报告一题目：Nearly Gorenstein rings and the trace of the canonical module

报告人：Jürgen Herzog  德国杜伊斯堡-埃森大学教授

报告时间：2017年3月21日（周二）下午2:30-3:30

报告地点：维格堂113

报告摘要:  According to a famous paper of Bass, Gorenstein rings are ubiquitous.

Apart from this, they admit beautiful symmetryproperties: self dual resolutions, symmetric h-vectors, and,by a theorem of Kunz, among the numerical semigroup rings they are those withsymmetric semigroup.There have been several nice ideas to define properties ofrings which are sightly weaker than that of being Gorenstein. Barucci and Fröberg, forexample, defined almost symmetric numerical semigroups

and Goto, Takahashi and Taniguchi generalized thisconcept and introduced almost Gorenstein rings. Since then, almost Gorensteinrings have been studied in numerous papers.

In a recent paper, together with Stamate, weconsidered the trace of the canonical module w_R of a localCohen-Macaulay ring (R;m). The significance of this trace is that it describes the non-Gorensteinlocus of R. Thus R is Gorenstein ifand only if tr(w_R) = R. If the trace of w_R  comes very close to R, namely if mí tr(w_R), we call R nearly Gorenstein.

In this lecture we discuss nearly Goreenstein ringsfor Segre products and Veronese rings. The theory will be applied to give afull classification of all Hibi rings which are nearly Gorenstein in terms ofthe underlying poset. Furthermore, we discuss the relationship of almost Gorensteinto nearly Gorenstein and list several open problems.

报告二题目：Almost Gorenstein rings

报告人：Jürgen Herzog  德国杜伊斯堡-埃森大学教授

报告时间：2017年3月21日（周二）下午4:00-5:00

报告地点：维格堂113

报告摘要:  In my lecture I will talk about Cohen-Macaulay local rings of dimensionone; especially, about almost Gorenstein rings with slightly generalizeddefinition. The basic theory and the characterization of almost Gorensteinrings in terms of the first Hilbert coefficients e₁ (I) of canonicalideals I shall be described. Examples of almost Gorenstein rings which are notGorenstein will be given in the case where the rings are analytically ramified.

   Originally,almost Gorenstein rings were introduced in 1997 by Valentina Barucci

and Ralf Fröberg, in thecase where the rings are analytically unramified. They developed an interestingtheory of almost Gorenstein rings and gave many inspiring

results. In 2009, Valentina published one morepaper, and provided one of their

results with a counterexample. Valentina might feelsome gap in their proof. However, the counterexample itself is wrong, and afterslight modification in their proof, the beautiful result holds true in fullgenerality. To see this, we however need to generalize the notion of almost Gorensteinring to the case where the rings are not necessarily analytically unramified, whichI would like to talk about in my lecture.