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Bouchiba, Samir; Kabbaj, Salah

On the finiteness of Ext-indices of ring extensions. (English) Zbl 1507.13013

Rocky Mt. J. Math. 52, No. 6, 1969-1981 (2022).
Recall that Ext-index\((R)\) for a commutative ring \(R\), is defined as \(\sup\{ n\in \mathbb N \mid \{\mathrm{Ext}_{R}^{n}(M,N)\ne 0\}\), where the supremum is taken over all pairs \((M,N)\) of finite \(R\)-modules with Ext\(_{R}^{n}(M,N)= 0\) for all \(i \gg 0\). Quoting the authors: “The main goal of this paper is to investigate the finiteness of Ext-indices of ring extensions. We discuss some known related conjectures in the literature and observe the relationships among them within large classes of rings. This allows us to present interesting special cases verifying these conjectures.”
Here are some of the results in this paper. If \(R\) is a Noetherian ring, then \(\dim(R)\le \text{Ext-index}(R)\le\sup\{\text{Ext-index}(R_{\mathfrak m} )\mid \mathfrak m \in \text{Max}(R)\}\). The equality \(\text{Ext-index}(R) = \dim(R)\) holds if \(R\) is a locally AB ring, in particular, if \(R\) is a locally complete intersection. If \(R\) is a Gorenstein ring of finite Krull dimension, then \(\text{Ext-index}(R[X_{1}, X_{2},\dots, X_{n}])\) is finite if and only if so is \(\text{Ext-index}(R)\). The authors study in this context trivial ring extensions of Artinian rings.
Reviewer: Moshe Roitman (Haifa)

MSC:

13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H05 Regular local rings

Keywords:

Artinian ring; AB ring; locally AB ring; complete intersection ring; Ext-index; Gorenstein ring; Cohen-Macaulay ring
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References:

[1] D. D. Anderson and M. Winders, “Idealization of a module”, J. Commut. Algebra 1:1 (2009), 3-56. · Zbl 1194.13002 · doi:10.1216/JCA-2009-1-1-3
[2] T. Araya and Y. Yoshino, “Remarks on a depth formula, a grade inequality and a conjecture of Auslander”, Comm. Algebra 26:11 (1998), 3793-3806. · Zbl 0906.13002 · doi:10.1080/00927879808826375
[3] M. Auslander, Selected works of Maurice Auslander, edited by I. Reiten et al., Collected Works 10, American Mathematical Society, Providence, RI, 1999. · Zbl 1007.01022
[4] L. L. Avramov and R.-O. Buchweitz, “Support varieties and cohomology over complete intersections”, Invent. Math. 142:2 (2000), 285-318. · Zbl 0999.13008 · doi:10.1007/s002220000090
[5] L. L. Avramov and O. Veliche, “Stable cohomology over local rings”, Adv. Math. 213:1 (2007), 93-139. · Zbl 1127.13012 · doi:10.1016/j.aim.2006.11.012
[6] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993. · Zbl 0788.13005
[7] L. W. Christensen and H. Holm, “Algebras that satisfy Auslander’s condition on vanishing of cohomology”, Math. Z. 265:1 (2010), 21-40. · Zbl 1252.16008 · doi:10.1007/s00209-009-0500-4
[8] L. W. Christensen and H. Holm, “Vanishing of cohomology over Cohen-Macaulay rings”, Manuscripta Math. 139:3-4 (2012), 535-544. · Zbl 1255.13009 · doi:10.1007/s00229-012-0540-7
[9] D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995. · Zbl 0819.13001 · doi:10.1007/978-1-4612-5350-1
[10] R. Fossum, “Commutative extensions by canonical modules are Gorenstein rings”, Proc. Amer. Math. Soc. 40 (1973), 395-400. · Zbl 0271.13013 · doi:10.2307/2039380
[11] S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics 1371, Springer, 1989. · Zbl 0745.13004 · doi:10.1007/BFb0084570
[12] A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, II”, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5-231. · Zbl 0135.39701
[13] J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics 117, Marcel Dekker, New York, 1988. · Zbl 0637.13001
[14] C. Huneke and D. A. Jorgensen, “Symmetry in the vanishing of Ext over Gorenstein rings”, Math. Scand. 93:2 (2003), 161-184. · Zbl 1062.13005 · doi:10.7146/math.scand.a-14418
[15] D. A. Jorgensen and L. M. Şega, “Nonvanishing cohomology and classes of Gorenstein rings”, Adv. Math. 188:2 (2004), 470-490. · Zbl 1090.13009 · doi:10.1016/j.aim.2003.11.003
[16] S.-E. Kabbaj and N. Mahdou, “Trivial extensions defined by coherent-like conditions”, Comm. Algebra 32:10 (2004), 3937-3953. · Zbl 1068.13002 · doi:10.1081/AGB-200027791
[17] I. Kaplansky, Commutative rings, University of Chicago Press, 1974. · Zbl 0296.13001
[18] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986. · Zbl 0603.13001
[19] I. Mori, “Symmetry in the vanishing of Ext over stably symmetric algebras”, J. Algebra 310:2 (2007), 708-729. · Zbl 1134.16003 · doi:10.1016/j.jalgebra.2005.07.009
[20] M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics 13, Interscience Publishers, New York-London, 1962. · Zbl 0123.03402
[21] S. Nasseh and Y. Yoshino, “On Ext-indices of ring extensions”, J. Pure Appl. Algebra 213:7 (2009), 1216-1223. · Zbl 1163.13007 · doi:10.1016/j.jpaa.2008.11.034
[22] B. Olberding, “A counterpart to Nagata idealization”, J. Algebra 365 (2012), 199-221. · Zbl 1262.13031 · doi:10.1016/j.jalgebra.2012.05.002
[23] I. Palmér, “The global homological dimension of semi-trivial extensions of rings”, Math. Scand. 37:2 (1975), 223-256. · Zbl 0327.16018 · doi:10.7146/math.scand.a-11603
[24] M. Raynaud and L. Gruson, “Critères de platitude et de projectivité: Techniques de “platification” d’un module”, Invent. Math. 13 (1971), 1-89. · Zbl 0227.14010 · doi:10.1007/BF01390094
[25] I. Reiten, “The converse to a theorem of Sharp on Gorenstein modules”, Proc. Amer. Math. Soc. 32 (1972), 417-420. · Zbl 0235.13016 · doi:10.2307/2037829
[26] H. Schoutens, “Existentially closed models of the theory of Artinian local rings”, J. Symbolic Logic 64:2 (1999), 825-845. · Zbl 1060.03056 · doi:10.2307/2586504
[27] R. Y. Sharp, “The dimension of the tensor product of two field extensions”, Bull. London Math. Soc. 9:1 (1977), 42-48. · Zbl 0345.13007 · doi:10.1112/blms/9.1.42
[28] B. Stenström, “Coherent rings and \[FP\]-injective modules”, J. London Math. Soc. (2) 2 (1970), 323-329. · Zbl 0194.06602 · doi:10.1112/jlms/s2-2.2.323
[29] M. Tousi and S. Yassemi, “Tensor products of some special rings”, J. Algebra 268:2 (2003), 672-676. · Zbl 1087.13506 · doi:10.1016/S0021-8693(03)00105-4
[30] E. Valtonen, “Some homological properties of commutative semitrivial ring extensions”, Manuscripta Math. 63:1 (1989), 45-68 · Zbl 0669.13003 · doi:10.1007/BF01173701
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