## Local cohomology of Rees algebras and Hilbert functions

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- by Bernard Johnston and Jugal Verma
- Proc. Amer. Math. Soc.
**123**(1995), 1-10 - DOI: https://doi.org/10.1090/S0002-9939-1995-1217453-X
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## Abstract:

Let*I*be an ideal primary to the maximal ideal in a local ring. We utilize two well-known theorems due to J.-P. Serre to prove that the difference between the Hilbert function and the Hilbert polynomial of

*I*is the alternating sum of the graded pieces of the graded local cohomology (with respect to its positively-graded ideal) of the Rees ring of

*I*. This gives new insight into the higher Hilbert coefficients of

*I*. The result is inspired by one due to J. D. Sally in dimension two and is implicit in a paper by D. Kirby and H. A. Mehran, where very different methods are used.

## References

- Shiro Goto,
*Buchsbaum rings with multiplicity $2$*, J. Algebra**74**(1982), no. 2, 494–508. MR**647250**, DOI 10.1016/0021-8693(82)90035-7 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157**, DOI 10.1007/978-1-4757-3849-0 - William Heinzer, Bernard Johnston, David Lantz, and Kishor Shah,
*Coefficient ideals in and blowups of a commutative Noetherian domain*, J. Algebra**162**(1993), no. 2, 355–391. MR**1254782**, DOI 10.1006/jabr.1993.1261 - William Heinzer, Bernard Johnston, David Lantz, and Kishor Shah,
*The Ratliff-Rush ideals in a Noetherian ring: a survey*, Methods in module theory (Colorado Springs, CO, 1991) Lecture Notes in Pure and Appl. Math., vol. 140, Dekker, New York, 1993, pp. 149–159. MR**1203805** - William Heinzer, Bernard Johnston, and David Lantz,
*First coefficient domains and ideals of reduction number one*, Comm. Algebra**21**(1993), no. 10, 3797–3827. MR**1231634**, DOI 10.1080/00927879308824766 - M. Herrmann, S. Ikeda, and U. Orbanz,
*Equimultiplicity and blowing up*, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. MR**954831**, DOI 10.1007/978-3-642-61349-4 - Sam Huckaba,
*Reduction numbers for ideals of higher analytic spread*, Math. Proc. Cambridge Philos. Soc.**102**(1987), no. 1, 49–57. MR**886434**, DOI 10.1017/S0305004100067037 - Craig Huneke,
*Hilbert functions and symbolic powers*, Michigan Math. J.**34**(1987), no. 2, 293–318. MR**894879**, DOI 10.1307/mmj/1029003560 - David Kirby and Hefzi A. Mehran,
*Hilbert functions and the Koszul complex*, J. London Math. Soc. (2)**24**(1981), no. 3, 459–466. MR**635877**, DOI 10.1112/jlms/s2-24.3.459 - Thomas Marley,
*The coefficients of the Hilbert polynomial and the reduction number of an ideal*, J. London Math. Soc. (2)**40**(1989), no. 1, 1–8. MR**1028910**, DOI 10.1112/jlms/s2-40.1.1 - M. Morales,
*Polynôme d’Hilbert-Samuel des clôtures intégrales des puissances d’un idéal $m$-primaire*, Bull. Soc. Math. France**112**(1984), no. 3, 343–358 (French, with English summary). MR**794736**, DOI 10.24033/bsmf.2009 - Akira Ooishi,
*Genera and arithmetic genera of commutative rings*, Hiroshima Math. J.**17**(1987), no. 1, 47–66. MR**886981** - Judith D. Sally,
*Reductions, local cohomology and Hilbert functions of local rings*, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 231–241. MR**693638** - Peter Schenzel,
*Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe*, J. Algebra**64**(1980), no. 1, 93–101 (German). MR**575785**, DOI 10.1016/0021-8693(80)90136-2 - Jean-Pierre Serre,
*Faisceaux algébriques cohérents*, Ann. of Math. (2)**61**(1955), 197–278 (French). MR**68874**, DOI 10.2307/1969915

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 1-10 - MSC: Primary 13D45; Secondary 13A30, 13D40
- DOI: https://doi.org/10.1090/S0002-9939-1995-1217453-X
- MathSciNet review: 1217453