\documentclass[twoside,11pt]{article} \usepackage{latexsym,amssymb,times,mathptm} \setlength{\topmargin}{-1in} \setlength{\oddsidemargin}{-.4in} \setlength{\evensidemargin}{-.4in} \setlength{\textheight}{10in} \setlength{\textwidth}{7in} \title{\Large\bf Commutative Algebra: Hilbert functions, homological methods,\\ and combinatorial aspects} \author{Aldo Conca, Anna Guerrieri, Claudia Polini and Bernd Ulrich} \begin{document} \date{} \maketitle \section*{Saturdayday (June 15, 2002)} \bigskip \noindent {\bf Title:} F-signatures in local rings. \noindent {\bf Author:} Ian Aberbach* and Graham Leuschke. \noindent {\bf Abstract:} Let $R$ be a reduced Noetherian ring of positive prime characteristic $p$, and let $R^{1/q}$ be the ring of $q = p^e$th roots. Assume that $R^{1/p}$ is a finitely generated $R$-module. Then the F-signature of $R$, $s(R)$, introduced by Huneke and Leuschke, is an asymptotic measure of the number of summands in $R^{1/q}$ as an $R$-module. In joint work with Leuschke we show that $s(R) > 0$ if and only if $R$ is strongly $F$-regular. \bigskip \bigskip \bigskip \noindent {\bf Title:} Cohomology of dualizing complexes. \noindent {\bf Author:} Luchezar L.\ Avramov*, Ragnar-Olaf Buchweitz and Liana M. \ Sega \noindent {\bf Abstract:} Let $R$ be a commutative noetherian local ring having a dualizing complex $D$; shifting $D$, if neecessary, assume $\inf\{j\in{\mathbb Z}\mid \mbox{H}_j(D)\ne0\}=0$. If $R$ is Gorenstein, then $D$ is quasi-isomorphic to $R$, hence $\mbox{Ext}^n_R(D,R)$ vanishes for all $n>0$. Partly motivated by a long standing conjecture of Tachikawa on the cohomology of (not necessarily commutative) finite dimensional algebras over a field, we conjecture that if $\mbox{Ext}^n_R(D,R)=0$ for all $n>0$, then $R$ is Gorenstein. Proofs of this conjecture in several significant cases will be discussed. \bigskip \bigskip \noindent {\bf Title:} Unimodular covers of lattice polytopes \noindent {\bf Author:} Winfried Bruns* and Joseph Gubeladze \noindent {\bf Abstract:} Let $P$ be a $d$-dimensional lattice polytope. We show that there exists a natural number $\mathfrak{c}_d$, only depending on $d$, such that the multiples $cP$ have a unimodular cover for every $c\in\Bbb N$, $c\ge \mathfrak{c}_d$. Actually, an explicit upper bound for $\mathfrak{c}_d$ is provided, together with an analogous result for unimodular covers of rational cones. As an auxiliary result we bound on the complexity of equivariant resolutions of toric singularities. \bigskip \bigskip \noindent {\bf Title:} Castelnuovo-Mumford regularity: Some examples in dimensions one and two. \noindent {\bf Author:} Marc Chardin* and Clare V. D'Cruz \noindent {\bf Abstract:} The behaviour of Castelnuovo-Mumford regularity under ``geometric'' transformations is not well understood. We try to shed some light on few questions concerning this behaviour by looking at examples. Simple questions as: does the regularity improve by passing to radical or removing embedded primes~? seemed open. We will show examples that answers (negatively) to these questions, and consider other related issues. For example we provide examples of defining ideals of curves in ${\bf P}^{3}$ (resp. in ${\bf P}^{4}$) such that the regularity of their radical is essentially the square (resp. the cube) of the one of the ideal. Removing points (embedded or not) the regularity improves, so we have to go to surfaces in order to find examples where removing an embedded prime increases the regularity. More surprising is the fact that we find an irreducible surface such that we may improve the depth of its coordinate ring by embedding a line into it! Another important issue is to understand the limits of validity of Kodaira type vanishing theorems. The canonical module of a reduced curve have Castelnuovo-Mumford regularity 2. This type of results extends in particular to higher dimensionnal varieties with isolated singularities (in characteristic zero), thanks to Kodaira vanishing, and as a corollary one may derive bounds for Castelnuovo-Mumford regularity. In ``Pathologies III'', Mumford proves that for an ample line bundle ${\mathcal L}$ on a normal surface ${\mathcal S}$, $H^{1}({\mathcal S} ,{\mathcal O}_{{\mathcal S}}\otimes {\mathcal L}^{-1})=0$. He remarks that this is false if ${\mathcal S}$ is not Cohen-Macaulay and asks if ever the $S_{2}$ condition is sufficient. A first counter-example was given by Arapura and Jaffe. We here show that it is sufficient to construct counter-examples satisfying $S_{1}$, and provide some monomial surfaces whose canonical module have rather big Castelnuovo-Mumford regularity, so that this vanishing fails. We also give a simple proof that if ${\mathcal S}$ satisfies $R_{1}$, then $H^{1}({\mathcal S},{\omega}_{{\mathcal S}}\otimes {\mathcal L})=0$ and $H^{1}({\mathcal S} ,{\mathcal O}_{{\mathcal S}}\otimes {\mathcal L}^{i})=0$ have constant dimension for $i<0$ if ${\mathcal L}$ is very ample. This dimension measures the defect of Macaulyness of ${\mathcal S}$. These monomial surfaces also provides examples of complete intersection surfaces with two reduced irreducible components such that one of them have a regularity bigger than the complete intersection, this doesn't happend in dimension one. \bigskip \bigskip \noindent {\bf Title:} Koszul homology and integral closure of ideals. \noindent {\bf Author:} Alberto Corso \noindent {\bf Abstract:} If $I$ is an equidimensional ideal of a Cohen--Macaulay local ring, then the annihilators of the nonzero Koszul homology modules $H_i(I)$ are contained in $\sqrt{I}$. A number of preliminary results by C. Huneke, D. Katz, W.V. Vasconcelos and ourselves suggest that these annihilators are always contained in the integral closure of $I$. In the talk, we will discuss some of these results (focusing on the annihilator of $H_1(I)$). \bigskip \bigskip \noindent {\bf Title:} Ulrich Modules -- their existence and uses. \noindent {\bf Author:} David Eisenbud \noindent {\bf Abstract:} An Ulrich Module over a standard graded commutative algebra R = S/I, with S a polynomial ring, is a finitely generated graded maximal Cohen-Macaulay R-module M whose free resolution as an S-module is linear. There are many other characterizations, and the condition is rather natural when one regards the module as a sheaf on the associated projective variety. Schreyer and I discovered that Ulrich modules play a key role in producing the nicest determinantal formulas for Chow forms and resultants, and I will explain this connection, which generalizes the notion of matrix factorizations over hypersurfaces. The existence of Ulrich modules in general is open, but there are many cases in which it is known, and I will report on recent progress in this by Hanes, myself, Schreyer, and Weyman. \bigskip \bigskip \noindent {\bf Title:} Arf characters of algebroid curves. \noindent {\bf Author:} Valentina Barucci, Marco D'Anna and Ralf Fr\"oberg* \noindent {\bf Abstract:} An algebroid branch is a one-dimensional ring of the form $R=k[[x_1,\ldots,x_n]]/P$, where $P$ is a prime ideal. The blowup of $R=R_0$ is $R_1=\cup(m^n:m^n)$, where $m$ is the maximal ideal of $R$. The blowup $R_1$ is a local overring of $R$ and, if $R_{i+1}$ denotes the blowup of $R_i$, then $R_j=k[[t]]$ if $j>>0$. The multiplicity sequence of $R$ is $e_0=e(R_0),e_1=e(R_1),\ldots$. Two algebroid branches are said to be equivalent if they have the same multiplicity sequence. (For plane curves, this is topological equivalence.) There is a smallest so called Arf ring $R'$ containing $R$, the Arf closure of $R$, and $R'$ is equivalent to $R$. Since the integral closure of $R$ is $k[[t]]$, every element of $R$ has a value, and the set of values constitute a semigroup $v(R)$. Now $v(R')$ is a so called Arf semigroup. There is a smallest semigroup $T$, such that $v(R')$ is the smallest Arf semigroup containing $T$. The generators of $N$ was called the characters of the curve by Arf. In the semigroup setting these set of generators has been studied by Rosales, Garcia-Sanches, Garcia-Garcia, and Branco. We generalize the results above to algebroid curves, i.e. one-dimensional reduced rings $k[[x_1,\ldots,x_n]]/P_1\cap\cdots\cap P_d$. \bigskip \bigskip \noindent {\bf Title:} Toric varieties with huge Grothendieck group. \noindent {\bf Author:} Joseph Gubeladze \noindent {\bf Abstract:} In each dimension $n\geq3$ there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly. \bigskip \bigskip \noindent {\bf Title:} On the regularity of simplicial semigroup rings with isolated singularity. \noindent {\bf Author:} J\"{u}rgen Herzog* and Takayuki Hibi \noindent {\bf Abstract:} Let $S = K[x_1, \ldots, x_n]$ be the polynomial ring in $n \geq 2$ variables over a field $K$ and $m$ its graded maximal ideal. Let $f_1,\ldots, f_m \in S$ be homogeneous polynomials of degree $d-1\geq 2$ generating an $m$-primary ideal, and let $g_1,\ldots,g_r\in S$ be arbitrary homogeneous polynomials of degree $d$. It will be proved that the Castelnuovo--Mumford regularity of the standard graded $K$-algebra $A=K[\{f_ix_j\}_{i=1,\ldots,m\atop j=1,\ldots,n}, g_1,\ldots, g_r]$ is at most $(d-2)(n-1)$. By virtue of this result, it follows that the regularity of a simplicial semigroup ring $K[C]$ with isolated singularity is bounded by $e(K[C]) - codim(K[C])$. Hence in this particular case the Eisenbud-Goto conjectured bound for the regularity of a standard graded domain holds. \bigskip \bigskip \noindent {\bf Title:} Rings with countable representation type. \noindent {\bf Author:} Craig Huneke* and Graham Leuschke \noindent {\bf Abstract:} This talk is a preliminary report on Cohen-Macaulay local rings having countably many isomorphism classes of maximal Cohen-Macaulay modules. Such rings are said to have countable CM representation type. Schreyer conjectured that such rings have singular locus of dimension at most one. We prove this conjecture for complete rings or rings with uncountable residue field. We further prove that this property localizes and discuss some questions related to the structure of such rings. \bigskip \bigskip \noindent {\bf Title:} Action of the Borel group on monomial ideals. \noindent {\bf Author:} Alessandro Logar \noindent {\bf Abstract:} Let $\mathbb T \subseteq {\mathrm{GL}(n,K)}$ be the Borel group of upper triangular matrices. In this paper we want to study the action of $\mathbb T$ on the set of monomial ideals of ${K[x_1,\dots,x_n]}$ ($K$ a field of characteristic zero) from a computational point of view. More specifically, we show that the stabilizer of a monomial ideal $M \subseteq {K[x_1,\dots,x_n]}$ in $\mathbb T$ is a purely combinatorial object and we give an algorithm for computing it. Then we characterize the subgroups of $\mathbb T$ that are stabilizers of monomial ideals, we give an algorithm which finds if a given ideal $J$ is in the orbit of a monomial ideal $M$ under the action of $\mathbb T$ and in the affirmative case, finds the matrices $W\in \mathbb T$ such that $W\cdot J=M$. We show that the entries of $W$ can be directly obtained from the coefficients of the generators of $J$, so in particular no solutions of polynomial equations are required. \bigskip \bigskip \noindent {\bf Title:} Movable components of intersection cycles. \noindent {\bf Author:} Mirella Manaresi \noindent {\bf Abstract:} St\"uckrad-Vogel intersection cycle for projective varieties over an algebraic closed field k contains some fixed parts, defined over the field k, and some movable parts, defined over trascendental extensions of k. Van Gastel proved that the fixed parts are the distinguished varieties of Fulton-McPherson. Flenner and Manaresi showed that movable parts appear as ramification loci of geneic projections and, using this, gave some estimations of their variation (in terms of their transcendence degree over k) and their contribution to the intersection cycle. The talk will be a survey on some recent results of Flenner-Manaresi, Achilles-Manaresi and Achilles-St\"uckrad on movable components and their intersection numbers. \par References \par [1] R.Achilles-M.Manaresi: Multiplicities of bigraded rings and intersection theory. Math. Ann. 309, n. 4, 573-591 (1997) \par [2] R.Achilles - M.Manaresi: Self-intersection of surfaces and Whitney stratifications.: Self-intersections of surfaces and Whitney stratifications. In: F. Norguet (ed.), S. Ofman (ed.), Complex Geometry 1998, Proceedings of the Conference, Paris June 29-July 3, International Press, Boston, MA, to appear. \par [3] R.Achilles-J.St\"uckrad: Generic residual intersections and intersection numbers of movable components. Preprint 2002 \par [4] H.Flenner-M.Manaresi: Intersections of projective varieties and generic projections. Manuscripta Math. 92, 273-286 (1997) \par [5] H.Flenner-M.Manaresi: Variation of the ramification loci of generic projections. Math. Nachr. 194 (1998), 79-92 \par [6] H.Flenner - M.Manaresi: A numerical characterization of reduction ideals Math. Zeitschrift 238 (2001), 205-214. \par [7] H.Flenner - M.Manaresi: A length formula for the multiplicity of distinguished components of intersections J. Pure and Applied Algebra 165 (2001) 155-168. \bigskip \bigskip \noindent {\bf Title:} Some new results on the Hilbert coefficients. \noindent {\bf Author:} Maria Evelina Rossi \noindent {\bf Abstract:} Let $I$ be an ${\mathfrak m}$-primary ideal of a local Cohen--Macaulay ring $(R, {\mathfrak m})$ of dimension $d.$ The {\it Hilbert--Samuel function} of $I$ is the numerical function that measures the growth of the length of $R/I^n$ for all $n \geq 1$. For $n \gg 0$ this function $\lambda(R/I^n)$ is a polynomial $P_I(n) $ in $n$ of degree $d$ $$ P_I(n) = e_0(I) {{n+d-1}\choose d} - e_1(I) {{n+d-2}\choose d-1} + \ldots + (-1)^d e_d(I), $$ where $e_0(I), e_1(I), \ldots, e_d(I) $ are the {\it Hilbert coefficients} of $I$. These integers give asymptotic information on the Hilbert function of $I$ and hence geometric information on the singular locus of $R.$ In particular $e_0(I) $ is the multiplicity, $ e_d(I) $ and $ (-1)^d (P_I(1) - \lambda(R/I)) $ are the so called genus and arithmetic genus of $I.$ Classically one has sought knowledge about the Hilbert coefficients in presence of good depth properties of the associated graded ring $G= \oplus_{n \ge 0} I^n/I^{n+1} $ of $I.$ Conversely, numerical information on $e_i(I)'s, $ have been used to obtain information on the depth of $G,$ for example this has been a constant theme in the work of J.D. Sally. A very useful technique is to consider the generating function of $\lambda(R/\widetilde {I^n}) $ or $\lambda(R/\overline {I^n}) $ instead of $\lambda(R/{I^n}) $ where $ \widetilde {I^n} $ denotes the Ratliff-Rush ideal and $ \overline {I^n} $ the integral closure of $ {I^n} $ (under the assumption $R $ is analytically unramified). The advantage is that $ \widetilde{G}=\oplus_{n\ge 0}(\widetilde{I^{n}}/\widetilde{I^{n+1}})$ and $ \overline{G}=\oplus_{n\ge 0}(\overline{I^{n}}/\overline{I^{n+1}})$ have positive depth, but unfortunately in general they are not standard graded algebras. In this talk we present some result obtained with A.Corso, C.Polini and G.Valla in different papers involving the use of the above techniques. In the first paper we study the interplay between the integral closedness -- or the normality -- of an ideal $I$ and conditions on the Hilbert coefficients of $I$. We relate these properties to the Cohen--Macaulayness of the associated graded ring of $I.$ We generalize some results of Itoh and Narita and, for normal ideals, we give a positive answer to a conjecture stated by Valla in the case $e_2(I) $ is minimal. If $I$ is a ${\mathfrak m}-$primary ideal of a one-dimensional Cohen-Macaulay local ring, Valla and myself proved that the Hilbert function of $\widetilde{G} $ is strictly increasing up to reach the multiplicity $e_0(I), $ the same behaviour which the Hilbert function of $G$ has in the case $G$ is Cohen-Macaulay. As a numerical consequence of the described properties of $\widetilde{G}, $ we improve the upper bound for $e_1(\mathfrak m) $ proved by Elias by showing that for a $\mathfrak m $-primary ideal of a one-dimensional Cohen-Macaulay local ring $R, $ one has $$e_1(I) \le {e_0(I) \choose 2} -{v(\widetilde{I})-1\choose 2}-\lambda(R/I)+1 $$ where $ v(\ \ ) $ denotes the minimal number of the ideal. This result can be used to give strict constraints on the Hilbert function of a $\mathfrak m $-primary ideal in a one-dimensional Cohen-Macaulay local ring. This inequality can be extended to the higher dimensional case when $I$ is an integrally closed $\mathfrak m-$primary ideal. \bigskip \bigskip \noindent {\bf Title:} A class of birational transformations. \noindent {\bf Author:} Aron Simis \noindent {\bf Abstract:} We will briefly review some aspects of birational maps (mainly Cremona transformations). A special class to be envisaged is that of monomial rational representations of projective space. This brings a definite combinatorial flavor into the subject along with various questions. \bigskip \bigskip \noindent {\bf Title:} Multiplicities and the Number of Generators of Cohen--Macaulay Ideals. \noindent {\bf Author}: Wolmer V. Vasconcelos \noindent {\bf Abstract:} Let $(R, \mathfrak{m})$ be a Cohen--Macaulay local ring of dimension $d$ and let $I$ be a Cohen--Macaulay ideal of codimension $g$. There are several invariants of $R$ and of $I$ (multiplicities, embedding dimension, index of nilpotency, etc) that may affect the number of generators $\nu(I)$ of $I$. After reviewing results of Sally, Valla and others, we derive estimates for $\nu(I)$ depending on another mix of some of these invariants. \bigskip \bigskip \noindent {\bf Title:} Symmetric and Rees algebras of Koszul cycles. \noindent {\bf Author}: J\"{u}rgen Herzog, Zhongming Tang and Santiago Zarzuela* \noindent {\bf Abstract:} Let $k$ be a field, $P=k[x_1, \dots , x_n]$ the polynomial ring in $n$ variables, and $E_i$ the $i$-th syzygy module of $k$. We study the symmetric algebra $S(E_i)$ and the Rees algebra $R(E_i)=S(E_i)/T_P(S(E_i))$ of $E_i$. By using SAGBI basis techniques the case $E_2$ is fully described. In particular, we prove that $x_1, \dots , x_n$ is an unconditioned $d$-sequence on $S(E_2)$ and $R(E_2)$. We also show that the same result holds for the case $E_{n-2}$. Preliminary report. \end{document}