Deconstructible classes of modules are among the main sources of approximations in relative homological algebra. They also occur in connection with abstract elementary classes (AECs). The latter were introduced by Shelah as far-reaching generalizations of classic first-order structures [2]. A direct connection is provided by the ‘AECs of roots of Ext’: these are the AECs of the form P = (𝒜,≼) where 𝒜 = { M in Mod-R such that ExtiR(M,N) = 0 for all i > 0 and all N in 𝒞 } for a class of modules 𝒞, and ≼ is a partial order on 𝒜 satisfying X ≼ Y iff Y/X is in 𝒜. By [1], P is an AEC iff 𝒜 is a deconstructible class closed under arbitrary direct limits. A major open problem concerning AECs is Shelah’s Categoricity Conjecture (SCC). It claims that categoricity of an AEC is a large enough cardinal λ (= existence of a unique structure in 𝒜 of cardinality λ up to isomorphism) is equivalent to its categoricity in a tail of cardinals. After recalling the role of deconstructible classes of modules, we will prove SCC for the AECs of roots of Ext, and more in general, for all ‘deconstructible’ AECs (𝒟,≤), i.e., such that 𝒟 is a deconstructible class of modules [4]. We will also consider the open problem of whether for all deconstructible AECs, the class 𝒟 is necessarily closed under direct limits. We will show that it is consistent with ZFC that 𝒟 is closed under countable direct limits provided that 𝒟 is closed under direct summands and ≤ refines direct summands [3].
References:
[1] J.T.Baldwin, P.C.Eklof, J.Trlifaj. ⊥N as an abstract elementary class. Annals of Pure Appl. Logic 149(2007), 25-39.
[2] S.Shelah. Classification Theory for Abstract Elementary Classes, vols. 1 and 2, Studies in Logic, no. 18 & 20, College Publ., London 2009.
[3] J.Saroch, J.Trlifaj. Deconstructible abstract elementary classes of modules. manuscript.
[4] J.Trlifaj. Categoricity for transfinite extensions of modules. arXiv:2212.04433v1.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
