Approximation of subcategories by abelian subcategories

arXiv:1006.0048v4 [math.CT] 7 May 2023 APPROXIMATION OF SUBCATEGORIES BY ABELIAN SUBCATEGORIES. A. SALCH 1. Introduction. The earliest circulated versions of this paper date from 2010, and stem from ideas I had earlier, while I was a student. Several versions of this preprint have been made available, but the main idea in each of them is a very simple statement: given an ideal I of a commutative ring R satisfying very mild hypotheses, the category of L0- complete R-modules is the smallest abelian subcategory of Mod(R) containing all the I-adically complete R-modules and satisfying a few reasonable conditions1. As a slogan, “the category of L0-complete modules is the best abelian approximation to the category of I-adically complete modules.” See Theorem 3.3, below, for a precise statement. In this short note, I prove the main theorem in close to its original level of generality. That level of generality is sufficient for every application I have ever had for the theorem, and it is also sufficient for every application I have ever seen anyone else have for the theorem. In this level of generality, the theorem is dramatically easier to prove than in the more general settings, as you can see from how short this note is! To me it seems that this short note is not worth sending to a journal, but I think it is worth having on the arXiv. None of the various versions of this paper were published, but in one version of this paper (the most general version) that I circulated some years ago, I believe there is a hypothesis missing from some of the statements, so I think it is worthwhile to post this short note, to serve as the simple, straightforward, and easily-seen-to-be-correct “version of record” for this theorem. 2. The relevant definitions. Definition 2.1. Given an ideal I in a commutative ring R, we write Λ for the I-adic completion functor Λ : Mod(R) →Mod(R), i.e., Λ(M) = limn M/InM. We write L0Λ for the zeroth left-derived functor L0Λ : Mod(R) →Mod(R) of Λ. Since I-adic completion is, in general, not right exact, L0Λ(M) does not neces- sarily coincide with Λ(M). Definition 2.2. We say that an R-module M is L0-complete if the canonical map ηM : M →L0ΛM is an isomorphism. We write L0Λ Mod(R) for the full subcate- gory of Mod(R) whose objects are the L0-complete modules. See sections A.2 and A.3 of [3] for an excellent introduction to L0-completion and L0-complete modules, including proofs of many basic properties. 1You can take those conditions to be “replete,” “exact,” and “full.” It is not difficult to also prove variants of that result, replacing exactness with reflectivity, for example. 1 2 A. SALCH 3. The theorem. Throughout, let R be a commutative ring, and let I be a weakly pro-regular ideal in R. Many early references on local homology and derived completion, such as [2] and appendix A of [3], assumed that R is Noetherian, or that I is generated by a regular sequence. In [1] and [6], it was established that weak pro-regularity of I is sufficient for the proofs of most of the fundamental results in the area. Every weakly pro-regular ideal is finitely generated. Rather than reproduce the rather technical definition of weak pro-regularity here, I prefer to simply cite the result of [6] which states that, in a Noetherian commutative ring, every ideal is weakly pro-regular. Consequently, in most practical situations, one knows that the ideals one encounters in examples are weakly pro-regular. The papers [5] and [4] have valuable treatments of properties of L0Λ when I is weakly pro-regular. For example, Theorem 3.9(a) of [4] establishes that: Theorem 3.1. If I is weakly pro-regular, then the full subcategory L0Λ Mod(R) of Mod(R) is abelian, and the inclusion functor ι : L0Λ Mod(R) →Mod(R) is exact. Theorem 3.1 also appears in references that pre-date [4], although generally with stronger assumptions: for example, compare Theorem A.6 of [3], which is similar but includes the assumptions that R is Noetherian and that I is regular. It is also straightforward (e.g. see Proposition 3.7 in [4]) to prove that: Lemma 3.2. If I is weakly pro-regular, then every I-adically complete R-module is L0-complete. With Theorem 3.1 and Lemma 3.2 in hand, we now have little trouble in proving the main theorem: Theorem 3.3. Let I be a weakly pro-regular ideal in a commutative ring R. Then the category L0Λ Mod(R) of L0-complete modules is the unique smallest replete2 exact3 full subcategory of Mod(R) containing all the I-adically complete R-modules. Proof. Theorem 3.1 establishes that L0Λ Mod(R) is a replete exact full subcategory of Mod(R), while Lemma 3.2 establishes that L0Λ Mod(R) contains the I-adically complete R-modules. So suppose that A is a replete exact full subcategory of Mod(R) which contains the I-adically complete R-modules. We must then prove that A contains L0Λ Mod(R). Suppose that X is an L0-complete R-module. Choose an exact sequence P1 d −→P0 −→X →0 in Mod(R), with P0, P1 projective R-modules. Applying Λ, we have a short e

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