Tight Closure Theory and Characteristic p Methods

2004 ◽  
pp. 181-210 ◽  
Author(s):  
Melvin Hochster ◽  
Graham J. Leuschke
Keyword(s):  
Tight Closure ◽  
Characteristic P ◽  
Closure Theory

scholarly journals A dual to tight closure theory

2014 ◽  
Vol 213 ◽  
pp. 41-75
Author(s):  
Neil Epstein ◽  
Karl Schwede
Keyword(s):  
Existence Results ◽  
Finite Ring ◽  
Vanishing Theorem ◽  
Tight Closure ◽  
Transformation Rules ◽  
Test Ideal ◽  
Dual Notion ◽  
Closure Theory

AbstractWe introduce an operation on modules over anF-finite ring of characteristicp. We call this operationtight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

scholarly journals Length of Local Cohomology in Positive Characteristic and Ordinarity

2018 ◽  
Vol 2020 (7) ◽  
pp. 1921-1932 ◽  
Author(s):  
Thomas Bitoun
Keyword(s):  
Characteristic Zero ◽  
Local Cohomology ◽  
Differential Operators ◽  
Positive Characteristic ◽  
Isolated Singularity ◽  
Local Cohomology Module ◽  
Perfect Field ◽  
Tight Closure ◽  
Characteristic P

Abstract Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.

scholarly journals A dual to tight closure theory

2014 ◽  
Vol 213 ◽  
pp. 41-75
Author(s):  
Neil Epstein ◽  
Karl Schwede
Keyword(s):  
Existence Results ◽  
Finite Ring ◽  
Vanishing Theorem ◽  
Tight Closure ◽  
Transformation Rules ◽  
Test Ideal ◽  
Dual Notion ◽  
Closure Theory

AbstractWe introduce an operation on modules over anF-finite ring of characteristicp. We call this operationtight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

scholarly journals Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

2021 ◽  
Vol 9 ◽  
Author(s):  
Benjamin Antieau ◽  
Bhargav Bhatt ◽  
Akhil Mathew
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Characteristic P ◽  
De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

Effect of oxygen vacancies and crystal symmetry on piezocatalytic properties of Bi2WO6 ferroelectric nanosheets for wastewater decontamination

2021 ◽  
Author(s):  
Zihan Kang ◽  
Enzhu Lin ◽  
Ni Qin ◽  
Jiang Wu ◽  
Baowei Yuan ◽  
...  
Keyword(s):  
Organic Pollutants ◽  
Oxygen Vacancies ◽  
Mechanical Energy ◽  
Crystal Symmetry ◽  
Characteristic P ◽  
Novel Technique ◽  
Effect Of Oxygen

Piezocatalysis emerged as a novel technique to make use of mechanical energy in dealing with organic pollutants in wastewater. In this work, the ferroelectric Bi2WO6 (BWO) nanosheets with a characteristic...

scholarly journals Mechanical colon obstruction due to the alimentary limb after Roux-en-Y gastric bypass: a case report

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Caspar Joyce Peterson ◽  
Jennifer Klasen ◽  
Tarik Delko ◽  
Romano Schneider
Keyword(s):  
Gastric Bypass ◽  
Bowel Obstruction ◽  
Transverse Colon ◽  
Bypass Surgery ◽  
Revisional Surgery ◽  
Gastric Bypass Surgery ◽  
Morbidly Obese ◽  
Tight Closure ◽  
Alimentary Limb ◽  
Colon Obstruction

Abstract Background Small bowel obstruction is a known and potentially lethal complication after gastric bypass surgery, in both the early and the late postoperative course. Colon or large bowel obstruction, on the other hand, seems to be rare after gastric bypass surgery and thus is not routinely considered. Case presentation We present the case of a 21-year old morbidly obese caucasian patient who underwent laparoscopic Roux-en-Y gastric bypass surgery and developed an early severe transverse colon obstruction due to compression of the transverse colon by the antecolic alimentary limb. Emergency revisional surgery showed a short and tense alimentary limb mesentery and possibly tight closure of Petersen’s space contributing to the compression. Through opening of Petersen’s space and mobilization of alimentary limb mesentery, decompression was achieved, and the patient fully recovered. Conclusions This is a rare case of colon obstruction caused by direct compression of the transverse colon by the antecolic alimentary limb. We propose that a combination of short tense alimentary limb mesentery and perhaps tight closure of Petersen’s space was responsible for the obstruction in this case. Surgeons and treating physicians need to be aware of such rare causes of early postoperative bowel obstruction and take these into consideration when evaluating patients.

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