Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Adic reduction to the diagonal and a relation between cofiniteness and derived completion


Author: Liran Shaul
Journal: Proc. Amer. Math. Soc. 145 (2017), 5131-5143
MSC (2010): Primary 13B35, 13C12, 13H15
DOI: https://doi.org/10.1090/proc/13662
Published electronically: June 8, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove two results about the derived functor of $ \mathfrak{a}$-adic completion: (1) Let $ \mathbb{K}$ be a commutative noetherian ring, let $ A$ be a flat noetherian $ \mathbb{K}$-algebra which is $ \mathfrak{a}$-adically complete with respect to some ideal $ \mathfrak{a}\subseteq A$, such that $ A/\mathfrak{a}$ is essentially of finite type over $ \mathbb{K}$, and let $ M,N$ be finitely generated $ A$-modules. Then adic reduction to the diagonal holds: $ A\otimes ^{\mathrm {L}}_{ A\widehat {\otimes }_{\mathbb{K}} A } (M\widehat {\otimes }^{\mathrm {L}}_{\mathbb{K}} N ) \cong M \otimes ^{\mathrm {L}}_AN$. A similar result is given in the case where $ M,N$ are not necessarily finitely generated. (2) Let $ A$ be a commutative ring, let $ \mathfrak{a}\subseteq A$ be a weakly proregular ideal, let $ M$ be an $ A$-module, and assume that the $ \mathfrak{a}$-adic completion of $ A$ is noetherian (if $ A$ is noetherian, all these conditions are always satisfied). Then $ \textrm {Ext}^i_A(A/\mathfrak{a},M)$ is finitely generated for all $ i\ge 0$ if and only if the derived $ \mathfrak{a}$-adic completion $ \mathrm {L}\widehat {\Lambda }_{\mathfrak{a}}(M)$ has finitely generated cohomologies over $ \widehat {A}$. The first result is a far-reaching generalization of a result of Serre, who proved this in case $ \mathbb{K}$ is a field or a discrete valuation ring and $ A = \mathbb{K}[[x_1,\dots ,x_n]]$.


References [Enhancements On Off] (What's this?)

  • [1] Leovigildo Alonso Tarrío, Ana Jeremías López, and Joseph Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 1, 1-39. MR 1422312, https://doi.org/10.1016/S0012-9593(97)89914-4
  • [2] Nicolas Bourbaki, Commutative algebra. Chapters 1-7, translated from the French, reprint of the 1972 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. MR 979760
  • [3] A. Grothendieck and J. A. Dieudonné, Eléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
  • [4] Tracy Dawn Hamilton and Thomas Marley, Non-Noetherian Cohen-Macaulay rings, J. Algebra 307 (2007), no. 1, 343-360. MR 2278059, https://doi.org/10.1016/j.jalgebra.2006.08.003
  • [5] Leif Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668. MR 2125457, https://doi.org/10.1016/j.jalgebra.2004.08.037
  • [6] Marco Porta, Liran Shaul, and Amnon Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31-67. MR 3160712, https://doi.org/10.1007/s10468-012-9385-8
  • [7] Marco Porta, Liran Shaul, and Amnon Yekutieli, Cohomologically cofinite complexes, Comm. Algebra 43 (2015), no. 2, 597-615. MR 3274024, https://doi.org/10.1080/00927872.2013.822506
  • [8] Peter Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), no. 2, 161-180. MR 1973941, https://doi.org/10.7146/math.scand.a-14399
  • [9] Sean Sather-Wagstaff and Richard Wicklein, Support and adic finiteness for complexes, Comm. Algebra 45 (2017), no. 6, 2569-2592. MR 3594539, https://doi.org/10.1080/00927872.2015.1087008
  • [10] Sean Sather-Wagstaff and Richard Wicklein, Extended local cohomology and local homology, Algebr. Represent. Theory 19 (2016), no. 5, 1217-1238. MR 3551316, https://doi.org/10.1007/s10468-016-9616-5
  • [11] Jean-Pierre Serre, Local algebra, translated from the French by CheeWhye Chin and revised by the author, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1771925
  • [12] Liran Shaul, Tensor product of dualizing complexes over a field. To appear in J. Commut. Algebra. arXiv:1412.3759v2, 2016.
  • [13] Liran Shaul, Hochschild cohomology commutes with adic completion, Algebra Number Theory 10 (2016), no. 5, 1001-1029. MR 3531360, https://doi.org/10.2140/ant.2016.10.1001
  • [14] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121-154. MR 932640

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13B35, 13C12, 13H15

Retrieve articles in all journals with MSC (2010): 13B35, 13C12, 13H15


Additional Information

Liran Shaul
Affiliation: Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheim Campus, Middelheimlaan 1, 2020 Antwerp, Belgium
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
Email: LShaul@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/proc/13662
Received by editor(s): February 23, 2016
Received by editor(s) in revised form: October 24, 2016, December 6, 2016, and January 10, 2017
Published electronically: June 8, 2017
Additional Notes: The author acknowledges the support of the European Union via ERC grant No. 257004-HHNcdMir.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society