Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Detecting completeness from Ext-vanishing

Author(s): Anders J. Frankild; Sean Sather-Wagstaff
Journal: Proc. Amer. Math. Soc. 136 (2008), 2303-2312.
MSC (2000): Primary 13B35, 13D07, 13D25, 13D45, 13J10
Posted: February 28, 2008
MathSciNet review: 2390496
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and $\mathfrak{a}$ an ideal in the Jacobson radical of . Let $\widehat{\mathfrak{Ra}}$ be the $\mathfrak{a}$ -adic completion of . If is a finitely generated -module such that $\operatorname{Ext}^i_R(\,M)=0$ for all $i\neq 0$ , then is $\mathfrak{a}$ -adically complete.

References:

1.: L. Alonso Tarrıo, A. Jeremıas López, and J. Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 1, 1-39. MR 1422312 (98d:14028)
2.: L. L. Avramov and H.-B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), 129-155. MR 93g:18017
3.: R.-O. Buchweitz and H. Flenner, Power series rings and projectivity, Manuscripta Math. 119 (2006), no. 1, 107-114. MR 2194381 (2007a:13025)
4.: L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 2002e:13032
5.: -, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839-1883. MR 2002a:13017
6.: L. W. Christensen, A. Frankild, and H. Holm, On Gorenstein projective, injective and flat dimensions--a functorial description with applications, J. Algebra 302 (2006), no. 1, 231-279. MR 2236602
7.: H.-B. Foxby, Isomorphisms between complexes with applications to the homological theory of modules, Math. Scand. 40 (1977), no. 1, 5-19. MR 0447269 (56:5584)
8.: A. Frankild, Vanishing of local homology, Math. Z. 244 (2003), no. 3, 615-630. MR 1992028 (2004d:13027)
9.: S. I. Gelfand and Y. I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. MR 1438306 (97j:18001)
10.: J. P. C. Greenlees and J. P. May, Derived functors of -adic completion and local homology, J. Algebra 149 (1992), no. 2, 438-453. MR 1172439 (93h:13009)
11.: R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, Berlin, 1966. MR 36:5145
12.: C. U. Jensen, On the vanishing of $\underset{\longleftarrow}{\lim}^{(i)}$ , J. Algebra 15 (1970), 151-166. MR 0260839 (41:5460)
13.: -, Les foncteurs dérivés de $\varprojlim$ et leurs applications en théorie des modules, Lecture Notes in Mathematics, vol. 254, Springer-Verlag, Berlin, 1972. MR 0407091 (53:10874)
14.: J. Lipman, Lectures on local cohomology and duality, Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 39-89. MR 1888195 (2003b:13027)
15.: M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de ``platification'' d'un module, Invent. Math. 13 (1971), 1-89. MR 0308104 (46:7219)
16.: N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121-154. MR 932640 (89m:18013)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13B35, 13D07, 13D25, 13D45, 13J10

Retrieve articles in all Journals with MSC (2000): 13B35, 13D07, 13D25, 13D45, 13J10

Additional Information:

Anders J. Frankild
Affiliation: Department of Mathematics, Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København, Denmark

Sean Sather-Wagstaff
Affiliation: Department of Mathematics, California State University, Dominguez Hills, 1000 E. Victoria Street, Carson, California 90747
Address at time of publication: Department of Mathematics, North Dakota State University, 300 Minard Hall, Fargo, North Dakota 58105-5075
Email: sean.sather-wagstaff@ndsu.edu

DOI: 10.1090/S0002-9939-08-09199-5
PII: S 0002-9939(08)09199-5
Keywords: Completions, completeness, ext, local cohomology, local homology
Received by editor(s): June 28, 2006,
Received by editor(s) in revised form: March 28, 2007
Posted: February 28, 2008
Additional Notes: This research was conducted while the first author had a Steno Stipend from the Danish Research Council.
Dedicated: Dedicated to Lex Remington
Communicated by: Bernd Ulrich
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|