A locally F-finite Noetherian domain that is not F-finite
aa r X i v : . [ m a t h . A C ] J un A LOCALLY F-FINITE NOETHERIAN DOMAIN THAT IS NOTF-FINITE
TIBERIU DUMITRESCU AND CRISTODOR IONESCU
Abstract.
Using an old example of Nagata, we construct a Noetherian ringof prime characteristic p, whose Frobenius morphism is locally finite, but notfinite. Introduction
Let p > p, that is they contain a field of characteristic p. For such a ring A, one can define theFrobenius morphism: F A : A → A, F A ( a ) = a p , a ∈ A. The Frobenius morphism is playing a major role in studying the properties of A. In[4, Th. 2.1] Kunz proved that a Noetherian local ring of characteristic p is regularif and only if the Frobenius morphism of A is flat. This was the starting point inthe study of singularities in characteristic p . An important property of Frobeniusis its finiteness. A ring A is called F-finite if the Frobenius morphism of A is afinite morphism, that is A is a finite A -module via F . Rings that are F -finite haveimportant properties. For example such rings are excellent, as it is proved by Kunz[5, Th. 2.5]. Conversely, a reduced Noetherian ring with F -finite total quotientring is excellent if and only if it is F -finite [2, Cor. 2.6]. Recall that an excellentring is a Noetherian ring A such that (see [6, Def. p. 260]):i) the formal fibers of A are geometrically regular;ii) any A -algebra of finite type has open regular locus;iii) A is universally catenary.Excellent rings were invented by Grothendieck, in order to avoid pathologies in thebehaviour of Noetherian local rings.In [1] Datta and Murayama asked the following: Question 1.
Let A be a Noetherian domain of prime characteristic p > . Supposethat for any prime ideal p of A , the localization A p is F-finite. Does it follow that A is F-finite? We show that a certain specialization of Nagata’s example [7, § p. All the rings will be commutative and unitary. Moreover, we fix a prime number p > . The example
Assume first that p is odd. Let K be an algebraically closed field of characteristic p and X an indeterminate. Consider the ring A := K [ X, p ( X + a ) + √ b ; a, b ∈ K, b = 0] . For each square radical above, we choose one of its two values. Hence the denomi-nator p ( X + a ) − √ b is not in our list. Remark 2. i) Note that p ( X + a ) = 1 p ( X + a ) + √ b (( X + a ) − b ) + √ b ∈ A. ii) It follows from i) that A is a fraction ring of K [ X, p ( X + a ) ; a ∈ K ], whichin turn is an integral extension of K [ X ].iii) By ii) A is at most one dimensional. Lemma 3.
The factor ring A/ ( X, √ X ) is isomorphic to K , while the factor ring A/ ( X ) is isomorphic to K [ Y ] / ( Y ) , where Y is an indeterminate.Proof. From Kneser’s Theorem [3, Th. 5.1] we obtain that √ X + c / ∈ K ( X, √ X + a, a ∈ K, a = c )for every c ∈ K (this also follows by adapting the well-known argument for √ p n / ∈ Q ( √ p , . . . , √ p n − ) , when p , . . . , p n are distinct primes). Consequently, we get aring isomorphism K [ X, T a , a ∈ K ]( T a − ( X + a ) , a ∈ K ) ≃ K [ X, p ( X + a ) , a ∈ K ]sending each indeterminate T a into p ( X + a ) , which extends to an isomorphism K [ X, T a , ( T a + √ b ) − , a, b ∈ K, b = 0]( T a − ( X + a ) , a ∈ K ) ≃ A. It follows that A/ ( X ) ≃ K [ T a , ( T a + √ b ) − , a, b ∈ K, b = 0]( T a − a , a ∈ K ) ≃ K [ T a , ( T a + √ b ) − , a, b ∈ K, b = 0]( T , T a − √ a , a ∈ K, a = 0) ≃ K [ T , ( √ a + √ b ) − , ( T + √ b ) − , a, b ∈ K − { } ]( T ) ≃ K [ T ]( T )so A/ ( X, √ X ) is isomorphic to K . Note that √ a + √ b is nonzero for a, b ∈ K − { } , by our initial one-value-choice for √ b . Also note that T + √ b is a unitmodulo T . (cid:3) Lemma 4.
The nonzero prime ideals of A are ( X + a, p ( X + a ) ) A with a ∈ K .In particular, A is a Noetherian domain of dimension one. LOCALLY F-FINITE NOETHERIAN DOMAIN THAT IS NOT F-FINITE 3
Proof.
By Lemma 3, ( X, √ X ) is the only prime ideal of A containing X . Bypart ii) of Remark 2, every nonzero prime ideal of A contains some X + a , so ourassertion follows from Lemma 3, performing a translation in K . The final assertionfollows from Cohen’s Theorem [6, Th. 3.4]. (cid:3) Proposition 5. A m is F-finite for each maximal ideal m of A .Proof. Performing a translation in K , it suffices to work with m = ( X, √ X ). Asnoted in Remark 2, ii), A is a fraction ring of B := K [ X, p ( X + a ) ; a ∈ K ] . For a ∈ K − { } , we have that X + a is a unit of A m so p ( X + a ) = ( p ( X + a ) ) p ( X + a ) pk ( X + a ) ( p − k ∈ ( A m ) p [ X, √ X ]where k is the integer 3( p − /
2. Hence B ⊆ ( A m ) p [ X, √ X ] ⊆ A m Let n = m A m ∩ B . Since A is a fraction ring of B , we get A m = B n ⊆ ( A m ) p [ X, √ X ] , therefore A m = ( A m ) p [ X, √ X ] . (cid:3) Lemma 6.
The regular locus of A is { (0) } .Proof. Suppose that A m is regular for some maximal ideal m of A . By our trans-lation argument, it follows that A m is regular for each maximal ideal m of A . Then A is normal, so √ X ∈ A. Hence X divides √ X in A , which contradicts Lemma3. (cid:3) Proposition 7. A is not F-finite.Proof. Appy Lemma 6 and [4, Cor. 2.3]. (cid:3)
From Propositions 5 and 7 we get:
Theorem 8.
A is a one-dimensional Noetherian domain that is not F-finite, suchthat A p is F-finite for any prime ideal p of A. Remark 9.
In characteristic two, a similar example can be constructed as follows.Let K be an algebraically closed field of characteristic two and X an indeterminate.Consider the ring B := K [ X, p ( X + a ) , p ( X + a ) + p ( X + a ) √ b + √ b ; a, b ∈ K, b = 0] . For each cube radical above, we choose one of its three values. By Kneser’s Theorem[3, Th. 5.1], it follows that[ K ( X, p X + a , ..., p X + a n ) : K ( X )] = 3 n whenever a ,..., a n are distinct elements of K , so we get a ring isomorphism K [ X, T a , ( T a + T a √ b + √ b ) − ; a, b ∈ K, b = 0]( T a − ( X + a ) , a ∈ K ) ≃ B TIBERIU DUMITRESCU AND CRISTODOR IONESCU sending each indeterminate T a into p ( X + a ) . It easily follows that B/ ( X ) ≃ K [ T ] / ( T ) and B/ ( X, √ X ) ≃ K . Now all arguments used above can be adaptedto show that B is locally F -finite but not F -finite. Acknowledgement.
We thank Monica Lewis for pointing us out a misleadingargument in a previous version on this paper.
References [1] R. Datta, T. Murayama, The property of being japanese (aka N-2) is not local,https://rankeya.people.uic.edu/Japanese is not local.pdf[2] R. Datta, K. Smith, Excellence in prime characteristic, Local and global methods inalgebraic geometry, Contemp. Math., 712(2018), 105-116[3] G. Karpilovsky, Topics in Field Theory, Notas de Matematica, 155, North Holland, 1989.[4] E. Kunz, Characterizations of regular local rings of characteristic p , Amer. J. Math.,91(1969), p. 772-784[5] E. Kunz, On noetherian rings of characteristic p , Amer. J. Math., 98(1976), p. 999-1013[6] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986.[7] M. Nagata, On the closedness of singular loci, Publ. Math. I.H.´E.S., 2 (1959), p. 5-12 Facultatea de Matematica si Informatica,University of Bucharest,14 Academiei Str.,Bucharest, RO 010014,Romania
E-mail address : tiberiu [email protected], [email protected], Simion Stoilow Institute of Mathematics of the Romanian Academy, P. O. Box 1-764,RO-014700 Bucharest, Romania
E-mail address ::