ON THE FINITENESS OF EXT-INDICES OF RING EXTENSIONS SAMIR BOUCHIBA AND SALAH KABBAJ Abstract . 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. Finally, we tackle questions on the vanishing of Ext-index of trivial extensions of Artinian rings. We put the new results into use to construct new families of examples subject to the Gorenstein or Cohen-Macaulay conditions with finite Ext-index. 1. Introduction Throughout, all rings considered are commutative with identity elements and all modules are unital. A Noetherian local ring (R, m) is regular if its (Krull) dimen- sion, dim(R), and embedding dimension, embdim(R), coincide, where embdim(R) denotes the dimension of m/m 2 as an R/m-vector space. The ring R is a complete intersection if the completion ˆ R of R with respect to the m-adic topology is the quo- tient ring of a local regular ring modulo an ideal generated by a regular sequence. The ring R is Gorenstein if its injective dimension (as an R-module) is finite; and Cohen-Macaulay if the grade and height coincide for every ideal of R. All these notions are globalized by carrying over to localizations with respect to the prime ideals. For basic details on these notions, we refer the reader to [7, 10, 13, 20, 23]. In 1958, Auslander conjectured that every Artinian algebra R satisfies the fol- lowing condition (called Auslander’s condition): (ac) For every finite R-module M, there exists an integer n M ≥ 0 such that for every finite R-module N: Ext i R (M, N) = 0 ∀ i ≫ 0 = ⇒ Ext i R (M, N) = 0 ∀ i ≥ n M + 1. This conjecture generated numerous research works and gave birth to several related homological problems and homologically defined classes of commutative rings in the literature over the last decades [3, p. 795] (see also [2, 8, 9, 18]). It’s reported in [18] that all commutative local rings, in which this conjecture is valid satisfy the following stronger condition (called the uniform Auslander condition): (uac) There exists an integer n ≥ 0 such that for all finite R-modules M and N: Ext i R (M, N) = 0 ∀ i ≫ 0 = ⇒ Ext i R (M, N) = 0 ∀ i ≥ n + 1. In 2000, Avramov & Buchweitz [4] developed geometric methods for the study of finite modules over locally complete intersection rings. Their results yielded unexpected and remarkable properties of finite modules over these rings [4]. They Date: July 30, 2021. 2010 Mathematics Subject Classification. 13H05, 13F20, 13B30, 13E05, 13D05, 14M05, 16E65. Key words and phrases. Artinian ring, AB ring, locally AB ring, complete intersection ring, Ext-index, Gorenstein ring, Cohen-Macaulay ring. 1 22 Sep 2021 08:13:37 PDT 210922-Bouchiba Version 1 - Submitted to Rocky Mountain J. Math.