aa r X i v : . [ m a t h . A C ] F e b A note on factorial A -forms with retractions Prosenjit Das
Stat-Math Unit, Indian Statistical Institute,203, B.T. Road, Kolkata 700 108, India.
Abstract
Let k be a field. In this paper we will show that any factorial A -form A overany k -algebra R is trivial if A has a retraction to R . Keywords: A -form, Factorial domain, Retraction AMS Subject classifications (2000) . Primary 13B25; Secondary 12F05, 13B10
Throughout this paper, by “ring”, we shall mean “commutative ring withunity”.
Definitions.
Let k be a field, R a k -algebra and A an R -algebra. A is saidto be an A -form over R (with respect to k ) if there exists an algebraic fieldextension k ′ | k such that A ⊗ k k ′ ∼ = ( R ⊗ k k ′ )[ X ]. An A -form is called purelyinseparable (resp. separable ), if we can take the extension k ′ | k to be a purelyinseparable (resp. a separable) extension.A ring homomorphism Φ : A −→ R is said to be a retraction if Φ is R -linear. If a retraction Φ : A −→ R exists, we say R is a retract of A .It is well known that any separable A -form over any field is trivial.More generally, it has been shown that a separable A -form over an arbitrarycommutative algebra is trivial ([Dut00], Theorem 7), i.e., Theorem 1.
Let k be a field, L a separable field extension of k , R a k -algebra and A an R -algebra such that A ⊗ k L ∼ = Sym ( R ⊗ k L ) ( P ′ ) for a finitelygenerated rank one projective module P ′ over R ⊗ k L . Then A ∼ = Sym R ( P ) for a finitely generated rank one projective module P over R . If k is not perfect, there exist non-trivial purely inseparable A -forms.Asanuma gave a complete structure theorem for purely inseparable A -formsover a field k of characteristic p > A -form over a field k with a k -rational point is trivial, i.e., Email address: [email protected] (Prosenjit Das)
Preprint submitted to Elsevier February 7, 2020 heorem 2.
Let k be a field and A an A -form over k such that (1) A is a UFD. (2) A has a k -rational point.Then A ∼ = k [ X ] . In this paper we shall show that Theorem 2 has a generalization, in thespirit of Theorem 1, to A -forms over k -UFDs (see Corollary 4 for preciseformulation). The generalization turns out to be a special case of our maintheorem (Theorem 3) which may be envisaged as an example of “faithfullyflat descent”. Theorem 3.
Let R be a ring and A be an R -algebra such that (1) A is a UFD. (2) There is a retraction
Φ : A −→ R . (3) There exists a faithfully flat ring homomorphism η : R −→ R ′ suchthat A ⊗ R R ′ ∼ = R ′ [ X ] .Then A ∼ = R [ X ] .Proof. Let A ′ = A ⊗ R R ′ = R ′ [ f ], P = Ker Φ, P ′ = P A ′ (= P ⊗ R R ′ ), andlet Φ ′ = Φ ⊗ A ′ to R ′ . Then P is a primeideal of A and we have a short exact sequence of R -modules0 −→ P −→ A Φ −→ R −→ R ′ -modules0 −→ P ′ −→ A ′ (= R ′ [ f ]) Φ ′ −→ R ′ −→ . Then P ′ = Ker Φ ′ = ( f − Φ ′ ( f )) R ′ [ f ]. Replacing f − Φ ′ ( f ) by f , weassume P ′ = f A ′ = f R ′ [ f ]. Since A ′ is faithfully flat over A , going-downtheorem holds between A and A ′ ([Mat89], Pg. 68, Theorem 9.5) and also P ′ ∩ A = P ([Mat89], Pg. 49, Theorem 7.5). As ht ( P ′ ) = 1, it follows that ht ( P ) = 1.Now, since A is a UFD, there exists g ∈ A such that P = gA . Thus weget gA ′ = P ′ = f A ′ . Since f is a non-zero divisor in A ′ (= R ′ [ f ]), it followsthat g = λf for some unit λ in A ′ . Let λ = a + a f + a f + · · · + a n f n where a is a unit in R ′ and a i is nilpotent in R ′ for 1 ≤ i ≤ n . Let I = ( a , a , · · · , a n ) R ′ . Then I is a nilpotent ideal of R ′ . Let N be the leastpositive integer such that I N = (0). Since g ≡ a f (mod I ), we have R ′ [ f ] = R ′ [ g ] + IR ′ [ f ] = · · · = R ′ [ g ] + I N R ′ [ f ] = R ′ [ g ] . R [ g ] ⊆ A and R [ g ] ⊗ R R ′ = R ′ [ g ] = A ⊗ R R ′ . Since R ′ isfaithfully flat over R , it follows that A = R [ g ] ∼ = R [ X ].As an immediate consequence of Theorem 3, we get the following corol-lary: Corollary 4.
Let k be a field, R a k -algebra and A an R -algebra such that (1) A is a UFD. (2) R is a retract of A . (3) A is an A -form over R .Then A ∼ = R [ X ] . The following two well-known examples ([KMT74], Pg. 70–71, Remark6.6(a), Examples (i) and (ii)) respectively show that in Theorem 2 (andhence in Theorem 3), the hypothesis on the existence of a retraction and thehypothesis “ A is a UFD” are necessary. Example 1.
Let F p be the prime field of characteristic p and let k = F p ( t, u )be a purely transcendental extension of F p with variables t and u . Then A = k [ X, Y ] / ( Y p − t − X − uX p ) is a factorial non-trivial A -form over k which does not have a retraction to k . Example 2.
Let k be a field of characteristic p ≥ A = k [ X, Y ] / ( Y p − X − aX p ) where a ∈ k \ k p . Then A is a non-trivial A -form over k with aretraction to k . Here A is not a UFD. Acknowledgement:
In the earlier version of the article, Corollary 4 was presented as the maintheorem. I thank the referee for his observation that the arguments give themore general statement Theorem 3.I thank Amartya K. Dutta for suggesting the problem and for his usefulcomments and suggestions. I am grateful to T. Asanuma for his lectures onpurely inseparable A -forms in Indian Statistical Institute, Kolkata. I alsothank S. M. Bhatwadekar and N. Onoda for helpful discussions. References [Asa05] Teruo Asanuma,
Purely inseparable k -forms of affine algebraiccurves , Affine algebraic geometry, Contemp. Math., vol. 369,Amer. Math. Soc., Providence, RI, 2005, pp. 31–46.[Dut00] Amartya K. Dutta, On separable A -forms , Nagoya Math. J. (2000), 45–51. 3KMT74] Tatsuji Kambayashi, Masayoshi Miyanishi, and MitsuhiroTakeuchi, Unipotent algebraic groups , Lecture Notes in Mathemat-ics, Vol. 414, Springer-Verlag, Berlin, 1974.[Mat89] Hideyuki Matsumura,