Support and Injective Resolutions of Complexes over Commutative Rings

Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. These also give examples of semiinjective complexes whose localization need not be homotopically injective. Let R be a commutative noetherian ring. Recall that the support of a finitely generated R-module M is the set of prime ideals p in R such that Mp 6= 0. For arbitrary modules and, more generally, for complexes of modules, there are various possible notions of support. Among them it is by now clear that the right definition, from a homological perspective, is the one introduced by Foxby in [3], and recalled below. With this notion, Foxby [3, 2.8,2.9] proved that when X is a complex with H(X) = 0 for n ≪ 0, a prime p is in the support of X if and only if the injective hull of R/p appears in the minimal semi-injective resolution of X . The purpose of this note is to describe examples that show that such a result does not extend to arbitrary complexes, contrary to expectation; see Remark 2. Support. We write SpecR for the set of all the prime ideals in R. For each p in SpecR, the residue field Rp/pRp of the local ring Rp is denoted by k(p). The support of a complex X of R-modules is the subset suppR X = {p ∈ SpecR | H(X ⊗ L R k(p)) 6= 0} . This notion was introduced by Foxby [3, p.157], under the name ‘small support’, to distinguish it from the ‘big support’, namely, the set {p ∈ SpecR | H(X) p 6= 0}. They coincide when the R-module H(X) is finitely generated—see [3, 2.1]—but not in general. Also, suppR X and suppR H(X) need not coincide; see [2, 9.4]. For each R-module M we write assR M for the set of its associated primes and ER(M) for its injective hull; see Matsumura’s book [9, §§6,18]. Injective modules. Using [9, 18.4], it is easy to verify that the support ofER(R/p) equals {p}, which also equals assR ER(R/p). By the structure theorem for injective R-modules [9, 18.5] any injective R-module is of the form ⊕ p∈SpecR E(R/p) , where each μ(p) is a non-negative integer (possibly ∞) which depends only on E. It then follows that there are equalities: suppR E = {p ∈ SpecR | μ(p) 6= 0} = assR E . Date: 12th May 2009.