$D$-module and $F$-module length of local cohomology modules
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- by Mordechai Katzman, Linquan Ma, Ilya Smirnov and Wenliang Zhang PDF
- Trans. Amer. Math. Soc. 370 (2018), 8551-8580 Request permission
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
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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
- MR Author ID: 1050700
- Email: lquanma@math.utah.edu
- Ilya Smirnov
- Affiliation: Department of Mathematics, University of Michigan , Ann Arbor, Michigan 48109-1043
- MR Author ID: 949874
- Email: ismirnov@umich.edu
- Wenliang Zhang
- Affiliation: Department of Mathematics, University of Illinois at Chicago , Chicago, Illinois 60607
- MR Author ID: 805625
- Email: wlzhang@uic.edu
- 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. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 8551-8580
- MSC (2010): Primary 13D45; Secondary 13A35, 13C60
- DOI: https://doi.org/10.1090/tran/7266
- MathSciNet review: 3864387