Browsing Department of Mathematics (MU) by Author "Aberbach, Ian M."
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The depth of the associated graded ring of ideals with any reduction number
Aberbach, Ian M.; Ghezzi, Laura; Ha, Huy Tai (200212)Let R be a local CohenMacaulay ring, let I be an Rideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily CohenMacaulay. We assume that I is either equimultiple, ... 
Extension of weakly and strongly Fregular rings by flat maps
Aberbach, Ian M. (200208)Throughout this paper all rings will be Noetherian of positive characteristic p. Hence tight closure theory [HH14] takes a prominent place (see §2 for tight closure definitions and terminology). The purpose of this note ... 
Frational rings and the integral closure of ideals
Aberbach, Ian M.; Huneke, C. (Craig) (200010)The history of the BriançonSkoda theorem and its ensuing avatars in commutative algebra has been welldocumented in many papers. A theorem proved by Briançon and Skoda for convergent power series over the complex numbers ... 
The Fsignature and strong Fregularity
Aberbach, Ian M.; Leuschke, Graham J. (200211)We show that the Fsignature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly Fregular. 
Homology multipliers and the relation type of parameter ideals
Aberbach, Ian M.; Ghezzi, Laura; Ha, Huy Tai (200501)We study the relation type question, raised by C. Huneke, which asks whether for a complete equidimensional local ring R there exists a uniform bound for the relation type of parameter ideals. Wang gave a positive answer ... 
The structure of Fpure rings
Aberbach, Ian M.; Enescu, Florian (200310)For a reduced Ffinite ring R of characteristic p >0 and q=p^e one can write R^{1/q} = R^{a_q} \oplus M_q, where M_q has no free direct summands over R. We investigate the structure of Ffinite, Fpure rings R by studying ... 
Test ideals and flat base change problems in tight closure theory
Aberbach, Ian M.; Enescu, Florian (200210)Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably ... 
The vanishing of Tor_1^R(R^+,k) implies that R is regular
Aberbach, Ian M. (200309)Let (R,m,k) be an excellent local ring of positive prime characteristic. We show that if Tor_1^R(R^+,k) = 0 then R is regular. This improves a result of Schoutens, in which the additional hypothesis that R was an isolated ... 
When does the Fsignature exist?
Aberbach, Ian M.; Enescu, Florian (200502)We show that the Fsignature of an Ffinite local ring R of characteristic p > 0 exists when R is either the localization of an Ngraded ring at its irrelevant ideal or QGorenstein on its punctured spectrum. This extends ...