Recall that a derivation d of an algebra A is called locally ﬁnite if A, as a vector space, is a sum of ﬁnite dimensional d-invariant subspaces

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(1)COMMUNICATIONS IN ALGEBRA, 29(11), 5115–5130 (2001). ON SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO Andrzej Nowicki* and Andrzej Tyc{ N. Copernicus University, Faculty of Mathematics and Informatics, ul. Chopina 12718, 877100 Torun´, Poland. 1. INTRODUCTION. Let k be an algebraically closed field of characteristic 0. All algebras are supposed to be commutative k-algebras. Recall that a derivation d of an algebra A is called locally finite if A, as a vector space, is a sum of finite dimensional d-invariant subspaces. A derivation d is said to be semisimple (resp., locally nilpotent) if dðai Þ 2 kai ; i 2 I, for some linear basis fai ; i 2 Ig of A (resp., if given an a 2 A, there exists an n with dn ðaÞ ¼ 0). It is clear that each semisimple or locally nilpotent derivation d is locally finite. As usual, Ad stands for the algebra of constants of d, that is, Ad ¼ Ker d. By a k-domain we mean an algebra A such that ab 6¼ 0 for a; b 2 A n f0g. An algebra A is called reduced if it has no nonzero nilpotents. The main objective of this paper is to prove the following results. 1.. Let d be a derivation of the polynomial algebra k½x1 ; . . . ; xn such Pj¼n that dðxi Þ ¼ j¼1 aij xj þ bi for some aij ; bi 2 k, i; j ¼ 1; . . . ; n. Then the algebra of constants k½x1 ; . . . ; xn d is finitely generated.. *E-mail: anow@mat.uni.torun.pl { E-mail: atyc@mat.uni.torun.pl 5115 Copyright # 2001 by Marcel Dekker, Inc.. www.dekker.com.

(2) 5116. NOWICKI AND TYC. 2.. Let d be a semisimple derivation of a noetherian algebra A. (i) If P is a prime ideal in A and dP is the induced derivation of the localization AP , then the algebra of constants ðAP ÞdP is also noetherian. (ii) If A is UFD, S is an arbitrary multiplicative system in A and dS is the induced derivation of the localization AS , then the d algebra of constants ðAS Þ S is noetherian. 3. Let d be a locally finite derivation of a finitely generated k-domain A such that dðM Þ

(3) M for some maximal ideal M in A. Then the following conditions are equivalent. (1) Ad ¼ k. (2) There exists a k-subspace V M such that: dimk V < 1, A ¼ k½V , dðV Þ V , and the set of eigenvalues of djV is linearly independent over N. (3) The set of eigenvalues of djM is linearly independent over N. P Moreover, if d is semisimple and Ad ¼ k, then M ¼ t2knf0g At with At ¼ fa 2 A; dðaÞ ¼ tag is the unique maximal d-invariant ideal in A. 4.. Let d be a locally finite derivation of a reduced algebra A, and let dðhÞ ¼ ph for some h; p 2 A, where h is not a zero divisor of A. Then p 2 Ad . Moreover, if A is a domain, then p 2 k.. Result 1 is a generalization of a fact from [5] (see Corollary 3.11 in [10]). For locally nilpotent derivations Result 4 has been proved in [7] (for domains) and in [6] (for reduced algebras).. 2. PRELIMINARIES. Throughout the paper we assume that k is an algebraically closed field of characteristic zero and N is the set of all non-negative integers. If A is an algebra, then A the set A n f0g. L will stand for Let A ¼ l2k Al be a kþ -gradation of an algebra A (1 2 A0 , each Al is a subspace of A and Al Am Alþm for all l; m 2 k). Since charðkÞ ¼ 0, the group ðk; þÞ is torsion-free and hence, by [1], there exists a total order on ðk; þÞ satisfying the condition: a b ¼) a þ c b þ c; for all a; b; c 2 k. Let us fix such an order on ðk; þÞ and let P us consider the functions deg; v : A ! k defined, for each nonzero a ¼ l al , by degðaÞ ¼ maxfl; al 6¼ 0g;. vðaÞ ¼ minfl; al 6¼ 0g:.

(4) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. Proposition 2.1. (1) (2). If A is a domain, then:. degðabÞ ¼ degðaÞ þ degðbÞ, vðabÞ ¼ vðaÞ þ vðbÞ, for a; b 2 A ; degða þ bÞ maxfdegðaÞ; degðbÞg, vða þ bÞ minfvðaÞ; vðbÞg, for a; b 2 A with a þ b 2 A .. In particular, v is a valuation of A. Proof.. 5117. u. The proof of this proposition is easy and we omit it.. Below we will frequently use the following. Theorem 2.2 ð½13 ; ½10 Þ: (1) (2) (3). Let d be a locally finite derivation of an algebra A.. There exist unique derivations ds and dn of A such that d ¼ ds þ dn , ds is semisimple, dn is locally nilpotent and ds dn ¼ dn ds . If f : A ! A is a linear map with fd ¼ df , then fds ¼ ds f and fdn ¼ dn f . If V0 V1 A are subspaces and dðV1 Þ V0 , then ds ðV1 Þ V0 and dn ðV1 Þ V0 . u. The decomposition d ¼ ds þ dn is called the Jordan-Chevalley decomposition of d (for short, J-Ch decomposition of d). If we apply part ð3Þ of this theorem to V0 ¼ 0 and V1 ¼ Ad , then we get the following. Corollary 2.3. If d is a locally finite derivation of an algebra A, then u Ad ¼ Ads \ Adn . Assume that d is a locally finite derivation of an algebra A and let d ¼ ds þ dn be its J-Ch decomposition. If l 2 k, then we denote by Al the subspace of A defined as: n. Al ¼ fa 2 A; 9n0 ðd lIÞ ðaÞ ¼ 0g: In particular, A0 ¼ fa 2 A; 9n0 dn ðaÞ ¼ 0g: It is well known that Al Am Alþm and that Al ¼ Ker ðds lIÞ. Hence ðAl Þl2k is a kþ -gradation of the algebra A. This gradation will be called the gradation associated with d. PObserve that if V is a d-invariant subspace of A (i.e., dðVÞ V) and v ¼ l2k vl ; vl 2 Al , is an element of V, then vl 2 V for all l 2 k. This can be easily proved by induction of the number of nonzero vl in the sum P v ¼ vl , using the fact that dðvÞ v 2 V. Note also that ds ðVÞ V and dn ðVÞ V, by part (3) of Theorem 2.2..

(5) 5118. NOWICKI AND TYC. In the next section we shall need the following. Lemma 2.4. Let d be a locally finite derivation of an algebra A and let d ¼ ds þ dn be the J-Ch decomposition of d. Assume that V is a d-invariant subspace of A and let Vs ¼ ds ðV Þ. Then Vs is a d-invariant subspace of V and the restriction djVs : Vs ! Vs is a bijection. Proof. The space Vs is d-invariant, because dðVs Þ ¼ dds ðV Þ ¼ ds dðV Þ ds ðV Þ ¼ Vs . Let v 2 V. We shall show that there exists w 2 Vs such that ds ðvÞ ¼ dðwÞ. For this aim we can assume that v is homogeneous with respect to the gradation ðAl Þ associated with d. Let us assume that v 2 Al for some l 2 k. If l ¼ 0, then ds ðvÞ ¼ 0 ¼ dð0Þ and we are done. Now let l 6¼ 0 and let m be such m a natural number that ðd lIÞ ðvÞ ¼ 0. Then v ¼ dðuÞ, where u ¼ ðlÞ. m. m1 X m. i. i¼0. i. ðlÞ dm1i ðvÞ:. Hence ds ðvÞ ¼ ds dðuÞ ¼ dds ðuÞ, that is, ds ðvÞ ¼ dðwÞ for w ¼ ds ðuÞ 2 Vs . Therefore, Vs ¼ dðVs Þ, which means that djVs is surjective. Now suppose that for a given v 2 V there exist two elements w; w0 2 Vs such L that dðwÞ ¼ ds ðvÞ ¼ dðw0 Þ. Then w w0 2 Ker d A0 and w w0 2 Vs l6¼0 Al . Hence w w0 ¼ 0. Therefore, djVs is an injection. u 3. AFFINE DERIVATIONS. The classical result of Weitzenbo¨ck [12] states that every linear action of the additive group ðk; þÞ on A_> has a finitely generated algebra of invariants. A modern proof of this result is due to C. S. Seshadri [9]; (cf. [5] pp. 36740, [11]). In the vocabulary of derivations this is equivalent to the following. Theorem 3.1. Let d be a derivation of the polynomial algebra k½X ¼ k½x1 ; . . . ; xm such that m X dðxi Þ ¼ aij xj for i ¼ 1; . . . ; m; with aij 2 k: ð3:1Þ j¼1. If the matrix ½aij is nilpotent, then the algebra of constants k½X d is finitely generated. u It turns out that the assumption of nilpotency of the matrix ½aij is not essential. Let us call a derivation d of k½x1 ; . . . ; xm linear, if it satisfies the condition ð3:1Þ. Then holds the following..

(6) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5119. Theorem 3.2 ð½5 ; ½10 Þ: Every linear derivation of the polynomial algebra k½x1 ; . . . ; xm has finitely generated algebra of constants. u Now we shall prove the following. Theorem 3.3. xm such that dðxi Þ ¼. If d is a derivation of the polynomial algebra k½X ¼ k½x1 ; . . . ; m X. aij xj þ bi for i ¼ 1; . . . ; m;. with aij 2 k and bi 2 k;. ð3:2Þ. j¼1. then the algebra of constants k½X d is finitely generated over k. Proof. Consider the polynomial algebra k½Y ¼ k½y0 ; y1 ; . . . ; ym and the derivation D : k½Y ! k½Y defined by m X Dðy0 Þ ¼ 0; Dðyi Þ ¼ aij yj þ bi y0 ; for i ¼ 1; . . . ; m: j¼1. Let j : k½Y ! k½X be the homomorphism of algebras such that jðy0 Þ ¼ 1 D d and jðyi Þ ¼ xi for i ¼ 1; . . . ; m. Then dj ¼ jD and hence jðk½Y Þ k½X . d If h 2 k½X , then denote by H the polynomial from k½Y defined by y1 ym deg h Hðy0 ; y1 ; . . . ; ym Þ ¼ y0 h ;...; : y0 y0 D. It is clear that h ¼ jðHÞ and it is easy to check that H 2 k½Y . This means D d d that jðk½Y Þ ¼ k½X . Hence, k½X is finitely generated, because, by D Theorem 3.2, k½Y is finitely generated. u We say that a derivation d of k½X ¼ k½x1 ; . . . ; xm is affine if it is of the form ð3:2Þ. Notice that each affine derivation d of k½X is locally finite. Now we are going to find the J-Ch decomposition of d. Let us fix derivations d and d of k½X defined by ð3:1Þ and ð3:2Þ, respectively, and let V ¼ kx1 þ þ kxm , V ¼ k þ kx1 þ þ kxm . Denote by A the m m matrix ½aij , and put 2 3 2 3 2 3 x1 0 0 0 0 b1 6 b2 7 6 x2 7 6 b1 7 6 7 6 7 6 7 B ¼ 6 . 7; X ¼ 6 . 7; A ¼ 6 . 7: 4 .. 5 4 .. 5 4 .. 5 A bm xm bm The vector space V

(7) k½X is d-invariant and AT (the transposition of A) is the matrix of the restriction djV in the basis fx1 ; . . . ; xm g. The vector space V

(8) k½X is d-invariant and 3 AT is the matrix of the restriction djV in the basis f1; x1 ; . . . ; xm g..

(9) 5120. NOWICKI AND TYC. With the above notations the following properties hold.. Proposition 3.4. (1) (2). The derivation d is locally nilpotent if and only if the matrix A is nilpotent. The derivation d is semisimple if and only if the matrix A is diagonalizable and the system AX ¼ B has a solution in k m .. Proof. (1) It is obvious that the derivation d is locally nilpotent if and only T is nilpotent. But A T is if the mapping djV is nilpotent, that is, if the matrix A nilpotent if and only if A is nilpotent. (2) Assume that d is semisimple. Then the matrix AT is diagonalizable and so, the matrix A is also diagonalizable. Thus, there exists an invertible ðm þ 1Þ ðm þ 1Þ matrix U (with coefficients in k) such that UA ¼ DU, where D is a diagonal matrix. Now it is easy do deduce that the system AX ¼ B has a solution in km . Since dðkÞ ¼ 0, the subspace W ¼ k

(10) k½X 1 is d-invariant and hence, the induced linear mapping d : V=W ! V=W is semisimple. This implies that AT , as the matrix of d, is diagonalizable and consequently, A is diagonalizable. Now assume that A is diagonalizable and that the system AX ¼ B has a solution u ¼ ðu1 ; . . . ; um Þ 2 km . Put yi ¼ xi ui for i ¼ 1; . . . ; m. Then, for each i, dðyi Þ ¼ dðxi ui Þ ¼. m X. aij xj þ bi ¼. j¼1. ¼. m X. aij ðxj uj Þ ¼. j¼1. m X j¼1. m X. aij xj . m X. aij uj. j¼1. aij yj ;. j¼1. P that is, dðyi Þ ¼ m j¼1 aij yj for i ¼ 1; . . . ; m. It follows that d is semisimple, because A is diagonalizable and k½x1 ; . . . ; xm ¼ k½y1 ; . . . ; ym . u Let d ¼ ds þ dn be the J-Ch decomposition of the derivation d. By Theorem 2.2 (3), ds ðxi Þ ¼. m X. ðsÞ. aij xj ;. dn ðxi Þ ¼. j¼1. m X. ðnÞ. aij xj ;. i ¼ 1; . . . ; m;. j¼1 ðsÞ. ðnÞ. for some matrices As ¼ ½aij and An ¼ ½aij (over k) such that A ¼ As þ An , As An ¼ An As , As is diagonalizable and An is nilpotent. Let D be the linear of k½x1 ; . . . ; xm defined by the transpose Pderivation m of A, that is, Dðxi Þ ¼ j¼1 aji xj , i ¼ 1; . . . ; m. The decomposition of PJ-Ch ðsÞ m D has the form D ¼ D þ D , where D ðx Þ ¼ a s n s i j¼1 ji xj and Dn ðxi Þ ¼ Pm ðnÞ a x , for i ¼ 1; . . . ; m. j j¼1 ji.

(11) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5121. Consider the vector b ¼ b1 x1 þ þ bm xm , belonging to the D-invariant vector space V ¼ kx1 þ þ kxm . It follows from Lemma 2.4 that there exists a unique a ¼ a1 x1 þ þ am xm 2 Ds ðVÞ such that Ds ðbÞ ¼ DðaÞ. This means that for each i ¼ 1; . . . ; m we have the equality m X. aij aj ¼. j¼1. m X. ðsÞ. ð3:3Þ. aij bj :. j¼1. Now we define the derivations ds and dn as follows: ds ðxi Þ ¼ ds ðxi Þ ai ;. dn ðxi Þ ¼ dn ðxi Þ þ bi þ ai ;. i ¼ 1; . . . ; m;. that is, ds ðxi Þ ¼. m X. ðsÞ. aij xj ai ;. dn ðxi Þ ¼. j¼1. m X. ðnÞ. aij xj þ bi þ ai ;. i ¼ 1; . . . ; m:. j¼1. Then d ¼ ds þ dn and we have: Proposition 3.5.. The sum d ¼ ds þ dn is the J-Ch decomposition of d.. Proof. By Proposition 3.4 (1) it is clear that the derivation dn is locally nilpotent. Since a ¼ a1 x1 þ þ am xm 2 D s ðVÞ, there exists u ¼ u1 x1 þ þ P ðsÞ m um xm 2 V such that a ¼ Ds ðuÞ. Then i¼1 aij uj ¼ ai for i ¼ 1; . . . ; m, that Pm ðsÞ is, the system of linear equations i¼1 aij xj þ ai ¼ 0 has a solution in km . Recall also that the matrix As is diagonalizable. Therefore, by Proposition 3.4 (2), the derivation ds is semisimple. We must prove yet that ds dn ¼ dn ds . But this is a consequence of the equalities ð3:3Þ and the equality As An ¼ An As . u Corollary 3.6. If d ¼ ds þ dn is the J-Ch decomposition of an affine derivation d of k½x1 ; . . . ; xm , then ds and dn are also affine derivations of k½x1 ; . . . ; xm . u 4. LOCALLY FINITE DERIVATIONS WITH TRIVIAL CONSTANTS. In this section we investigate locally finite derivations of finitely generated k-domains having trivial the algebras of constants. If A is an algebra, then we denote by UðAÞ the group of units of A. For a given linear mapping f : V ! V of a k-vector space V we denote by Wð Þ the set of all eigenvalues of f. First we consider semisimple derivations. O.

(12) 5122. NOWICKI AND TYC. Theorem 4.1. Let d be a nonzero semisimple derivation of a finitely generated k-domain A, and let ðAl Þ be the gradation associated with d. The following conditions are equivalent. (1) (2) (3) (4) (5). Ad ¼ k and U ðAÞL ¼ k . d A ¼ k and M ¼ l2W ðdÞnf0g Al is the unique d-invariant maximal ideal of A. Ad ¼ k and there exists a maximal ideal M of A such that dðM Þ M. There exists a d-invariant subspace V A such that: dimk V < 1, A ¼ k½V , and the set WðjVÞ is linearly independent over N. Ad ¼ k and the set WðÞ n fKg is linearly independent over N.. ?. Proof. Put W ¼ WðÞ, W L0 ¼ W n f0g. ð1Þ ) ð2Þ. If M ¼ l2W0 Al is an ideal of A, then M is maximal and it is easy to see that it is the unique d-invariant maximal ideal in A. Thus, we must show that M is an ideal of A. A proof of this fact reduces to a proof that W0 þ W W0 . Let l 2 W0 , m 2 W and suppose that l þ m ¼ 0. Then m ¼ l 6¼ 0 and there exist nonzero elements a; b such that a 2 Al and b 2 Am . Hence 0 6¼ ab 2 Al Am Alþm ¼ A0 ¼ k, which means that a is invertible in A, that is, a 2 UðAÞ ¼ k . This implies that a 2 A0 and we get a contradiction, because 0 6¼ a 2 Al and l 6¼ 0. ð2Þ ) ð3Þ. It is obvious. ð3Þ ) ð4Þ. Let M be a maximal ideal of A such that dðMÞ M and let fa1 ; . . . ; an g be a finite set of nonzero generators of the algebra A. As A=M ¼ k, we can assume that a1 ; . . . ; an are homogeneous and belong to M. Let a1 2 Al1 ; . . . ; an 2 Aln and let V ¼ ka1 þ þ kan . Then clearly A ¼ k½V ; dðVÞ V, and WðjVÞ ¼ flj= ; . . . ; l g. It remains to prove that the elements l1 ; . . . ; ln are linearly independent over N. To this end, suppose b1 l1 þ þ bn ln ¼ 0, for some b1 ; . . . ; bn 2 N with ðb1 ; . . . ; bn Þ 6¼ b b ð0; . . . ; 0Þ. Let b ¼ a1 1 an n . Then b 2 M; b 6¼ 0 and dðbÞ ¼ 0, and hence we have a contradiction: 0 6¼ b 2 M \ Ad ¼ M \ k ¼ 0. ð4Þ ) ð5Þ. Let a subspace V A satisfy ð4Þ. Then V is spanned by homogeneous elements v1 ; . . . ; vn such that its eigenvalues l1 ; . . . ; ls are linearly independent over N. Furthermore, the equality k½V ¼ A implies that W ¼ Nlj= þ þ Nl , and hence the set W0 is linearly independent over N. Now we prove that Ad ¼ k. If i ¼ ði1 ; . . . ; in Þ 2 N , then we denote by i i i v the element v11 vnn . Since the set fbi ; i 2 N g generates A over k, there exists a subset T P N such that the set fvi ; i 2 Tg is a basis of A over k. Let d x 2 A . Then x ¼ i2T bi vi for some bi 2 k, and we have: X X 0 ¼ dðxÞ ¼ bi dðvi Þ ¼ bi ði1 l1 þ þ in ln Þvi : ?. i2T. i2T.

(13) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5123. Thus, bi ði1 l1 þ þ in ln Þ ¼ 0 for all i 2 T. If bi 6¼ 0, then i1 l1 þ þ in ln ¼ 0, and hence i1 ¼ ¼ in ¼ 0, because l1 ; . . . ; ln are linearly independent over N. This implies that x 2 k and consequently, Ad ¼ k. ð5Þ ) ð1Þ. Let us fix a total order on ðk; þÞ and let deg; v : A ! k be the functions defined in Section 2. Observe that for each nonzero a 2 A the elements degðaÞ and vðaÞ belong to W. Let a; b 2 A and ab ¼ 1. Then 0 ¼ degð1Þ ¼ degðabÞ ¼ degðaÞ þ degðbÞ and 0 ¼ vð1Þ ¼ vðabÞ ¼ vðaÞþ vðbÞ, and hence, by the assumption that the set W0 is linearly independent over N, degðaÞ ¼ degðbÞ ¼ 0 and vðaÞ ¼ vðbÞ ¼ 0. This implies that a; b 2 A0 ¼ Ad ¼ k, and therefore, UðAÞ ¼ k . This completes the proof. u Corollary 4.2. If d is a semisimple derivation of A ¼ k½x1 ; . . . ; xn with Ad ¼ k, then A has a unique d-invariant maximal ideal. u Now we consider locally finite derivations. Proposition 4.3. Let d be a locally finite derivation of a finitely generated algebra A and let d ¼ ds þ dn be the J-Ch decomposition of d. Assume that there exists a maximal ideal M of A such that dðM Þ M . Then Ad ¼ k if and only if Ads ¼ k. Proof. By Corollary 2.3, Ad ¼ Ads \ Adn . Hence Ad ¼ k whenever Ads ¼ k. Now assume that Ad ¼ k and suppose that Ads 6¼ k. Then A ¼ M k, so that there exists 0 6¼ m 2 M with ds ðmÞ ¼ 0. Let p 0 be such an integer that dnp ðmÞ 6¼ 0 and dnpþ1 ðmÞ ¼ 0. Put c ¼ dnp ðmÞ. Since dn ðMÞ M, 0 6¼ c 2 M \ Adn . Moreover, ds ðcÞ ¼ ds dnp ðmÞ ¼ dnp ds ðmÞ ¼ dnp ð0Þ ¼ 0 so, c 2 Ads \ Adn ¼ Ad ¼ k and we have a contradiction: 0 6¼ c 2 M \ k ¼ 0. u The following example shows that in the above proposition the assumption on the existence of a d-invariant maximal ideal M is important. Example 4.4. Let A ¼ k½t and let d ¼ @t@ . The derivation d is locally nilpotent, so dn ¼ d and ds ¼ 0. Hence Ad ¼ k and Ads ¼ k½t 6¼ k. u Theorem 4.5. Let d be a locally finite derivation of a finitely generated k-domain A. Assume that there exists a maximal ideal M of A such that dðMÞ M. Then the following conditions are equivalent. (1) (2). (3). Ad ¼ k. There exists a subspace V M such that: dimk V < 1; A ¼ k½V ; dðV Þ V , and the set WðjVÞ is linearly independent over N. The set WðjMÞ is linearly independent over N..

(14) 5124. NOWICKI AND TYC. ?. Proof. Let d ¼ ds þ dn be the J-Ch decomposition of d. By Theorem 2.2 (3), ds ðMÞ M and dn ðM Þ M , because dðM Þ M . ð1Þ ) ð2Þ. Assume that Ad ¼ k. We know that A ¼ k½a1 ; . . . ; an for some a1 ; . . . ; an 2 A. Since A ¼ M k, we may assume that a1 ; . . . ; an 2 M. Since d is locally finite, there exists a finite-dimensional subspace V M such that dðVÞ V and a1 ; . . . am 2 V. Hence A ¼ k½V and it remains to prove that the set WðjVÞ ¼ flj= ; . . . ; l> g is linearly independent over N. Let v1 ; . . . ; vm be eigenvectors of l1 ; . . . ; lm , respectively. Now, if i i i1 l1 þ þ im lm ¼ 0 for some i1 ; . . . ; im 2 N, then 0 6¼ b ¼ v11 . . . vmm 2 M and dðbÞ ¼ ði1 l1 þ þ im lm Þb ¼ 0, and we get a contradiction, because Ad ¼ k. ð2Þ ) ð1Þ. Let a subspace V M satisfy ð2Þ. Then ds ðVÞ V, and the set Wð jVÞ ¼ WðjVÞ is linearly independent over N. Hence Ads ¼ k, by Theorem 4.1, which implies that Ad ¼ k, by Proposition 4.3. ð1Þ ) ð3Þ. This is an easy consequence of Theorem 4.1, because WðjMÞ ¼ Wð jMÞ. ð3Þ ) ð1Þ. If 0 6¼ m 2 M, then clearly dðmÞ 6¼ 0, because the set WðjMÞ is linearly independent over N. Lest a 2 Ad . As A ¼ M þ k; a ¼ m þ a for some m 2 M; a 2 k. Hence dðmÞ ¼ dðaÞ ¼ 0, which implies m ¼ 0 and a ¼ a 2 k. Therefore, Ad ¼ k. This completes the proof. u Using a similar argument as in the above proof we get the following. Proposition 4.6. Let d be a locally finite derivation of a finitely generated k-domain A and let M be a maximal ideal of A such that dðM Þ M . Assume that Ad ¼ k. If V is a subspace of M such that dimk V < 1; A ¼ k½V and dðV Þ V , then the set WðjVÞ is linearly independent over N. u The following example shows that if A is not a domain, then the above proposition need not be true. Example 4.7. Let A ¼ k½X ; Y =ðXY Þ and let d be the derivation of A such that dðxÞ ¼ x, dðyÞ ¼ y, where x ¼ X þ ðXY Þ, y ¼ Y þ ðXY Þ. Then d is locally finite, the ideal M ¼ ðx; yÞ is maximal and d-invariant, and it is easy to show that Ad ¼ k. The subspace V ¼ kx þ ky is such that dimk V ¼ 2 < 1; A ¼ k½V and dðV Þ V . However, the eigenvalues of djV are not linearly independent over N. u 5. ON THE EQUALITY dðhÞ¼ ph. In this section K is a subfield of k. Without loss of generality we will that is, k is an algebraic closure of K. Let us recall the assume that k ¼ K, following theorem..

(15) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5125. Theorem 5.1 ð½7 ; ½6 Þ: Let d be a locally nilpotent derivation of a reduced K-algebra A. If dðhÞ ¼ ph for some h; p 2 A, then dðhÞ ¼ 0. u In this section we generalize this fact for locally finite derivations. Let us start with the following Theorem 5.2. Let d be a locally finite derivation of a k-domain A. If dðhÞ ¼ ph for some h; p 2 A with h 6¼ 0, then p 2 k. Proof. First assume that d is semisimple. Let ðAl Þ be the gradation associated with d and let dðhÞ ¼ ph, 0 6¼ h; p 2 A. We shall show that p 2 A0 . For this aim let us fix a total order on ðk; þÞ and consider the functions deg; v : A ! k defined in Section 2. Observe that degðdðaÞÞ degðaÞ and vðdðaÞÞ vðaÞ, for all a 2 A with dðaÞ 6¼ 0. If p ¼ 0, then of course p 2 A0 . Assume that p 6¼ 0. Then dðhÞ 6¼ 0 and we have: degð pÞ þ degðhÞ ¼ degð phÞ ¼ degðdðhÞÞ degðhÞ; vð pÞ þ vðhÞ ¼ vð phÞ ¼ vðdðhÞÞ vðhÞ: This implies that degð pÞ 0, vð pÞ 0. Hence 0 degð pÞ vð pÞ 0, so that vð pÞ ¼ degð P pÞ ¼ 0, that is, p 2 A0 . In particular, p is homogeneous. If h ¼ m hm is the decomposition of h, then we have the decompositions X X dðhm Þ ¼ dðhÞ ¼ ph ¼ phm ; m. m. and hence dðhm Þ ¼ phm for each m. Thus, we can assume that h is also homogeneous. Let h 2 Am for some m 2 k. Then ph ¼ dðhÞ ¼ mh and so, p ¼ m 2 k. Now assume that d is locally finite and d ¼ ds þ dn is the J-Ch decomposition of d. Since dðhÞ ¼ ph, the ideal ðhÞ is d-invariant and hence, by Theorem 2.2 (3), this ideal is also ds -invariant and dn -invariant. So, ds ðhÞ ¼ ps h and dn ðhÞ ¼ pn h for some ps ; pn 2 A. But pn ¼ 0, by Theorem 5.1 and, ps 2 k, by the first part of this proof. Therefore, p ¼ ps 2 k. This completes the proof. u Corollary 5.3. Let d be a locally finite derivation of a k-domain A and let h be a unit of A. Then dðhÞ ¼ lh for some l 2 k. Proof.. Let l ¼ dðhÞh1 . Then dðhÞ ¼ lh and, by Theorem 5.2, l 2 k.. u. The following examples show that all the assumptions to Theorem 5.2 are necessary..

(16) 5126. NOWICKI AND TYC. Example 5.4. Assume that charðkÞ ¼ p > 0 and consider the derivation d : k½t ! k½t defined by dðtÞ ¼ t p . Then d is locally finite, dðtÞ ¼ ðt p1 Þt and t p1 62 k. Thus, Theorem 5.2 is not valid in positive characteristics. u Example 5.5. Let R and C be the fields of real and complex numbers, respectively. Consider the polynomial algebra C½ as an R-algebra in the natural way. Let d be the derivation of C½ defined by dðtÞ ¼ it and dðiÞ ¼ 0 (where i2 ¼ 1). Then d is semisimple, C½ is an R-domain, dðtÞ ¼ it, and i 62 R. u We see from the above example that Theorem 5.2 is not valid, when the basic field k is not algebraically closed. The next proposition shows that sometimes this assumption is not important. Proposition 5.6. Let d be a locally finite derivation of a polynomial algebra K½X ¼ K½x1 ; . . . ; xn . If dðhÞ ¼ ph, for some h; p 2 K½X with h 6¼ 0, then p 2 K. Proof. Let d be the derivation of the algebra k½X ¼ k½x1 ; . . . ; xn given by i Þ ¼ dðxi Þ for i ¼ 1; . . . ; n. Then d is locally finite and dðhÞ dðx ¼ ph. Hence, by Theorem 5.2, p 2 k \ K½x ¼ K. u The next example shows that Theorem 5.2 is not true in the case where the algebra A is not a domain. Example 5.7. Let d be the derivation of the polynomial algebra k½t such that dðtÞ ¼ t. Consider A ¼ k½t k½t , the product of algebras. It is an algebra with the scalar multiplication lða; bÞ ¼ ðla; lbÞ for all l 2 k, a; b 2 k½t . Let d : A ! A be the mapping defined by dða; bÞ ¼ ðdðaÞ; dðbÞÞ for all ða; bÞ 2 A. It is clear that d is a semisimple derivation of A. Let h ¼ ðt; t 2 Þ, p ¼ ð1; 2Þ. Then k; p 2 A, h 6¼ 0 (h is even not a zero-divisor of A), dðhÞ ¼ ph and p 62 k. u Now we shall prove the following. Theorem 5.8. Let d be a locally finite derivation of a reduced K-algebra A, and let dðhÞ ¼ ph, for some h; p 2 A, where h is not a zero-divisor of A. Then dð pÞ ¼ 0. Proof. Replacing A by A K k and d by d 1 we can assume that K ¼ k. Since d is locally finite, there exists a finite dimensional d-invariant subspace V containing p and h. Thus A0 ¼ k½V is a reduced noetherian k-algebra and d can be viewed as a locally finite derivation of A0 . Consequently we may assume that A is a noetherian algebra over an algebraically closed field. Let P1 ; . . . ; Pn be the all prime ideals of A associated with the ideal 0. Then the sum P1 [ [ Pn coincides with the set zðAÞ of all zero-divisors of A.

(17) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5127. and the intersection P1 \ \ Pn is equal to 0 (because it is the nilradical of A). We know (see [8]) that the ideals P1 ; . . . ; Pn are d-invariant. Let di (for each i ¼ 1; . . . ; n) be the derivation of the K-domain A=Pi defined by: di ða þ Pi Þ ¼ dðaÞ þ Pi . It is clear that each di is locally finite. Fix an i belonging to f1; . . . ; ng and put hi ¼ h þ Pi ; pi ¼ p þ Pi . Then hi 6¼ 0 (because h 62 zðAÞ) and di ðhi Þ ¼ pi hi , so, by Theorem 5.2, pi 2 K, that is, p ai 2 Pi for some ai 2 K. Hence dð pÞ 2 Pi , because dð pÞ ¼ dð p ai Þ 2 dðPi Þ Pi . Therefore, dð pÞ 2 P1 \ \ Pn ¼ 0, that is, dð pÞ ¼ 0. u Let d be a derivation (not necessarily locally finite) of a k-domain A and let NilðdÞ ¼ fa 2 A; 9n2N dn ðaÞ ¼ 0g. It is obvious that ðdÞ is a subalgebra of A containing Ad . It is known (see [6] page 95) that if dðhÞ ¼ ph for some h 2 A and 0 6¼ p 2 NilðdÞ, then h is transcendental over NilðdÞ. As a consequence of this fact and Theorem 5.8 we get Corollary 5.9. Let d be a locally finite derivation of a K-domain A. If dðhÞ ¼ ph for some h; p 2 A , then h is transcendental over NilðdÞ. u Let us end this section with the following. Proposition 5.10. Let d be a locally finite derivation of a K-algebra A. If A is a field, then d ¼ 0. Proof. Suppose that dðaÞ 6¼ 0 for some (nonzero) a 2 A. Since dðaÞ ¼ ðdðaÞa1 Þa, Theorem 5.8 implies that dðaÞa1 2 Ad . Repeating the 1 1 same for the element a þ 1 we get: dðaÞða þ 1Þ ¼ dða þ 1Þða þ 1Þ 2 Ad . Hence a1 þ 1 2 Ad , because a1 þ 1 ¼ ða þ 1Þa1 ¼ ðða þ 1Þ1 dðaÞÞ1 a1 dðaÞ : Thus, a 2 Ad , which is a contradiction.. 6. u. FRACTIONS AND SEMISIMPLE DERIVATIONS. Let us recall that if S is a multiplicative subset of an algebra A, then each derivation d : A ! A induces a derivation d : S1 A ! S1 A (S1 A is the algebra of fractions of A with respect to S) given by the formula: dða=sÞ ¼ ðdðaÞs adðsÞÞ=s2 for a 2 A, s 2 S. If A is a domain, then all the induced derivations d : S1 A ! S1 A are restrictions of the induced derivation d : A0 ! A0 , where A0 is the field of fractions of A. Proposition 6.1. Let d be a semisimple derivation of a k-domain A and let associated with d. Further, let T A be a multiplicative ðAl Þ be the gradation S subset such that T l2k Al and let A ¼ T 1 A. Then:.

(18) 5128. NOWICKI AND TYC. (1) (2). d : A ! A is semisimple; then if ðAl Þ is the gradation associated with d : A ! A, na o a 2 Am ; t 2 An ; l ¼ m n 2 A; Al ¼ for all l 2 k; t. (3). if A is noetherian, then so is Ad .. Proof. The proof of parts (1) and (2) is simple and we omit it. Part (3) follows from ð1Þ and the fact that the ring of constants of a semisimple derivation of a noetherian algebra is noetherian (see [10]). u Theorem 6.2. Let d be a semisimple derivation of a noetherian k-domain A and let ðAl Þ be the associated gradation of A. Moreover, S be a multiplicative subset of A. Suppose that A is either a unique factorization domain ðUFDÞ or S satisfies the condition: 8s2S 9l2k sl 2 S;. ðÞ. (where sl denotes the homogeneous component of s belonging to Al ). Then the algebra of constants of the derivation d : S 1 A ! S 1 A is noetherian. S Proof. Let T ¼ t 2 l2k Al ; 9s2S 9a2A at ¼ s : Then T is a multiS plicative subset of A satisfying the condition T l2k Al . Denote by A the algebra T 1 A. It follows from Proposition 6.1 that d : A ! 3A is semisimple, na o 9m2k a 2 Am ; t 2 Am ; ; 2 A; Ad ¼ A0 ¼ t d d that Ad ¼ S1 A . and the algebra A is noetherian. So, it suffices to verify d As clearly A ¼ T1 A S1 A, we have Ad S1 A . It remains to prove d d 1 that ðS AÞ A . d First assume that A is UFD. If x ¼ ða=sÞ 2 S1 A , then clearly dðaÞs ¼ adðsÞ. Let g ¼ gcdða; sÞ, a ¼ a1 g, s ¼ s1 g, where a1 ; s1 2 A, gcdða1 ; s1 Þ ¼ 1. Then dðaÞs1 ¼ a1 dðsÞ so, dða1 gÞs1 ¼ a1 dðs1 gÞ and so, a1 dðs1 Þ ¼ s1 dða1 Þ. Since gcdða1 ; s1 Þ ¼ 1, there exists p 2 A such that dða1 Þ ¼ pa1 and dðs1 Þ ¼ ps1 . It follows from Theorem 5.2 that p 2 k. Hence, a1 ; s1 2 Ap and moreover, s1 2 T, because gs1 ¼ s 2 S and s1 2 Ap . Thus, =s1 Þ 2 Ad , whence x ¼ ða=sÞ ¼ ðga1 =gs1 Þ ¼ ða1 =s1 Þ 2 Ad . Therefore, ða11 d S A Ad . assume that S satisfies the condition ðÞ and let x ¼ a=s 2 1 Now d S A . One simply checks that I ¼ fy 2 A; yx 2 Ag is a d-invariant ideal in A. Obviously s 2 I, which implies that every homogeneous component of s belongs also to I. Hence, in view of the condition ðÞ, there exist l 2 k and.

(19) SEMISIMPLE DERIVATIONS IN CHARACTERISTIC ZERO. 5129 d. s0 2 S \ Al T such that s0 x ¼ b 2 A. This means that x ¼ b=s0 2 ðT1 AÞ 1 d and consequently S A Ad . u Corollary 6.3. Let d be a semisimple derivation of a noetherian k-domain A and let S ¼ A n P where P is a prime ideal of A. Then the algebra of constants of the derivation d : S 1 A ! S 1 A is noetherian. Proof. It is a consequence of Theorem 6.2, because the multiplicative set S ¼ A n P satisfies the condition ðÞ. u Remark 6.4. We do not know if Theorem 6.2 holds for any multiplicative subset S

(20) A.. ACKNOWLEDGMENT Supported by KBN Grant 2 PO3A 017 16.. REFERENCES 1. Bourbaki, N. E´le´ments de Mathe´matique, Alge´bre, Livre II; Hermann: Paris, 1961. 2. van den Essen, A. Locally Nilpotent Derivations and their Applications III. J. Pure Appl. Algebra 1995, 98, 15723. 3. Humphreys, J.E. Introduction to Lie Algebras and Representation Theory; Springer-Verlag: New York, Heidelberg, Berlin, 1972. 4. Matsumura, H. Commutative Algebra; Benjamin, 1980 (second edn.). 5. Nagata, M. Lectures on the Fourteenth Problem of Hilbert; Lect. Notes 31, Tata Institute: Bombay, 1965. 6. Nowicki, A. Polynomial Derivations and their Rings of Constants; N. Copernicus University Press: Torun´, 1994. 7. Rentschler, R. Ope´rations du Groupe Additif Sur Le Plan Affine. C. R. Acad. Sc. Paris 1968, 267, 3847387. 8. Seidenberg, A. Differential Ideals and Rings of Finitely Generated Type. Amer. J. Math. 1967, 89, 22742. 9. Seshadri, C.S. On a Theorem of Weitzenbo¨ck in Invariant Theory. J. Math. Kyoto Univ. 1961, 1, 4037409. 10. Tyc, A. Jordan – Chevalley Decomposition and Invariants for Locally Finite Actions of Commutative Hopf Algebras. J. Algebra 1996, 182, 1237139. 11. Tyc, A. An Elementary Proof of the Weitzenbo¨ck Theorem. Colloq. Math. 1998, 78, 1237132..

(21) 5130. 12. 13.. NOWICKI AND TYC. Weitzenbo¨ck, R. U¨ber Die Invarianten Von Linearen Gruppen. Acta. Math. 1932, 58, 2307250. Zurkowski, V.D. Locally Finite Derivations. Preprint 1993.. Received May 2000 Revised March 2001.

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