Finitistic weak dimension of commutative arithmetical rings

It is proven that each commutative arithmetical ring R has a finitistic weak dimension ≤ 2. More precisely, this dimension is 0 if R is locally IF, 1 if R is locally semicoherent and not IF, and 2 in the other cases.

• R is locally coherent [7,Theorem 3.3]; • R contains a prime ideal L such that L R L is non-zero and T-nilpotent (a slight generalization of [2, Theorem 6.4] using [1, Theorems P and 6.3]).
They conjectured that these conditions can be removed. Recall that each arithmetical ring is Gaussian.
In this paper, we only investigate the finitistic weak dimension of arithmetical rings (it seems that it is more difficult for Gaussian rings). Main results are summarized in the following theorem: Let P be a ring property. We say that a ring R is locally P if R P satisfies P for each maximal ideal P. As in [8], a ring R is said to be semicoherent if Hom R (E, F) is a submodule of a flat R-module for any pair of injective R-modules E, F. A ring R is said to be IF (semi-regular in [8]) if each injective R-module is flat. If R is a chain ring, we denote by P its maximal ideal, Z its subset of zerodivisors which is a prime ideal and Q(= R Z ) its fraction ring. If x is an element of a module M over a ring R, we denote by (0 : x) the annihilator ideal of x and by E(M) the injective hull of M.
Since flatness is a local module property, Theorem 1 is an immediate consequence of the following theorem that we will prove in the sequel. Theorem 2 Let R be a chain ring. Then:

Proposition 3 A ring R is IF if and only if it is coherent and self FP-injective.
The following proposition will be used frequently in the sequel.

Proposition 4 Let R be a chain ring. The following conditions are equivalent:
(1) R is semicoherent; (2) Q is an IF-ring; (3) Q is coherent; (4) either Z = 0 or Z is not flat. The first assertion of Theorem 2 is an immediate consequence of the following proposition:

Lemma 6 Let R be a chain ring. Then, for each non-zero element a of P, (0 : a) is a module over Q and it is a flat R/(a)-module.
Proof Let x ∈ (0 : a) and s a regular element of R. Since R is a chain ring x = sy for some y ∈ R, and 0 = ax = say. It follows that ay = 0. Hence the multiplication by s in (0 : a) is bijective.
If c ∈ R we denote byc the coset c + (a). Any element of (c) ⊗ R/(a) (0 : a) is of the formc ⊗ x where x ∈ (0 : a). Suppose that c / ∈ (a) and cx = 0. There exists t ∈ R such that a = ct = 0. Since R is a chain ring, from cx = 0 and ct = 0 we deduce that x = t y for some y ∈ R. We have ay = ct y = cx = 0. So, y ∈ (0 : a) andc ⊗ x =ā ⊗ y = 0. Hence (0 : a) is flat over R/(a). Conversely, let 0 = a ∈ R. By Lemma 6 (0 : a) is a module over Q. So, from the exact sequences we deduce for each integer q ≥ 1 the following isomorphisms: So, if p > 1 we get that w.d.(M Z ) ≤ p implies w.d.(M) ≤ p. Now assume that p = 1. Then, for each a ∈ R we have: In the following commutative diagram

Lemma 8 Let R be a chain ring which is not semicoherent. Then Z ⊗ R M is flat for each R-module M satisfying w.d.(M) ≤ 2.
Proof Consider the following flat resolution of M: If a ∈ Z we have Tor R p ((0 : a), M) ∼ = Tor R p+2 (R/(a), M) = 0 for each integer p ≥ 1. So, the following sequence is exact:  (0 : a). So, if S is the sequence By [6,Theorem I.8.13(a)] S is pure-exact. By Proposition 4 Z is flat. Consequently Z ⊗ R F 0 is flat and so is Z ⊗ R M. Now, we can prove the third assertion of Theorem 2.

Proof of Theorem 2(3)
Assume that R is not semicoherent. By [3,Proposition 14], the exact sequence 0 → Z → Q → Q/Z → 0 induces the following one for each non-zero element a of Z : From these two exact sequences we deduce the following one: It is well known that Q is flat. By (1) R/A is semicoherent; (2) A is either prime or the inverse image of a non-zero proper principal ideal of R A by the natural map R → R A ; Using [3, Theorem 11] we deduce the following: Corollary 10 Let A be a non-zero proper ideal of a chain ring R. The following conditions are equivalent: (2) either A = P or A is finitely generated; From these corollaries we deduce the following examples: Example 11 Let R be a valuation domain which is not a field. Let a be a non-zero element of its maximal ideal P. Then R/a R is IF, so f.w.d.(R/a R) = 0. Assume that R contains an idempotent prime ideal L = P. If a is a non-zero element of L, then aL is a non-finitely generated ideal of R L . So, R/aL is not semicoherent and f.w.d.(R/aL) = 2. It is well known that f.w.d.(R) = w.gl.d(R) = 1. Moreover, assume that R contains a non-zero prime ideal L = P. Then, if 0 = a ∈ L then a R L is an ideal of R and R/a R L is semicoherent and not IF. So, f.w.d.(R/a R L ) = 1.
Recall that a chain ring R is strongly discrete if there is no non-zero idempotent prime ideal. From Theorem 1 and [4, Corollary 5.4] we deduce the following: Corollary 12 Let R be an arithmetical ring. The following conditions are equivalent: (1) R is locally strongly discrete; (2) for each proper ideal A, R/A is locally semicoherent; (3) for each proper ideal A, f.w.d.(R/A) ≤ 1.
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