- Email: [email protected]
Corrigendum to “On the localization theorem for F -pure rings” [J. Pure Appl. Algebra 213 (2009) 1133–1139]
Journal of Pure and Applied Algebra 218 (2014) 504–505
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
Corrigendum
Corrigendum to ‘‘On the localization theorem for F -pure rings’’ [J. Pure Appl. Algebra 213 (2009) 1133–1139] K. Shimomoto a,∗ , W. Zhang b a
Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tama-Ku, Kawasaki 214-8571, Japan
b
Department of Mathematics, The University of Nebraska, Lincoln, NE, 68588, USA
article
info
Article history: Received 3 June 2013 Available online 25 July 2013 Communicated by A.V. Geramita
The assumption
ϕ : R → S satisfies the going-up property is missing from the statements of both Proposition 4.2 and Theorem 4.4. Before we explain why this assumption is needed, we recall our notation from our paper. Notation. R and S are noetherian excellent rings. ϕ : R → S is a flat map of finite type that satisfies the going-up property. Let N be an R-flat finite S-module. Let S ∗ N denote the trivial extension. Let ϕN : R → S → S ∗ N denote the composite map, and let ϕN∗ denote the induced map Spec(S ∗ N ) → Spec(R). Let U denote the MCM locus of S ∗ N and let W denote the complement of U in Spec(S ∗ N ). Let W ′ denote Spec(R)\ϕN∗ (W ). During the proof of Proposition 4.2 (on page 1138), the following was claimed: Proposition 3.7 implies that W ′ is stable under generization. It turns out that this claim requires the aforementioned assumption on ϕ : R → S in order to apply Proposition 3.7. And we include a proof here. Claim 1. W ′ is stable under generization. Proof. Assume that p ∈ W ′ and p′ ⊂ p. We wish to prove that p′ ∈ W ′ , i.e. the fiber ring of R → S ∗ N at p′ is MCM over S ⊗ κ(p′ ). If there is no prime of S lying over p′ , then we will have S ⊗ κ(p′ ) = 0 and hence the fiber ring at p′ will be MCM over S ⊗ κ(p′ ). Hence we may assume that there is a prime q′ of S lying over p′ . Since ϕ : R → S satisfies the going-up property, there is a prime q of S containing q′ and lying over p. Consider Rp → Sq → Sq ∗ Nq which is a flat local map. Since p ∈ W ′ , we know that the closed fiber of Rp → Sq ∗ Nq is MCM over S ⊗ κ(p). Since the fiber of R → S ∗ N at p′ is also a fiber of Rp → Sq ∗ Nq , Proposition 3.7 implies that the fiber ring of R → S ∗ N at p′ is MCM over S ⊗ κ(p′ ). This completes the proof of our claim. More generally, Proposition 4.2 and Theorem 4.4 hold true for a proper surjective morphism of excellent schemes, which is in a more geometric setting. Also, it was pointed out to us very recently by Karl Schwede, whom we thank, that Theorem 3.10 in our paper had been obtained by Mitsuyasu Hashimoto in [1].
∗
DOI of original article: http://dx.doi.org/10.1016/j.jpaa.2008.11.047. Corresponding author. E-mail addresses: [email protected] (K. Shimomoto), [email protected] (W. Zhang).
0022-4049/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jpaa.2013.06.018
K. Shimomoto, W. Zhang / Journal of Pure and Applied Algebra 218 (2014) 504–505
505
References [1] M. Hashimoto, Cohen–Macaulay F -injective homomorphisms, in: Geometric and Combinatorial Aspects of Commutative Algebra, Messina, 1999, in: Lecture Notes in Pure and Appl. Math., vol. 217, Dekker, New York, 2001, pp. 231–244.
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
Corrigendum
Corrigendum to ‘‘On the localization theorem for F -pure rings’’ [J. Pure Appl. Algebra 213 (2009) 1133–1139] K. Shimomoto a,∗ , W. Zhang b a
Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tama-Ku, Kawasaki 214-8571, Japan
b
Department of Mathematics, The University of Nebraska, Lincoln, NE, 68588, USA
article
info
Article history: Received 3 June 2013 Available online 25 July 2013 Communicated by A.V. Geramita
The assumption
ϕ : R → S satisfies the going-up property is missing from the statements of both Proposition 4.2 and Theorem 4.4. Before we explain why this assumption is needed, we recall our notation from our paper. Notation. R and S are noetherian excellent rings. ϕ : R → S is a flat map of finite type that satisfies the going-up property. Let N be an R-flat finite S-module. Let S ∗ N denote the trivial extension. Let ϕN : R → S → S ∗ N denote the composite map, and let ϕN∗ denote the induced map Spec(S ∗ N ) → Spec(R). Let U denote the MCM locus of S ∗ N and let W denote the complement of U in Spec(S ∗ N ). Let W ′ denote Spec(R)\ϕN∗ (W ). During the proof of Proposition 4.2 (on page 1138), the following was claimed: Proposition 3.7 implies that W ′ is stable under generization. It turns out that this claim requires the aforementioned assumption on ϕ : R → S in order to apply Proposition 3.7. And we include a proof here. Claim 1. W ′ is stable under generization. Proof. Assume that p ∈ W ′ and p′ ⊂ p. We wish to prove that p′ ∈ W ′ , i.e. the fiber ring of R → S ∗ N at p′ is MCM over S ⊗ κ(p′ ). If there is no prime of S lying over p′ , then we will have S ⊗ κ(p′ ) = 0 and hence the fiber ring at p′ will be MCM over S ⊗ κ(p′ ). Hence we may assume that there is a prime q′ of S lying over p′ . Since ϕ : R → S satisfies the going-up property, there is a prime q of S containing q′ and lying over p. Consider Rp → Sq → Sq ∗ Nq which is a flat local map. Since p ∈ W ′ , we know that the closed fiber of Rp → Sq ∗ Nq is MCM over S ⊗ κ(p). Since the fiber of R → S ∗ N at p′ is also a fiber of Rp → Sq ∗ Nq , Proposition 3.7 implies that the fiber ring of R → S ∗ N at p′ is MCM over S ⊗ κ(p′ ). This completes the proof of our claim. More generally, Proposition 4.2 and Theorem 4.4 hold true for a proper surjective morphism of excellent schemes, which is in a more geometric setting. Also, it was pointed out to us very recently by Karl Schwede, whom we thank, that Theorem 3.10 in our paper had been obtained by Mitsuyasu Hashimoto in [1].
∗
DOI of original article: http://dx.doi.org/10.1016/j.jpaa.2008.11.047. Corresponding author. E-mail addresses: [email protected] (K. Shimomoto), [email protected] (W. Zhang).
0022-4049/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jpaa.2013.06.018
K. Shimomoto, W. Zhang / Journal of Pure and Applied Algebra 218 (2014) 504–505
505
References [1] M. Hashimoto, Cohen–Macaulay F -injective homomorphisms, in: Geometric and Combinatorial Aspects of Commutative Algebra, Messina, 1999, in: Lecture Notes in Pure and Appl. Math., vol. 217, Dekker, New York, 2001, pp. 231–244.