Generalizing Serre’s Splitting Theorem and Bass’s Cancellation Theorem via free-basic elements

Abstract

We give new proofs of two results of Stafford from [Sta81], which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras. Theorem B. Let R be a commutative Noetherian ring and M a finitely generated R-module such that, for each p ∈ Spec(R), M p contains a free R p summand of rank at least dim(R/p)+ 1. For any finitely generated projective R-module Q and any finitely generatedStafford proved analogue results in [Sta81, Corollaries 5.9 and 5.11] where, instead of looking at local number of free summands, he used the notion of r-rank. Stafford's results are actually very general in that he proved analogues of Theorem A and Theorem B for rings that are not necessarily commutative.Let M be a finitely generated R-module. Observe that the conclusion of Theorem A can be viewed as a statement about the existence of a surjective homomorphism inside Hom R (M, R). For several applications, it is useful to restrict the selection of homomorphisms M → R to those belonging to a given R-submodule E of Hom R (M, R). This is the scenario arising, for instance, from the study of the F-signature of Cartier subalgebras of C R = ⊕ e Hom R (F e * R, R), where R is an F-finite local ring of prime characteristic [BST12]. We can naturally generalize the definitions in this paper and results about free-basic elements to the setup of R-submodules E of Hom R (M, R). The main achievement in this direction is a new generalization of Theorem A.Theorem C. Let R be a commutative Noetherian ring, M a finitely generated R-module, and E an R-submodule of Hom R (M, R). Assume that, for each p ∈ Spec(R), M p contains a free E p -summand of rank at least dim(R/p) + 1. Then M contains a free E -summand.As the main application of Theorem C, the authors establish in [DSPY16] the existence of a global F-signature with respect to Cartier subalgebras of C R , where R is an F-finite ring of prime characteristic, not necessarily local. Section 2, we present a version of Theorem A via rather direct and elementary techniques. However, the methods we employ are not effective enough to prove the result in its full generality. In Section 3, we introduce the notion of freebasic element, which allows us to prove Theorem A and Theorem B in their full generality. We believe that free-basic elements are worth exploring and interesting on their own, as they share many good properties both with basic elements and unimodular elements. In Section 4, we generalize free-basic elements to the setup of R-submodules of Hom R (M, R), and prove Theorem C.

Surjective capacity and splitting capacity

“…We remain faithful to this decision in our presentation of the analogous results of the next section. 6. Proofs of Theorems 0.5 and 0.9…”

“…These theorems have conclusions that are weaker than those of [6, Theorems 3.12 and 4.8], respectively, but the hypotheses of [6, Theorems 3.9 and 4.5] are more general. Later in this section, we state an analogue of [6, Theorems 3.9 and 4.5] and a generalization of [6,Theorems 3.12 and 4.8] in Theorem 1.13. We generalize [6, Theorems 3.9 and 4.5] in a separate paper [2] where our goal is to extend Bass's Cancellation Theorem.…”

“…Below, dim X (p) refers to the Krull dimension of Var(p) ∩ X, where p ∈ X ⊆ Spec(R) and Var(p) := {q ∈ Spec(R) : p ⊆ q}. The symbol j-Spec(R) refers to the set of all p ∈ Spec(R) such that p can be expressed as an intersection of maximal ideals of R. The set j-Spec(R) was introduced by Swan in [16] and was used by Eisenbud and Evans in [9] and by De Stefani, Polstra, and Yao in [6]. In the corollary following Proposition 1 in [16], Swan notes that j-Spec(R) is Noetherian if and only if Max(R) is Noetherian.…”

“…Theorem 0.7 (De Stefani, Polstra, and Yao [6,Theorem 3.12]). Let R be a commutative Noetherian ring, and let M be a finitely generated R-module.…”

Globalizing F-invariants

“…We now introduce some terminology in order to recall another Theorem from [DSPY16a]. Let R be a commutative Noetherian ring, M a finitely generated R-module.…”