# KUMUNU-05 2003-09-13

## KUMUNU 5 - September 13, 2003

The talks will be given in 120 Snow Hall.

Time | Speaker | Title |
---|---|---|

11:00 - 11:40 | Claudia Polini University of Notre Dame |
Normalization and integral closure of ideals |

11:50 - 12:30 | Ray Heitmann University of Texas-Austin |
Closure Operations and Colon-Capturing |

12:30 - 2:20 | Lunch break in 406 Snow Hall | |

2:20 - 3:00 | Hema Srinivasan University of Missouri-Columbia |
Asymptotic behavior of local cohomology |

3:10 - 3:50 | Roger Wiegand University of Nebraska-Lincoln |
Direct-sum decompositions of non-Cohen-Macaulay modules |

4:00 - 4:40 | Laura Ghezzi University of Missouri-Columbia |
Completions of valuation rings |

4:50 - 5:30 | Daniel Katz University of Kansas |
Reduction criteria for ideals and modules |

6:30 - | Dinner at Craig Huneke's house |

**Normalization and integral closure of ideals**

Claudia Polini

This is joint work with Ulrich and Vasconcelos. Finding the integral closure of an ideal $I$ is a fundamental problem. The only theoretical approach is through the Rees algebra ${\mathcal R}$ of $I$ -- it requires to compute the normalization $\overline{\mathcal R}$ of ${\mathcal R}$. In the first part of the talk we measure the complexity of this construction by relating it to the Hilbert coefficients of the filtrations of the integral closures of the powers of $I$. We then bound these coefficients through the Brian\c{c}on-Skoda number of the ambient ring $R$. In the second part of the talk we present a strategy to compute the integral closure of $I$, or at least part of it, by a {\it direct} construction, i.e. via an algorithm whose steps take place entirely in the ambient ring. Indeed, by imposing strong residual properties on $I$, we approximate the integral closure of $I$ by a residual intersection.

**Closure Operations and Colon-Capturing**

Ray Heitmann

The objective of extending many homological theorems to the mixed characteristic case can be achieved by developing a mixed characteristic analogue of tight closure. Restricting consideration to integral domains, we would like a closure operation for all local domains whose residue field has characteristic p and which coincides with tight closure when the domain is equicharacteristic and excellent. We will look at several candidates and their properties, focusing most on the colon-capturing property.

**Asymptotic behavior of local cohomology**

Hema Srinivasan

R is a polynomial ring over a field $k$ in $d$ variables and $m$ is the irrelevant maximal ideal. For any proper homogenous ideal $I$ of $R$, we will discuss the growth of the length of local cohomology modules $H^0_m(R/I^n)$ for large $n$ and show that the ratio ${\lambda(H^0_m(R/I^n))}\over {n^d}$ has a limit as $n$ tends to infinity and that this limit is not always rational. Thus, $\lambda(H^0_m(R/I^n))$ will not in general be a polynomial in $n$.

**Direct-sum decompositions of non-Cohen-Macaulay modules**

Roger Wiegand

In the 1980's, Sylvia Wiegand and I developed machinery for studying the torsion-free cancellation problem in dimension one: When does $M \oplus L \cong N \oplus L$ imply $M \cong N$? Here $M, N$ and $L$ are finitely generated torsion-free modules over a one-dimensional Noetherian domain $R$ with finite integral closure. Over the next twenty years, Guralnick, Klingler, Levy and others made considerable progress on the analogous question for mixed modules (finitely generated modules that are neither torsion nor torsion-free), but no general theory emerged. Recently, Wolfgang Hassler and I, intrigued by the nagging question of whether a ring $R$ as above having torsion-free cancellation actually has cancellation for {\it all} finitely generated modules, developed analogous machinery for handling the mixed (non-CM) case. Although most of the theory of torsion-free modules extends to the general case, the answer to the question above is ``no''. While cancellation is really a global question, most of the work is at the local level, and a key to our negative answer to the nagging question is the construction of indecomposable mixed modules of rank two over rings like $k[[t^2,t^3]]$. (At this point it is unknown whether this ring has indecomposables of arbitrarily large rank, but we suspect that it does.)

**Completions of valuation rings**

Laura Ghezzi

This is joint work with S.D. Cutkosky.

Let $k$ be a field of characteristic zero, $K$ an algebraic function field over $k$, and $V$ a $k$-valuation ring of $K$. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings $R_i$ with quotient field $K$ which are dominated by $V$, and such that the direct limit $\cup R_i=V.$

We investigate the ring $T=\cup \hat R_i$. The ring $T$ is Henselian and thus can be considered to be a "completion" of the valuation ring $V$. We compare this completion with other notions of completion of a valuation ring. We give an example showing that $T$ is in general not a valuation ring.

**Reduction criteria for ideals and modules**

Daniel Katz

Let (R,m) be a quasi-unmixed local ring. The celebrated Rees multiplicity theorem states that if J and I are m primary ideals, with J contained in I, then J is a reduction of I if and only if J and I have the same multiplicity. This theorem has been generalized in a number of ways over the years by numerous authors, including Rees himself. The purpose of this talk is to give a survey of some of these generalizations, including some recent variations by Flenner-Manaresi and Katz-Trung.