Abstract
Let A be a regular ring containing a field of characteristic zero and let \(R = A[X_1,\ldots , X_m]\). Consider R as standard graded with \(\deg A = 0\) and \(\deg X_i = 1\) for all i. In this paper we present a comprehensive study of graded components of local cohomology s \(H^i_I(R)\) where I is an arbitrary homogeneous ideal in R. Our study seems to be the first in this regard.
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Bhattacharyya, R., Puthenpurakal, T.J., Roy, S., Singh, J.: Components of multigraded local cohomology modules. Work in progress
Björk, J.-E.: Rings of Differential Operators. North-Holland Mathematical Library, vol. 21. North Holland, Amsterdam (1979)
Brodmann, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics, vol. 60. Cambridge University Press, Cambridge (1998)
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39, Revised edn. Cambridge University Press, Cambridge (1997)
Cutkosky, S.D., Herzog, J.: Failure of tameness for local cohomology. J. Pure Appl. Algebra 211(2), 428–432 (2007)
Huneke, C., Sharp, R.: Bass numbers of local cohomology modules. Trans. AMS 339, 765–779 (1993)
Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131. Springer, New York (1991)
Lyubeznik, G.: Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra). Invent. Math. 113, 41–55 (1993)
Lyubeznik, G.: F-modules: applications to local cohomology and D-modules in characteristic \(p{\>}0\). J. Reine Angew. Math. 491, 65–130 (1997)
Ma, L., Zhang, W.: Eulerian graded D-modules. Math. Res. Lett. 21(1), 149–167 (2014)
Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Cambridge University Press, Cambridge (1989)
Puthenpurakal, T.J.: de Rham cohomology of local cohomology modules. In: Rizvi, S., Ali, A., Filippis, V. (eds.) Algebra and Its Applications. Springer Proceedings in Mathematics & Statistics, vol. 174, pp. 159–181. Springer, Singapore (2016)
Puthenpurakal, T.J.: De Rham cohomology of local cohomology modules–the graded case. Nagoya Math. J. 217, 1–21 (2015)
Puthenpurakal, T.J.: Graded components of local cohomology modules over invariant rings-II. arXiv:1712.09197
Puthenpurakal, T.J., Reddy, R.B.T.: de Rham cohomology of local cohomology modules II. Beitr. Algebra Geom. 60(1), 77–94 (2019)
Puthenpurakal, T.J., Reddy, R.B.T.: On a relation between de Rham cohomology of \(H^1_{(f)}(R)\) and the Koszul cohomology of \(\partial (f)\) in \(R/(f)\). Indian J. Pure Appl. Math. 49(1), 7–86 (2018)
Puthenpurakal, T.J., Roy, S.: Graded components of Local cohomology modules II. arXiv:1708.01396
Puthenpurakal, T.J., Roy, S.: Graded components of local cohomology modules of invariant rings. Commun. Algebra 48(2), 803–814 (2020)
Puthenpurakal, T.J., Singh, J.: On derived functors of graded local cohomology modules. Math. Proc. Camb. Philos. Soc. 167(3), 549–565 (2019)
Acknowledgements
In 2008 I asked Prof. G. Lyubeznik whether de Rham cohomology of local cohomology modules will be interesting. He told me that it will be of interest. I (and co-authors) developed techniques to study de Rham cohomology and Koszul cohomology of local cohomology modules in a series of papers [12, 13, 15, 16, 19]. These techniques have proved to be fantastically useful in this paper. I thank Prof. G. Lyubeznik for his advice and to him this paper is dedicated.I thank the referee for many pertinent comments. I thank Prof. C. Huneke for giving me extra time to submit a revised version of this paper.
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Dedicated to Prof. Gennady Lyubeznik.
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Puthenpurakal, T.J. Graded components of local cohomology modules. Collect. Math. 73, 135–171 (2022). https://doi.org/10.1007/s13348-020-00311-4
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DOI: https://doi.org/10.1007/s13348-020-00311-4
Keywords
- Local cohomology
- Graded local cohomology
- Ring of differential operators
- Weyl algebra
- De Rham (and Koszul)