Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ D$-module and $ F$-module length of local cohomology modules


Authors: Mordechai Katzman, Linquan Ma, Ilya Smirnov and Wenliang Zhang
Journal: Trans. Amer. Math. Soc. 370 (2018), 8551-8580
MSC (2010): Primary 13D45; Secondary 13A35, 13C60
DOI: https://doi.org/10.1090/tran/7266
Published electronically: July 5, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a polynomial or power series ring over a field $ k$. We study the length of local cohomology modules $ H^j_I(R)$ in the category of $ D$-modules and $ F$-modules. We show that the $ D$-module length of $ H^j_I(R)$ is bounded by a polynomial in the degree of the generators of $ I$. In characteristic $ p>0$ we obtain upper and lower bounds on the $ F$-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of $ R/I$. The obtained upper bound is sharp if $ R/I$ is an isolated singularity, and the lower bound is sharp when $ R/I$ is Gorenstein and $ F$-pure. We also give an example of a local cohomology module that has different $ D$-module and $ F$-module lengths.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13D45, 13A35, 13C60

Retrieve articles in all journals with MSC (2010): 13D45, 13A35, 13C60


Additional Information

Mordechai Katzman
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: M.Katzman@sheffield.ac.uk

Linquan Ma
Affiliation: Department of Mathematics University of Utah Salt Lake City, Utah 84102
Email: lquanma@math.utah.edu

Ilya Smirnov
Affiliation: Department of Mathematics University of Michigan Ann Arbor, Michigan 48109-1043
Email: ismirnov@umich.edu

Wenliang Zhang
Affiliation: Department of Mathematics University of Illinois at Chicago Chicago, Illinois 60607
Email: wlzhang@uic.edu

DOI: https://doi.org/10.1090/tran/7266
Received by editor(s): September 19, 2016
Received by editor(s) in revised form: April 18, 2017
Published electronically: July 5, 2018
Additional Notes: This material is based partly upon work supported by the National Science Foundation under Grant No.1321794.\endgraf Some of this work was done at the Mathematics Research Community (MRC) in Commutative Algebra in June 2015.
The second author was partially supported by the National Science Foundation through grant DMS #1600198, and partially by the National Science Foundation CAREER Grant DMS #1252860/1501102.
The second, third, and fourth authors would like to thank the staff and organizers of the MRC and the American Mathematical Society for the support provided.
The fourth author was partially supported by the National Science Foundation through grant DMS #1606414.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society