GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 3, 1999, 259-262

A REGULARITY CRITERION FOR SEMIGROUP RINGS

W. BRUNS AND J. GUBELADZE

Abstract. An analogue of the Kunz–Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings.

Let S be a commutative semigroup (we always assume that S contains a neutral element), and K a field. For every m ∈ ℤ_{+} the assignment x ↦ x^{m}, x ∈ S, induces a K-endomorphism π_{m} of the semigroup ring R = K[S]. Therefore we can consider R as an R-algebra via π_{m}, and especially as an R-module. Let R^{[m]} denote R with its R-module structure induced by π_{m}. If S is finitely generated, then R^{[m]} is obviously a finitely generated R-module.

In this note we want to give a regularity criterion for S in terms of the homological properties of R^{[m]} that is analogous to Kunz’s [1] characterization of regular local rings of a characteristic p > 0 in terms of the Frobenius functor. Our criterion, which generalizes the result of Gubeladze [2, 10.2], requires only a mild condition on S and we provide a ‘pure commutative algebraic’ proof. (In [2] the result was stated for seminormal simplicial affine semigroup rings and derived from the main result of [2] that K_{1}-regularity implies the regularity for such rings.)

Theorem 1. Let S be a finitely generated semigroup, K a field, R = K[S], and m ∈ ℤ_{+}, m > 0. Suppose that S has no invertible element ≠ 1 and is generated by irreducible elements. Then the following conditions are equivalent:

(a) R^{[m]} has a finite projective dimension;

(b) R^{[m]} is a free module;

(c) S is free, in other words, S ≅ ℤ_{+}^{n} for some n ∈ ℤ_{+}. 259 Proof. It is obvious that (c) implies (b) and (b) implies (a). Now assume that (a) is satisfied. We first reduce the problem to a question of local algebra.

1991 Mathematics Subject Classification. Primary 13D05, 20M25; Secondary 13A35.

Key words and phrases. Kunz–Frobenius criterion, regularity, commutative semigroup rings.

1072-947X/99/0500-0259$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

Set T = S \ {1}. The ideal 𝔪 = T R is a maximal ideal of R. Indeed, R/𝔪 ≅ K. Furthermore the only prime ideal of 𝔮 of R such that 𝔪 = π_{m}^{−1}(𝔮) is 𝔪. Therefore π_{m}⊗R_{𝔪} is an endomorphism of R_{𝔪} that makes R_{𝔪} a finitely generated R_{𝔪}-module of finite projective dimension. In particular, R_{𝔪} has the same depth considered as an R_{𝔪}-module via π_{m} ⊗ R_{𝔪} as it has in its natural R_{𝔪}-module structure (for example, see [3, 1.2.26]. The Auslander– Buchsbaum formula [3, 1.3.3] thus implies that R_{𝔪} is a finite free module over itself via π_{m} ⊗ R_{𝔪}. The lemma below shows that R_{𝔪} is a regular local ring.

Let x_{1}, …, x_{n} be the irreducible elements of S. We claim that their images in R_{𝔪} form a minimal system of generators of the maximal ideal 𝔪R_{𝔪}. Indeed, consider a presentation

R^{r} ⟶^{φ} R^{n} ⟶^{ψ} 𝔪 ⟶ 0,

where the i-th element e_{i} of the natural basis of R^{n} is mapped to x_{i}. We must show that all the entries of the matrix φ are in 𝔪. Suppose on the contrary that there is a relation

a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0

with, for example, a_{1} ∉ 𝔪. Then a_{1} = α_{1} + α_{2}s_{2} + ⋯ + α_{u}s_{u} with α_{i} ∈ K, α_{i} ≠ 0, and s_{2}, …, s_{u} ∈ T. Writing a_{2}, …, a_{m} similarly, we see that there are only two possibilities, (i) x_{1} = s_{i}x_{1} for some i, or (ii) x_{1} = vx_{j} for some v ∈ S and j > 0. Both cases are impossible because x_{1} is irreducible.

However, R_{𝔪} is a regular local ring. Especially it is a factorial ring, in which the (images of the) x_{i} are pairwise non-associated prime elements. Therefore all the elements x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}}, e_{1}, …, e_{n} ∈ ℤ_{+} are pairwise different, and it follows that S ≅ ℤ_{+}^{n}.

Remark 2. (a) If we omit the hypothesis that S be generated by irreducible elements, then the proof above shows just the following: the subsemigroup generated by x_{1}, …, x_{n} ∈ S such that x_{1}, …, x_{n} form a minimal system of generators of the ideal 𝔪R_{𝔪} is free of rank n.

(b) One can weaken the hypothesis of the theorem by requiring only that the group S_{0} of invertible elements of S be a free abelian group. Then T = S \ S_{0} generates a prime ideal 𝔭 in R, and part (c) of the theorem must be replaced by the condition that S ≅ ℤ_{+}^{n} × ℤ^{q} for some n, q ∈ ℤ_{+}.

The following lemma is just an abstract version of Herzog’s argument [4] characterizing the modules of finite projective dimension in terms of the Frobenius functor.

Lemma 3. Let R be a Noetherian local ring with maximal ideal 𝔪. If there exists an endomorphism π of R with π(𝔪) ⊂ 𝔪^{2} and such that R is a flat R-module via π, then R is a regular local ring.

Proof. According to the criterion of Auslander–Buchsbaum–Serre [3, 2.2.7] we must show that k = R/𝔪 has finite projective dimenion as an R-module. Write R′ for R considered as an R-module via π, and let P be the functor that takes an R-module M to M ⊗ R′ considered as an R-module via the identification R = R′. We choose a minimal free resolution F of k,

F: ⋯ ⟶ F_{i+1} ⟶^{φ_{i+1}} F_{i} ⟶ ⋯ ⟶ F_{1} ⟶ k ⟶ 0.

One has P(R) = R, P(F_{i}) = F_{i}, and P(F) is the complex that we obtain from F by replacing all entries in its matrices by their images under π. By hypothesis, P(F) is again exact, and the exactness is preserved by an e-fold iteration of this process. Especially, P^{e}(F) is a free resolution of P^{e}(k) for all e > 0.

Let x_{1}, …, x_{t} ∈ 𝔪 be a maximal R-sequence. Then R̄ = R/(x_{1}, …, x_{t}) has projective dimension t, and so Tor_{i}^{R}(R̄, P^{e}(k)) = 0 for all i > t and e > 0. On the other hand, one can compute Tor_{i}^{R}(R̄, P^{e}(k)) by tensoring P^{e}(F) with R̄. Let B_{i} be the kernel of φ_{i}. Then for sufficiently large i and all e > 0 we have an exact sequence

0 ⟶ R̄ ⊗ P^{e}(B_{i+1}) ⟶ R̄ ⊗ F_{i+1} ⟶ R̄ ⊗ P^{e}(B_{i}) ⟶ 0.

Since we have chosen a maximal R-sequence, depth R̄⊗F_{i+1} = 0 if F_{i+1} ≠ 0. On the other hand, for e sufficiently large, P^{e}(B_{i+1}) ⊂ 𝔪^{2^{e}} R̄ ⊗ F_{i+1} and P^{e}(B_{i}) ⊂ 𝔪^{2^{e}} R̄ ⊗ F_{i} have a positive depth or are zero according to [4, Lemma 3.2]. This is a contradiction.

Aknowledgement

Joseph Gubeladze was supported by Alexander von Humboldt Foundation and CRDF grant #GM1–115.

References

1. E. Kunz, Characterizations of regular local rings of characteristic p. Amer. J. Math. 91(1969), 772–784.

2. J. Gubeladze, Nontriviality of SK_{1}(R[M]). J. Pure Appl. Algebra 104(1995), 169–190.

3. W. Bruns and J. Herzog, Cohen–Macaulay rings. Cambridge University Press, 1993.

4. J. Herzog, Ringe der Charakteristik p und Frobeniusfunktoren. Math. Z. 140(1974), 67–78.

(Recieved 25.06.1997)

Authors’ addresses:

Winfried Bruns Universität Osnabrück Fachbereich Mathematik/Informatik 49069 Osnabrück Germany

Joseph Gubeladze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia