Arithmetic Analogues of Derivations

JOURNAL OF ALGEBRA 198, 9099 1997 ARTICLE NO. JA977177 Arithmetic Analogues of Derivations Alexandru Buium Department of Math and Statistics, Uniersity of New Mexico, Albuquerque, New Mexico 87131 Communicated by Susan Montgomery Received January 31, 1997 In 1 the author introduced an arithmetic analogue of derivations, called -derivations; this concept was used to prove a series of arithmetic analogues of results about algebraic differential equations 1,, 3. Although the usefulness of this concept is probably well illustrated by these papers, the question arises of how natural these -derivation operators are and what other choices one has in defining arithmetic analogues of usual derivations. The present note is an attempt to answer this question: we will introduce an a priori quite general notion of jet operator on a ring and prove that any such operator on a local domain of characteristic zero is equivalent to one of the following: a difference operator, a derivation operator, a -difference operator Žcf. the definition in the following text., or a -derivation operator in the sense of 1. The first three kinds of operators are always trivial on the rational integers; so one is left with the fourth kind, i.e., with the -derivations of 1, as the only possibility, within our paradigm, for an arithmetic analogue of usual derivations. Throughout this paper rings are always assumed to be commutative, with unit element, and all ring homomorphisms preserve units. DEFINITIONS. Let A be a ring. A set theoretic map : A A will be called an operator on A if there exist two polynomials S, P AX,X,Y,Y such that for any x, y A we have 0 1 0 1 001-869397 $5.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. Ž xy. SŽ x, x x,y. Ž xy. PŽ x, y,x,y.. 90

ARITHMETIC ANALOGUES OF DERIVATIONS 91 We say that and the pair Ž S, P. belong to each other. We say that is generic on A if, whenever F AX,X is a polynomial such that 0 1 FŽ x,x. 0, x A, we must have F 0. By a generic extension of we understand an operator on a ring extension A of A such that the following hold: 1. coincides with on A,. There exists a pair Ž S, P. belonging to both and, and 3. is generic on A. An operator on A will be called a jet operator if Ž. 0 Ž. 1 0 and admits a generic extension. ŽThe terminology will be justified by the Remark in the following text.. Two operators 1, : A A are said to be equialent if there exist an invertible element A and a polynomial f AX such that fž. 0 f10 Ž. and 1Ž x. Ž x. fž x., xa. If 1 and are equivalent then 1 is a jet operator if and only if is a jet operator. EXAMPLES. There are four remarkable Ž series of. operators which we want to single out; under very mild assumptions on A these operators are actually jet operators Ž cf. the Remark in the following text.. In what follows A will be any ring. Ž. a Recall that a map : A A is called a difference operator if it satisfies Ž x y. x y Ž xy. x y y x x y. If A has no nontrivial idempotents then is a difference operator if and only if the map x x x is a ring homomorphism. Difference operators are the basis for difference algebra 5, 4. ŽIn these references difference operators are simply defined to be ring homomorphisms; for A without nontrivial idempotents, the two definitions lead, of course, to equivalent theories. We preferred a definition in which Ž. 1 0.. Ž b. Recall that a map : A A is called a deriation if it satisfies Ž x y. x y Ž xy. x y y x Derivations are the basis for differential algebra 8, 6.

9 ALEXANDRU BUIUM Ž. c Let A be noninvertible. A map : A A will be called a -difference operator if it satisfies Ž x y. x y Ž xy. x y y x xy. If A has no nontrivial idempotents and is a nonzero divisor then is a -difference operator if and only if the map x x x is a ring homomorphism. Ž d. Let A be noninvertible, assume * A is such that * p where p is a prime integer, and let q 1 be an integer power of p. Consider the polynomial with integral coefficients, q q q C Ž X,Y. X Y Ž XY. p. q A map : A A will be called a -deriation 1 if it satisfies Ž x y. x y *CqŽ x, y., Ž xy. x q y y q x xy. If A has no nontrivial idempotents and is a nonzero divisor then is a -derivation if and only if the map x x q x is a ring homomorphism. Remark. If A has no nontrivial idempotents and is a nonzero divisor then all operators considered in the foregoing examples are jet operators. Indeed for any such one trivially checks that Ž. 0 Ž. 1 0. On the other hand, if is of one of the four kinds of operators considered in the previous examples, then we may consider the ring, Žn. AAT,T,T,...,T,..., of polynomials in infinitely many indeterminates and we may extend to an operator on A, of the same kind, such that T T, T T,...,T Žn. T Žn1.,.... The possibility of the extension is well known and trivial to check in examples Ž. a and Ž.Ž b cf. Cohn 5 and Ritt and Kolchin 8, 6., and it is also trivial to check in examples Ž. c and Ž. d. Indeed, consider the ring homo- q morphism : A A, x x x in case c and x x x in case Ž d., extend to a ring endomorphism of A by letting Ž T. T T, Žn. Žn. Žn1. Ž T. T T,...,Ž T. T T,...,

ARITHMETIC ANALOGUES OF DERIVATIONS 93 Ž. in case c and q T T T, q q Žn. Žn. Žn1. Ž T. Ž T. T,...,Ž T. Ž T. T,..., in case Ž d., and finally define : A A by the formula Ž f. Ž Ž f.. Ž q f in case c and f f f. in case Ž d.. These formulae make sense since we assumed that is a nonzero divisor in A. Now is generic on A. Indeed assume F AX,X is such that FŽ x,x. 0 1 0 for Žn. all x A. Then let n be such that F AT,T,T,...,T X, X 0 1 and take x T Ž n1. to conclude that F 0. Žn. The ring AT,T,T,...,T,... should be viewed as the ring of jets of the affine line ; this justifies our terminology. It is worth noting that, in case A Z, Z Ž p., Z p, p, of the four kinds of operators in the preceding text, p-derivations are the only nontrivial Ž i.e., nonidentically zero. ones. Here is our main result. THEOREM. Assume A is a local integral domain of characteristic zero. Then any jet operator on A is equialent to one of the following: a difference operator, a deriation operator, a -difference operator, or a -deriation operator. COROLLARY. Any jet operator on Z Ž or on Z. Ž p. p is equialent to either 0 or to an operator of the form, x x p m x. p The rest of the paper is devoted to the proof of the theorem. We start with a useful lemma. LEMMA 1. Let A be any ring. Assume : A A is an operator and : AA is a generic extension. Assume F AX,..., X, X,..., X 1, 0 m,0 1,1 m,1 is a polynomial in m ariables such that for all x,..., x A. Then F 0. 1 m ž 1 m 1 m/ F x,..., x,x,...,x 0, Proof. Trivial, by induction on m. Note that the lemma immediately implies that, in its hypothesis, there exists exactly one pair Ž S, P. belonging to both and.

94 ALEXANDRU BUIUM LEMMA. Let A be any ring. Assume : A A is a jet operator and : AA is a generic extension. Let Ž S, P. be the Ž unique. pair belonging to and. Then for any A-algebra B, the formulae, Ž x, x. Ž y, y. x y, SŽ x, y, x, y., Ž 1. 0 1 0 1 0 0 0 0 1 1 Ž x, x. Ž y, y. x y, PŽ x, y, x, y., Ž. 0 1 0 1 0 0 0 0 1 1 define a ring structure on B B such that the zero and unit elements in B B are Ž 0, 0. and Ž 1, 0., respectiely. Proof. Let us check for instance that the addition on B B is associative. We must check that the following two polynomials in 6 variables, S X Y, Z,SŽ X,Y, X,Y., Z and 0 0 0 0 0 1 1 1 S X,Y Z, X,SŽ Y, Z,Y, Z. 0 0 0 1 0 0 1 1 are equal. By Lemma 1, it is sufficient to check that the previous polynomials become equal when X 0, Y 0, Z 0, X 1, Y 1, Z1 are substituted by x, y, z,x,y,z, for any x, y, z A. However, under this substitution, both polynomials become Ž x y z.. The rest of the ring axioms, including the statement about the zero and the unit being Ž 0, 0. and Ž 1, 0., can be proved similarly. By the way, the inverse for addition is given by Ž. Ž x, x. x, P 1, x, Ž 1., x. 0 1 0 0 1 Remark. If in Lemma we start with one of the examples Ž.Ž.Ž. a, b, c, Ž d. then the ring structures on B B one gets are familiar ones. In the case of example Ž. a the ring structure on B B is isomorphic to the twofold product of B in the category of rings. In the case of example Ž b. the ring structure on B B is, of course, isomorphic to the ring structure on the dual numbers Btt Ž.. In the case of example Ž d. the ring structure on B B is a version of what one calls ramified Witt vectors. Case Ž. c leads to a similarly familiar ring. 1 Let A be any ring and let A Spec AX,A Spec AX,X A 0 A 0 1 be the affine line and the affine plane over A. Denote by Ž A. the set of all ring A-scheme structures on A A such that the first projection pr 1: A A A 1 A is a ring A-scheme homomorphism and such that the zero and the unit elements in A correspond to Ž 0, 0., Ž 1, 0.. We can identify Ž A. A with the set of all pairs Ž S, P. where S, P AX,X,Y,Y 0 1 0 1 such that the formulae Ž. 1 and Ž. in Lemma define, for any A-algebra B, a ring structure on BB whose zero and unit are Ž 0, 0. and Ž 1, 0..

ARITHMETIC ANALOGUES OF DERIVATIONS 95 On the other hand, consider the group Ž A. of all A-scheme automor- phisms of A that fix Ž 0, 0. and Ž 1, 0. A, and for which pr1 pr 1. Assume in addition that A is an integral domain of characteristic zero. Ž This will be the case we shall consider in applications.. Then looking at Jacobian matrices, one immediately sees that any such is given by Ž x, x. x, x fž x., 0 1 0 1 0 for some A and some f AX with fž. 0 fž. 1 0. So we can Ž identify A with A X X. AX, the latter equipped with the semidirect product group structure, Ž 1, f1. Ž, f. Ž,f 1 11f.. Now Ž A. acts on Ž A. by transport of structure. Explicitly, if Ž, f. Ž A. and Ž S, P. Ž A. then we have where Ž, f. Ž S, P. Ž S, P., 1 1 S S X 0,Y 0, X1fŽ X 0., Y1fŽ Y0. fž X0Y 0., Ž 3. P P X,Y, X fž X., Y fž Y. fž X Y.. Ž 4. 1 1 0 0 1 0 1 0 0 0 Lemma shows that, if Ž S, P. is the pair belonging to a jet operator on A and to a generic extension of it, then Ž S, P. Ž A. and the map AAA, xž x,x. is a ring homomorphism Žwhere the ring structure on A A is defined by Ž S, P. as in Lemma.. Conversely, if we are given an operator on A such that A A A, x Ž x, x. is a ring homomorphism Žwhere the ring structure on A A is defined by Ž S, P. as in Lemma. then belongs to Ž S, P.. So, in order to prove our theorem it is enough to prove the following PROPOSITION. Assume A is a local integral domain of characteristic zero. Then any element in Ž A. is Ž A. -conjugate to a pair Ž S, P. belonging to the following list S X Y, P XYY X XY, 1 1 0 1 0 1 1 1 SX Y, PXYY X, 1 1 0 1 0 1 SX Y, PXYY X XY, 1 1 0 1 0 1 1 1 SX Y *C X,Y, PX q Y Y q X XY. 1 1 q 0 0 0 1 0 1 1 1

96 ALEXANDRU BUIUM The element in the last two pairs in the preceding equations is as in the Examples Ž. c and Ž. d, respectively. The first step in the proof of the proposition is the following observation. Assume A is as in the proposition and let K be its field of fractions. LEMMA 3. Any element in Ž K. is Ž K. -conjugate to one of the pairs, S X Y, P XYY X XY, 1 1 0 1 0 1 1 1 SX Y, PXYY X. 1 1 0 1 0 1 Proof. Note that any element of Ž K. defines, in particular, a commu- tative algebraic group structure on A with zero element Ž 0, 0. K such that the first projection to A 1 K is a morphism of algebraic groups. Since the kernel of the first projection is isomorphic, as a variety, to A 1 K, and since, by the theory of algebraic groups, any algebraic group structure on A 1 K is isomorphic to the usual additive group structure, it follows that our algebraic group A is an extension of the additive group by itself. By K 9, p. 171, this extension is trivial. So any pair Ž S, P. Ž K. is conjugate, under Ž the action of K, to a pair S, P., where S X1 Y 1. By distributivity and commutativity in the ring structure axioms P must be a bilinear symmetric form, i.e., P XY 0 0 Ž XYXY 0 1 1 0. XY. 1 1 Because P Ž 1, Y,0,Y. 0 1 Y1 we get 0 and 1. If 0 then ŽS, P. Ž X Y, XYY X. and we are done. If 0 note that 1 1 0 1 0 1 Ž S, P. Ž 1, 1 X. Ž X1Y 1, XYY 0 1 0X1XY 1 1., and we are done again. Proof of the Proposition. By Lemma 3, there are two cases to examine for a pair Ž S, P. Ž A.. Case 1. Ž S, P. is Ž K.-conjugate to Ž X Y, XYY X. 1 1 0 1 0 1. In this case we claim that Ž S, P. is actually Ž A. -conjugate to Ž X1 Y 1, XYY 0 1 0X 1.. Indeed write Ž S, P. Ž, f.ž X1Y 1, XYY 0 1 0X 1., Ž where K and f X X. KX. By the formulae Ž.Ž. 3, 4 we have S X1 Y1 fž X0Y0. fž X0. fž Y 0., PXYY X f X Y X f Y Y f X. 0 1 0 1 0 0 0 0 0 0

ARITHMETIC ANALOGUES OF DERIVATIONS 97 The preceding formulae show that we may assume 1. Also, since P has coefficients in A, it follows immediately that f has coefficients in A. So Ž, f. Ž A. and our claim is proved. Case. Ž S, P. is Ž K.-conjugate to Ž X Y, XYY X XY. 1 1 0 1 0 1 1 1. Write Ž S, P. Ž, f.ž X1Y 1, XYY 0 1 0X1XY 1 1., Ž. Ž.Ž. where K and f X X K X. By the formulae 3, 4 we have S X1 Y1 fž X0Y0. fž X0. fž Y 0., Ž 5. PXYY 0 1 0X1X0fŽ Y0. Y0fŽ X0. Ž 6. 1 XY 1 1X1f Y0 Y1f X0 f X0 f Y0 f X0Y 0. Setting f * X f X X we obtain 1 P XY 1 1X1f* Y0 Y1f* X0 f* X0 f* Y0 f* XY 0 0. 7 Since P has coefficients in A, looking at the coefficients of XY 1 1 and i 1 1 XY in f * we get that A and f * XAX 1 0. If A we have Ž, f. Ž A. and we are done. Hence we may assume from now on that A, i.e., 1 M, where M is the maximal ideal of A. Let us write f f1xfx fmx m. Assume for a moment that f A for all i. Since fž. i 1 0 it follows that f1 A as well. In this case note that Ž 1,f.Ž S, P. Ž X1Y 1, XYY 0 1 0X1XY 1 1.. Since Ž 1, f. Ž A. we are done in this case. So from now on we may assume that the following condition holds There exists an index i0 such that fi 0 A. Ž 8. Recall that by a cocycle G AX,Y one understands a polynomial G that satisfies GŽ Y, Z. GŽ XY, Z. GŽ X,YZ. GŽ X,Y. 0.

98 ALEXANDRU BUIUM By the coboundary associated to a polynomial AX one understands the polynomial in two variables, Ž.Ž X, Y. Ž X Y. Ž X. Ž Y.. By Ž. 6, since S has coefficients in A, it follows that GŽ X 0,Y0. fž X0Y0. fž X0. fž Y0. is a cocycle in AX,Y 0 0. We claim that M Z 0, for if the contrary holds then Q is contained in A so by 7 p. 57, Lemma 3, we must have G for some AX. Hence f would be additive, hence f would be a monomial of degree one and this contradicts assumption Ž. 8. So we have M Z pz for some prime integer p. By 7 loc. cit. again, there exist a polynomial AX and there exist a 0, a 1, a,...a such that GŽ X 0,Y0. Ž X0Y0. Ž X0. Ž Y0. 1 p j p j p Ý j j 0 0 0 0 ap X Y X Y. j1 Again it follows that the polynomial f Ý ap 1 X p j j1 j is additive, hence it is a monomial of degree one. Hence we have pfi A for all i. 1 Ž. i 0 i Let. Since, by 7, the coefficient f f of X Y 0 i i 0 0 in P is in 0 0 A, and since A is local, we must have fi 0 A so we get p A. Set Ff*. As we have seen F AX and note that the reduction of F modulo, F Ž A A.X, has degree because the coefficient of i X 0 in F is invertible. Multiplying the equations Ž. 5 and Ž. 7 by and reducing modulo we find that FŽ X Y. FŽ X. FY 0 0 0 0 and FŽ X Y. FŽ X. FŽ Y.. The latter Ž multiplicativity. relation implies Ž 0 0 0 0 due. m to the fact that A A is local that F X for some m. By further reducing modulo M and using the additivity relation for F we get that m q 1 is an integer power of p. Consequently, we may write F X q g, gax. Then a straightforward computation shows that Ž S, P. equals 1 q q q 1,g ž X1Y1 X0 Y0 X0Y 0, X0 q Y1Y0 q X1XY 1 1/. This completes the proof of the proposition and hence of the theorem. ACKNOWLEDGMENTS The author would like to thank S. Zhang for interesting conversations and the NSF for support through grant DMS 9500331.

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