Commutative Algebra and Algebraic Geometry

Lawrenceville, New Jersey
April 17-18, 2004

## List of Talks

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• Homological invariants of modules over local homomorphisms
by Luchezar AVRAMOV, University of Nebraska at Lincoln
Abstract: Numerical invariants, such as Betti numbers and Bass numbers, will be defined for a finitely generated module N   over a local ring S, in the the presence of a local homomorphism j: R ® S. Properties and applications of these invariants will be discussed. This is joint work with Srikanth Iyengar and Claudia Miller.

• The integral closure of monomial rings
by Joseph BRENNAN, North Dakota State University of
Abstract: This talk will examine the question of computation of the integral closure of subrings of polynomial rings generated by monomials. Particular attention will be focused on the question of normality of Rees rings of ideals generated by monomials.

• Buchsbaum-Rim multiplicity formulas
by C.-Y. Jean CHAN, University of Arkansas
Abstract: Over a commutative Noetherian local ring A, the Buchsbaum-Rim multiplicity eBR(M), defined for a submodule M of a free A-module F of rank r such that F/M has finite length, is a generalization of the Hilbert-Samuel multiplicity of an m-primary ideal. In a special case where r=2, there exist two m-primary ideals I and J such that F/M @ I/J. We have the following multiplicity formula for eBR(M)
eBR(M) = e(J) - e(I) + e(Fitt0(I/J)) - e(Fitt0(I/J')),
where J' is a minimal reduction of J and the above expression is independent from the choices of minimal reductions. This also improves the result of E. Jones in Computation of Buchsbaum-Rim multiplicities in JPAA, vol. 162/1 (2001). We will discuss the extension of the above multiplicity formula for modules of an arbitrary rank over a two dimensional regular local ring. In such a case, eBR(M) can be expressed in terms of the Hilbert-Samuel multiplicities of ideals in the same linkage class with the Fitting ideals of F/M. This leads a possibility of relating the Buchsbaum-Rim multiplicity with the Chern classes of certain types of matrices. This is joint work with Jung-Chen Liu and Bernd Ulrich.

• Maximal rank conjectures and algorithms for infinitesimal neighborhoods
by Karen CHANDLER, University of Notre Dame
Abstract: Alexander and Hirschowitz have shown, asymptotically, an upper bound of the degree in which every general collection of infinitesimal neighbourhoods of points of given order in a projective space displays maximal rank with respect to ideal cohomology. Namely, for each dimension n and multiplicity m there exists d(n,m) so that the Hilbert function of such a scheme in each degree d ³ d(n,m) has the expected value. One would like to obtain concrete values (or at least bounds) for the function d(n,m). According to the Segre-Hartshorne-Hirshowitz, we should have d(2,m)=3m. This has been shown to hold for each d £ 20 by Ciliberto, Miranda, and Orecchia. Indeed, we have conjectured that in general d(n,m) £ 3m. We shall describe algorithmic methods on verifying this bound, as we have used, for example, in the case of each d(n,3).

• b-generic modules
by Hara CHARALAMBOUS, SUNY at Albany
Abstract: We will introduce and discuss the notion of b-generic ideals and modules and their free resolutions.

• Asymptotic behavior of length of local cohomology
by Dale CUTKOSKY, University of Missouri at Columbia
Abstract: Let k be a field of characteristic 0, R=k[x1,..., xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I Ì R be a homogeneous ideal in R. We show that
limn ® ¥ length(H0m(R/In))/nd = limn ® ¥ length(ExtdR(R/In,R(-d)))/nd
always exists. This limit has been shown to be e(I)/d! for m-primary ideals I in a local Cohen Macaulay ring, where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. This is joint work with Tai Ha, Hema Srinivasan, and Emanoil Theodorescu.

• The tracking number of an algebra
by Kia DALILI, Rutgers University
Abstract: We introduce the technique of tracking numbers of graded algebras and modules. It is a modified version of the first Chern class of its free resolution relative to any of its standard Noether normalizations. Several estimations are obtained which are used to bound the length of chains of algebras occurring in the construction of the integral closure of a graded domain. Noteworthy is a quadratic bound on the multiplicity for all chains of algebras that satisfy the condition S2 of Serre. This is joint work with Wolmer Vasconcelos.

• A tight closure analogue of analytic spread
by Neil EPSTEIN, University of Kansas
Abstract: Let R be an excellent commutative Noetherian local ring of positive characteristic. Inspired by the classical work of Northcott and Rees (1954) on minimal reductions and analytic spread, we first note that their proof of the existence of minimal reductions provides most of a proof that minimal *-reductions exist. (A *-reduction of an ideal I is an ideal J such that J Í I Í J*, where the * indicates tight closure.) The missing ingredient is a Nakayama lemma for tight closure,' a useful new tool. Then, using an exchange lemma' style of proof, we show that all minimal *-reductions of an ideal I have the same minimal number of generators, at least in the case where R is analytically irreducible and its normalization has perfect residue field. We call this common number the *-spread of the ideal, denoted ell*(I), by analogy with analytic spread.

• Monomial ideals via square-free monomial ideals
by Sara FARIDI, Université du Québec à Montréal
Abstract: Polarization is an operation that transforms a monomial ideal in a polynomial ring into a square-free monomial ideal in an extension of the original ring. This talk focuses on what algebraic and combinatorial properties are preserved under polarization. Using this operation, we extend combinatorial structures defined on square-free monomial ideals to all monomial ideals, and explore possible ways to deduce properties of monomial ideals and their Rees rings using this technique.

• Baxter-type operators and shuffle-type products
by Li GUO, Rutgers University, Newark
Abstract: A Baxter operator is a linear operator P on an algebra A that satisfies the relation P(x)P(y)=P(xP(y))+P(P(x)y)+t P(xy) for all x and y in A. Here t is a constant. When t=0, we get the integration by parts formula.
The study of Baxter operators originated in the work of G. Baxter in 1960 on fluctuation theory, and the algebraic study of Baxter operators was started by Rota in the 1970s. The study has experienced a renaissance in the last couple of years, with applications in Hopf algebras, multiple theta values, umbral calculus, dendriform algebras and quantum field theory. Much of these are related to the shuffle like structure of the free Baxter algebras.
We will present the basic algebraic theory of Baxter operators and several related operators and algebras, with emphasis on the free algebras. We will also discuss some applications mention above. This is joint work with Kurusch Ebrahimi-Fard, from IHES.

• The general plane section of a curve in P3
by Elisa GORLA, University of Notre Dame
Abstract: We will discuss some necessary and sufficient conditions for a space curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non arithmetically Cohen- Macaulay curve of P3. We will see that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non arithmetically Cohen-Macaulay curve of P3, arise also as degree matrices of some smooth, integral, non arithmetically Cohen-Macaulay curve, and we will describe the exceptions.

• Normalization of modules
by Jooyoun HONG, Purdue University
Abstract: Let R be a Noetherian normal domain and E a finitely generated torsionfree R-module having a rank e. The Rees algebra of an R-module E is defined as a subalgebra of the polynomial ring R[T1, ..., Te] generated by all linear form a1T1 + ... + aeTe in E. Our general goal is to study the normalization of the Rees algebra of a module. First we deal with a submodule E of free module Re with finite colength. In this case, we formulate relationships among the coefficients of Buchsbaum-Rim polynomials of E and of the integral closure \overline{E} by using the Briançon-Skoda number of E. With this result we can measure the length of the algorithmic procedures used to compute the integral closure of the the Rees algebra of E. Next, we discuss bounds for degrees of the generators of Cohen-Macaulay Rees algebra which is finite over the Rees algebra of E. This is joint work with Bernd Ulrich and Wolmer V. Vasconcelos.

by Andrew KUSTIN, University of South Carolina
Abstract: Let I be a perfect ideal in the local ring (R,k). We conjecture that there exists a collection of operations {hi} on the Tor-algebra Tor·R(R/I,k), such that I is in the linkage class of a complete intersection if and only if
å im hi= Tor2.
Our best results are obtained when I is a grade four Gorenstein ideal. This is joint work with Andreea N. Brezaie, Carrie E. Finch, Sara M. Gabrielli, Derek J. Owens, Frank A. Sanacory, William H. Streyer, Adela Vraciu, and Brooks D. Willet.

• Pencils of forms and level Artinian algebras
by Anthony IARROBINO, Northeastern University
Abstract: Consider a type two graded level algebra A, quotient of the polynomial ring R=K[x1, ..., xr] in r variables, defined by an inverse system R ° < F,G>, F,G in Dj where D = KDP[X1, ....,Xr] denotes the ring of divided powers, upon which R acts by contraction. We have A=R/Ann(F,G). For each l in K, Fl = F+ lG determines a socle-degree j Gorenstein algebra quotient of A, namely Al = R/Ann(Fl). The family Fl is the pencil of degree-j forms in D determined by the two-dimensional vector space < F,G>. We give lower bounds for the Hilbert function H(Al), when l is generic, in terms of the Hilbert functions H(R/Ann(F)) and H(R/Ann(G)) or in terms of the Hilbert function H(A). We give several applications showing the impossibility of some candidate sequences to be Hilbert functions for type two level Artinian algebras. We also give an example of a compressed type two level Artinian algebra not having a compressed Gorenstein quotient of the same socle degree.

• On the first infinitesimal neighborhood of a k-configuration
by Juan MIGLIORE, University of Notre Dame
Abstract: It is an open problem to classify the possible Hilbert functions of the first infinitesimal neighborhood of a finite set of points in P2 (fat points). This is in contrast to the case of reduced points, where the Hilbert functions are precisely the differentiable O-sequences. All such Hilbert functions arise as the Hilbert function of a set of points in a k-configuration. Hence we study sets Z of fat points supported on a k-configuration X in P2. In certain cases, the minimal free resolution of IZ is uniquely determined by the type of the k-configuration (and not on the actual positioning of the points). In other cases the minimal free resolution is not uniquely determined, but the Hilbert function is. And in yet other cases not even the Hilbert function is uniquely determined. The trick is to determine when Z can be realized as the result of a sequence of basic double links. In the other direction: given any differentiable O-sequence H, we can produce a reduced set of points X with Hilbert function H, such that we can explicitly give the minimal free resolution (and hence the Hilbert function) of the set of fat points Z supported on X. We also give an example of two sets of fat points with the same Hilbert function, but whose supports have different Hilbert functions. This joint work with Anthony V. Geramita and Lousindi R. Sabourin.

• A normal form for space curves in a double plane
by Uwe NAGEL, University of Kentucky
Abstract: We discuss space curves that are contained in some double plane, i.e. in a quadric defined by the square of a linear form. Hartshorne and Schlesinger have studied such curves from a geometric point of view. Our goal is to explicitly relate the geometric and algebraic properties of these curves. We show in particular that the minimal generators of the homogeneous ideal of such a curve can be written in a very specific form. As applications we describe the minimal free resolution and characterize the possible Hartshorne-Rao modules of curves in a double plane as well as the minimal curves in their even Liaison classes. This is joint work with Nadia Chiarli and Silvio Greco.

• Applications of the purity of non-standard Frobenius
by Hans SCHOUTENS, College of Technology, CUNY
Abstract: Originating in the work of Hochster-Roberts on the Cohen-Macaulyness of rings of invariants, and of Mehta-Ramanathan on vanishing theorems for Schubert varieties, the purity of the Frobenius has proven to be a powerful tool in singularity theory in positive characteristic. For instance, Smith, Hara et al., have given a characterization of log-terminal singularities in terms of the purity of Frobenius. Smith also uses it to obtain Kawamata-Viehweg vanishing in positive characteristic. To make use of this tool in zero characteristic, one uses reduction mod p. However, in this reduction process, some information gets lost (especially when dealing with quotients), so that the results in zero characteristic remained unsatisfactory. This led Smith to conjecture that any GIT quotient Y of a Fano variety admits Kawamata-Viehweg vanishing: for a numerically effective line bundle L on Y, its higher sheaf cohomology is zero, and, if L is moreover big, then Hi(Y,L-1) vanishes for i < dim Y. I will indicate a proof of this conjecture making direct use of a characteristic zero version of Frobenius. This is achieved by constructing a faithfully flat cover Y¥ ® Y where Y¥ is obtained from an ultraproduct of projective varieties in positive characteristic.

• Asymptotic limits of local cohomology modules
by Hema SRINIVASAN, University of Missouri at Columbia
Abstract: For a standard graded ring R of dimension d and a homogenous ideal I of R, we investigage the asymptotic growth of the length of Extd (R/In, R) with respect to n. We show that in characteristic zero, when R is a normal domain of dimension d>1 and I is a proper homogenous ideal then this grows asymptotically as cnd for some constant c. This number c may not always be rational and hence the lengths of H0m(R/In) may not be a polynomial in n in general. This is joint work with Dale Cutkosky, Tai Ha, and Emanoil Theodorescu.

• On Extensions of modules
by Janet STRIULI, University of Kansas
Abstract: In this talk we study closely Yoneda's correspondence between short exact sequences and the Ext1 group. We prove a main theorem which gives conditions on the splitting of a short exact sequence after taking the tensor product with R/I, for any ideal I of R. As an application we prove a generalization of Miyata's Theorem. We introduce the notion of sparse module and we show that ExtR1(M,N) is a sparse module provided that there are finitely many isomorphism classes of maximal Cohen-Macaulay modules having multiplicity the sum of the multiplicities of M and N. We prove that sparse modules are Artinian.

• Almost Gorenstein rings
by Adela VRACIU, University of South Carolina
Abstract: There are (at least) three equivalent characterizations for the Gorenstein property of an Artinian ring R:
• for every ideal I, 0:(0:I)=I
• the canonical module w is isomorphic to R
• the ideal (0) is irreducible.
Each of these properties can be generalized in a natural way, giving rise to classes of rings that can be considered close to being Gorenstein. Concrete instances where these rings arise naturally are as specializations of rings of countable CM type modulo parameter ideals, or as quotients of Gorenstein Artinian rings by their socle. We give a variety of examples of such rings, and we explore the question of whether our generalizations of the three properties listed above are equivalent. This joint work with Craig Huneke.