Hostname: page-component-6bb9c88b65-spzww Total loading time: 0 Render date: 2025-07-27T20:12:18.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert functions and the extension functor

Published online by Cambridge University Press:  04 October 2011

David Kirby
David Kirby
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Southampton S09 5NH
Get access Rights & Permissions [Opens in a new window]

Extract

Throughout R will denote a commutative ring with identity, A,B etc. will denote ideals of R, and E will denote a unitary R-module. We recall from [5] the definition of homological grade hgrR(A;E) as inf{r|ExtRr(R/A,E) ≠ 0}, and we allow both hgrR(A;E) = ∞ (i.e. ExtRr(R/A,E) = 0 for all r) and AE = E. For the most part E will be Noetherian, in which case hgrR(A;E) coincides with the usual grade grR(A;E) which is the supremum of the lengths of the (weak) E-sequences contained in A (see [7], for example).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 441 - 446
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Bass, H.. On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8–28.CrossRefGoogle Scholar
[2] Gröbner, W.. Über irreduzible Ideale in kommutativen Ringen. Math. Ann. 110 (1934), 197–222.CrossRefGoogle Scholar
[3] Gröbner, W.. Ein Irreduzibilitätskriterium für Primärideale in kommutativen Ringen. Monatsh. Math. 55 (1951), 138–145.CrossRefGoogle Scholar
[4] Kirby, D.. Artinian modules and Hilbert polynomials. Quart. J. Math. Oxford Ser. (2) 24 (1973), 47–57.CrossRefGoogle Scholar
[5] Kirby, D. and Mehran, H. A.. The homological grade of a module. Mathematika 35 (1988), 114–125.CrossRefGoogle Scholar
[6] Northcott, D. G.. On irreducible ideals in local rings. J. London Math. Soc. (2) 32 (1957), 82–88.CrossRefGoogle Scholar
[7] Northcott, D. G.. Finite Free Resolutions. Cambridge Tracts in Math. no. 71 (Cambridge University Press, 1976).CrossRefGoogle Scholar