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Rigidity of Ext and Tor with Coefficients in Residue Fields of a Commutative Noetherian Ring

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  16 November 2018

Lars Winther Christensen ,
Srikanth B. Iyengar  and
Thomas Marley
Lars Winther Christensen
Affiliation:
Texas Tech University, Lubbock, TX 79409, USA (lars.w.christensen@ttu.edu)
Srikanth B. Iyengar
Affiliation:
University of Utah, Salt Lake City, UT 84112, USA (iyengar@math.utah.edu)
Thomas Marley
Affiliation:
University of Nebraska-Lincoln, Lincoln, NE 68588, USA (tmarley1@unl.edu)
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Abstract

Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k (𝔭), M) = 0 for some n ⩾ R𝔭, where k(𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $(k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.

Keywords

ExtTorrigidityflat dimensioninjective dimension

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D05: Homological dimension

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 305 - 321
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Alonso Tarrío, L., Jeremías López, A. and Lipman, J., Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30(1) (1997), 1–39.Google Scholar
2.Avramov, L. L. and Foxby, H.-B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71(2–3) (1991), 129–155.Google Scholar
3.Benson, D. J., Iyengar, S. B. and Krause, H., Colocalizing subcategories and cosupport, J. Reine Angew. Math. 673 (2012), 161–207.Google Scholar
4.Chen, X.-W. and Iyengar, S. B., Support and injective resolutions of complexes over commutative rings, Homology, Homotopy Appl. 12(1) (2010), 39–44.Google Scholar
5.Chouinard, L. G. II, On finite weak and injective dimension, Proc. Amer. Math. Soc. 60 (1976), 57–60 (1977).Google Scholar
6.Christensen, L. W. and Iyengar, S. B., Tests for injectivity of modules over commutative rings, Collect. Math. 68 (2017), 243–250.Google Scholar
7.Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (2002), 199–220Google Scholar
8.Fossum, R., Foxby, H.-B., Griffith, P. and Reiten, I., Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193–215.Google Scholar
9.Foxby, H.-B. and Iyengar, S., Depth and amplitude for unbounded complexes, Commutative algebra (Grenoble/Lyon, 2001), Contemporary Mathematics, Volume 331, pp. 119–137 (American Mathematical Society, Providence, RI, 2003).Google Scholar
10.Gruson, L. and Jensen, C. U., Dimensions cohomologiques reliées aux foncteurs $\mathop{\lim}\limits_{\longleftarrow} (i)$, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980), Lecture Notes in Mathematics, Volume 867, pp. 234–294 (Springer, Berlin, 1981).Google Scholar
11.Jensen, C. U., On the vanishing of $\mathop{\lim}\limits_{\longleftarrow} (i)$, J. Algebra 15 (1970), 151–166.Google Scholar
12.Lam, T. Y. and Reyes, M. L., A prime ideal principle in commutative algebra, J. Algebra 319 (2008), 3006–3027.Google Scholar
13.Lipman, J., Lectures on local cohomology and duality, Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Applied Mathematics, Volume 226, pp. 39–89 (Dekker, New York, 2002).Google Scholar
14.Osofsky, B. L., Homological dimension and the continuum hypothesis, Trans. Amer. Math. Soc. 132 (1968), 217–230.Google Scholar
15.Raynaud, M. and Gruson, L., Critères de platitude et de projectivité. Techniques de ‘platification’ d'un module, Invent. Math. 13 (1971), 1–89.Google Scholar
16.Simon, A.-M., Some homological properties of complete modules, Math. Proc. Cambridge Philos. Soc. 108(2) (1990), 231–246.Google Scholar
17.Yassemi, S., Width of complexes of modules, Acta Math. Vietnam. 23(1) (1998), 161–169.Google Scholar