ring if $\dim R/I = 1.$

math.AC↗

arXiv2008

On endomorphism rings and dimensions of local cohomology modules

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, resp. an estimate of the dimension of $H^i_I(R).$

math.AC↗

arXiv2008

On cohomologically complete intersections

An ideal $I$ of a local Gorenstein ring $(R, \mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \not= \height I.$ Here $H^i_I(R), i \in \mathbb Z,$ denotes the local cohomology of $R$ with respect to $I.$ For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of $H^c_I(R), c = \height I.$ As a main result it is shown that the vanishing $H^i_I(R) = 0$ for all $i \not= c$ is completely encoded in homological properties of $H^c_I(R),$ in particular in its Bass numbers.

math.AC↗

arXiv2007

On the formal cohomology of local rings

Let $\mathfrak a$ denote an ideal of a local ring $(R, \mathfrak m).$ Let $M$ be a finitely generated $R$-module. There is a systematic study of the formal cohomology modules $\varprojlim \HH^i(M/\mathfrak a^nM), i \in \mathbb Z.$ We analyze their $R$-module structure, the upper and lower vanishing and non-vanishing in terms of intrinsic data of $M,$ and its functorial behavior. These cohomology modules occur in relation to the formal completion of the punctured spectrum $\Spec R \setminus V(\mathfrak m).$ As a new cohomological data there is a description on the formal grade $\fgrade(\mathfrak a, M)$ defined as the minimal non-vanishing of the formal cohomology modules. There are various exact sequences concerning the formal cohomology modules. Among them a Mayer-Vietoris sequence for two ideals. It applies to new connectedness results. There are also relations to local cohomological dimensions.

math.AC↗

arXiv2006

Arithmetic properties of projective varieties of almost minimal degree

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X \subset {\mathbb P}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $\tilde {X} \subset {\mathbb P}^{r + 1}$ from an appropriate point $p \in {\mathbb P}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $X$ by means of the projection $\tilde {X} \to X$. If $X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $B$ of the projecting variety $\tilde {X}$ is the endomorphism ring of the canonical module $K(A)$ of the homogeneous coordinate ring $A$ of $X.$ If $X$ is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay but not arithmetically normal $B$ is just the graded integral closure of $A.$ It turns out, that the geometry of the projection $\tilde {X} \to X$ is governed by the arithmetic depth of $X$ in any case. We study in particular the case in which the projecting variety $\tilde {X} \subset {\mathbb P}^{r + 1}$ is a cone (over a) rational normal scroll. In this case $X$ is contained in a variety of minimal degree $Y \subset {\mathbb P}^r$ such that $\codim_Y(X) = 1$. We use this to approximate the Betti numbers of $X$. In addition we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of $Δ$-genus 1.

math.AC↗

arXiv2006

On varieties of almost minimal degree in small codimension

The aim of the present exposition is to investigate varieties of almost minimal degree and of low codimension, in particular their Betti diagrams. Here minimal degree is defined as $°X = \codim X + 2.$ We describe the structure of the minimal free resolution of a variety $X$ of almost minimal degree of $\codim X \leq 4$ by listing the possible Betti diagrams. The most surprising fact is, that the non-arithmetically Cohen-Macaulay case of varieties of almost minimal degree can occur only in small dimensions (cf. Section 2 for the precise statements). Our main technical tool is a result shown by the authors (cf. \cite{BS}), which says that besides of an exceptional case, (that is the generic projection of the Veronese surface in $\mathbb P^5_K$) any non-arithmetically normal (and in particular non-arithmetically Cohen-Macaulay) variety of almost minimal degree $X \subset \mathbb P^r_K$ (which is not a cone) is contained in a variety of minimal degree $Y \subset \mathbb P^r_K$ such that $\codim(X,Y) = 1.

math.AC↗

arXiv1996

Descent from the form ring and Buchsbaum rings

There is a spectral sequence technique in order to estimate the local cohomology of a ring by the local cohomology of a certain form ring. As applications there are information on the descent of homological properties (Cohen-Macaulay, Buchsbaum etc.) from the form ring to the ring itself. In the case of Buchsbaum ring there is a discussion of the descent of the surjectivity of a natural map into the local cohomology.

alg-geom↗

arXiv1995

Applications of Koszul homology to numbers of generators and syzygies

Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. There are also investigations about the shifts and the dimension of Betti numbers.

alg-geom↗