Abstract
We study the subgroup of -automorphisms of which commute with a simple derivation of . We prove, for instance, that this subgroup is trivial when is a shamsuddin simple derivation. in the general case of simple derivations, we obtain properties for the elements of this subgroup.
dynamical degree; isotropy group; Shamsuddin derivations; simple derivations
Introduction
Let be an algebraically closed field of zero characteristic and be the ring of polynomials over in two variables.
A -derivation of is a -linear map such that
for any . We denote by the set of all -derivations of . Let . An ideal of is called -stable if . For example, the ideals and are always -stable. If these are the only -stable ideals, we say is -simple. Even in the case of two variable polynomials, only a few examples of simple derivations are known (see, for instance, Brumatti et al. ( 2003r5 BRUMATTI P, LEQUAIN Y AND LEVCOVITZ D. 2003. Differential simplicity in Polynomial Rings and Algebraic Independence of Power Series. J London Math Soc 68(3): 615-630., Saraiva 2012r15 SARAIVA C. 2012. Sobre Derivações Simples e Folheações holomorfas sem Solução Algébrica, Tese de Doutorado., Nowicki 2008r13 NOWICKI A. 2008. An Example of a Simple Derivation in Two Variables. Colloq Math 113(1): 25-31., Baltazar and Pan 2015r3 BALTAZAR R AND PAN I. 2015. On solutions for derivations of a Noetherian k-algebra and local simplicity. Commun Algebra 43(7): 2739-2747., Kour and Maloo 2013r9 KOUR S AND MALOO AK. 2013. Simplicity of Some Derivations of k[x, y]. Commun Algebra 41(4): 1417-1431., Lequain 2011r11 LEQUAIN Y. 2011. Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization. J Pure Appl Algebra 215(4): 531-545.)).
We denote by the group of -automorphisms of . Let act on by:
Fix a derivation . The isotropy subgroup,with respect to this group action, is defined as
We are interested in the following question proposed by I.Pan (see Baltazar ( 2014r2 BALTAZAR R. 2014. Sobre soluções de derivações em k-algebras Noetherianas e simplicidade. Tese de Doutorado, Universidade Federal do Rio Grande do Sul.)):
Conjecture 1 If is a simple derivation of , then is finite.
Initially, in Section 2, we prove Theorem 6, which shows that the conjecture is true for a family of derivations, namely Shamsuddin derivations. For this purpose, we use a theorem due to Shamsuddin ( 1977r17 SHAMSUDDIN A. 1977. Automorphisms and Skew Polynomial Rings. Ph.D. thesis, Univesity of Leeds.) (see also Nowicki 1994r14 NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf.
http://www-users.mat.umk.pl/anow/psdvi/p...
, Theorem 13.2.1.) that gives a necessary and sufficient condition for a derivation to be extended to , with an indeterminate, and preserving simplicity. We observe Shamsuddin derivations is a reasonable class of objects. For instance, they have been previously used by Lequain ( 2011r12 LEQUAIN Y. 2008. Simple Shamsuddin derivations of K[X; Y1; :::; Yn]: An algorithmic characterizarion. J Pure Appl Algebra 212(4): 801-807.) in order to establish a conjecture about the Weyl algebra over .
In Section 3, to understand the isotropy of a simple derivation of , we give necessary conditions for an automorphism to belong to the isotropy of a simple derivation. We prove in Proposition 7 that if such an automorphism has a fixed point, then it is the identity. Next, we present the definition of dynamical degree of a polynomial map and prove in Corollary 9 that for , the elements of , with a simple derivation, have dynamical degree . More precisely, the condition that the dynamical degree is greater than 1corresponds to exponential growth of the degree under iteration, and this may be viewed as a complexity of the automorphism in the isotropy (see Friedland and Milnor ( 1989r7 FRIEDLAND S AND MILNOR J. 1989. Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn Syst 9: 67-99.)).
SHAMSUDDIN DERIVATIONS
The aim of this section is study the isotropy group of a Shamsuddin derivation in . In Nowicki ( 1994r13 NOWICKI A. 2008. An Example of a Simple Derivation in Two Variables. Colloq Math 113(1): 25-31.), there are numerous examples of these derivations and a criterion for determining the simplicity. Furthermore, Lequain ( 2008r11 LEQUAIN Y. 2011. Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization. J Pure Appl Algebra 215(4): 531-545.) introduced an algorithm for determining whether a Shamsuddin derivation is simple. We begin with an example that shows that the isotropy of an arbitrary derivation can be quite complicated.
Example 1 Let and . Note that is not a simple derivation. Indeed, for any , the ideal generated by is always invariant. Consider
Since , we obtain two conditions:
1)
Thus,
Then, and , . We conclude that is of the type
2)
Analogously,
that is, , with . We also infer that is of the type
Thus, contains the affine automorphisms
with . In particular, the isotropy group of a derivation which is not simple can be infinite. Indeed, contains all automorphisms of the type , with . Actually, these are all the elements of . By conditions and ,
with . Since is an automorphism, the determinant of the Jacobian matrix must be nonzero. Thus, , . Therefore, , with and . Consequently, is not finite and its first component has elements with any degree.
The following is a well known lemma.
Lemma 2 Let be a commutative ring, a derivation of , and , with an indeterminate. Then, we can also extend to a unique derivation of such that .
We also use the following result of Shamsuddin ( 1977r17 SHAMSUDDIN A. 1977. Automorphisms and Skew Polynomial Rings. Ph.D. thesis, Univesity of Leeds.).
Theorem 3 Let be a ring containing and let be a simple derivation of . Extend the derivation to a derivation of the polynomial ring by setting where . Then the following two conditions are equivalent:
is a simple derivation.
There exist no elements such that .
Proof See (Nowicki 1994r14 NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf.
http://www-users.mat.umk.pl/anow/psdvi/p...
, Theorem 13.2.1.) for a detailed proof.
A derivation of is said to be a Shamsuddin derivation if is of the form
where .
Example 4 Let be a derivation of as follows
Writing , we know that is -simple and, taking and , we are exactly in the conditions of Theorem 3. Thus, we know that is simple if, and only if, there exist no elements such that ; but the right hand side of the equivalence is satisfied by the degree of . Therefore, by Theorem 3, is a simple derivation of .
Lemma 5. (Nowicki 1994r14 NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf.
http://www-users.mat.umk.pl/anow/psdvi/p...
, Proposition. 13.3.2) Let be a Shamsuddin derivation, where . Thus, if is a simple derivation, then and .
Proof If , then the ideal is -invariante. If , let such that , then the ideal is -invariante.
One can determine the simplicity of the a Shamsuddin derivation according the polynomials and (see Nowicki 1994r14 NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf.
http://www-users.mat.umk.pl/anow/psdvi/p...
, §13.3).
Theorem 6 Let be a Shamsuddin derivation. If is a simple derivation, then .
Proof Let us denote and . Let be a Shamsuddin derivation and
where . Since , we obtain two conditions:
Then, by condition , and since can be written in the form
with , we obtain
By comparing the coefficients of ,
which can not occur by simplicity. More explicitly, Lemma 5 implies . Thus, , that is, . Therefore, and , with constant.
By using condition ,
By the previous part, we can suppose that , because is a automorphism. Now, write . Thus,
By comparing the coefficients of , we obtain
Then . In this way, is a constant and, consequently, . Comparing the coefficients in the last equality, we obtain and then is constant. Moreover, if is not a constant, since , it is easy to see that . Indeed, if , we obtain that the polynomial has infinite distinct roots. If is constant, then is not a simple derivation (this is a consequence of Lequain 2008r12 LEQUAIN Y. 2008. Simple Shamsuddin derivations of K[X; Y1; :::; Yn]: An algorithmic characterizarion. J Pure Appl Algebra 212(4): 801-807., Lemma.2.6 and Theorem.3.2); thus, we obtain .
Note that and, using the condition ,
Considering the independent term of , we have
By (Nowicki 1994r14 NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf.
http://www-users.mat.umk.pl/anow/psdvi/p...
, Proposition. 13.3.3), if , we have that is a simple derivation if and only if , defined by
is a simple derivation. Furthermore, by Theorem 3, there exist no elements in such that
This contradicts equation “eqrefeq1.1. Then, and . Since is a simple derivation, we know that and consequently . This shows that .
ON THE ISOTROPY OF THE SIMPLE DERIVATIONS
The purpose of this section is to study the isotropy in the general case of a simple derivation. More precisely, we obtain results that reveal nice features of the elements of . For this, we use some concepts presented in the previous sections and the concept of dynamical degree of a polynomial map.
In Baltazar and Pan ( 2015r3 BALTAZAR R AND PAN I. 2015. On solutions for derivations of a Noetherian k-algebra and local simplicity. Commun Algebra 43(7): 2739-2747.), which was inspired by Brumatti et al. ( 2003r5 BRUMATTI P, LEQUAIN Y AND LEVCOVITZ D. 2003. Differential simplicity in Polynomial Rings and Algebraic Independence of Power Series. J London Math Soc 68(3): 615-630.), the authors introduce and study a general notion of solution associated to a Noetherian differential -algebra and its relationship with simplicity.
The following proposition has a geometrical flavour: it says that if an element in the isotropy of a simple derivation has fixed point, then it is the identity automorphism.
Proposition 7 Let be a simple derivation and be an automorphism in the isotropy. Suppose that there exists a maximal ideal such that , then .
Proof Let be a solution of passing through (see Baltazar and Pan 2015, Definition 1r3 BALTAZAR R AND PAN I. 2015. On solutions for derivations of a Noetherian k-algebra and local simplicity. Commun Algebra 43(7): 2739-2747.). We know that and . If , then
In other words, is a solution of passing through . Then, by the uniqueness of the solution (Baltazar and Pan 2015r3 BALTAZAR R AND PAN I. 2015. On solutions for derivations of a Noetherian k-algebra and local simplicity. Commun Algebra 43(7): 2739-2747., Theorem.7.(c)), . Because is -simple and is a nontrivial solution, we have that is one-to-one. Therefore, .
Lane ( 1975r10 LANE DR. 1975. Fixed points of affine Cremona transformations of the plane over an algebraically closed field. Amer J Math 97(3): 707-732.) proved that every -automorphism of leaves a nontrivial proper ideal invariant over an algebraically closed field, that is, . In Shamsuddin ( 1982r18 SHAMSUDDIN A. 1982. Rings with automorphisms leaving no nontrivial proper ideals invariant. Canadian Math Bull 25: 478-486.), Shamsuddin proved that this result does not extend to , proving that the -automorphism given by , , and has no nontrivial invariant ideal.
In addition, since is Noetherian, leaves a nontrivial proper ideal invariant if, and only if, . In fact, the ascending chain
must stabilize; thus, there exists a positive integer such that . Hence, .
Suppose that and that is a simple derivation of . By Proposition 7, if this invariant ideal is maximal, we have . Suppose that is radical and let be a primary decomposition, where the ideals are maximal and are prime ideals with height such that , with irreducible (see Kaplansky 1974r8 KAPLANSKY I. 1974. Commutative Rings. Chicago, (2nd edition)., Theorem 5). If
we claim that leaves invariant one maximal ideal for some . Indeed, we know that and since is a prime ideal, we deduce that , for some (Atiyah and Macdonald 1969r1 ATIYAH MF AND MACDONALD IG. 1969. Introduction to Commutative Algebra. Massachusetts: Addison-Wesley Publishing Company., Prop.11.1.(ii)). Then, , that is, leaves invariant the maximal ideal , for some . Thus, it follows from Proposition 7 that .
Note that . In fact, writing , with irreducible, we would like to choose such that . If such does not exist, we would obtain , then , for some (Atiyah and Macdonald 1969r1 ATIYAH MF AND MACDONALD IG. 1969. Introduction to Commutative Algebra. Massachusetts: Addison-Wesley Publishing Company., Prop.11.1.(ii)): a contradiction. Thus, since , we obtain . Therefore, . Likewise, the same conclusion holds for the other prime ideals , . Finally, .
In the next corollary, we obtain consequences on the case of radical ideals.
Corollary 8 Let , a simple derivation of , and an ideal with height such that , with reduced. If is singular or some irreducible component of has genus greater than two, then is an automorphism of finite order.
Proof Suppose that is not a smooth variety and let be a singularity of . Since the set of singular points is invariant by , there exists such that . Using that , we obtain, by Proposition 7, .
Let be a component irreducible of that has genus greater than two. Note that there exists such that . By (Farkas and Kra 1992r6 FARKAS HM AND KRA I. 1992. Riemann Surfaces. 2nd edition. Graduate Texts in Mathematics, Springer., Theorem Hunvitz, p.241), the number of elements in is finite; in fact, , where is the genus of . Then, we deduce that is an automorphism of finite order.
In the rest of this section, we let
Consider a polynomial map and define the degree of by . Thus, we may define the dynamical degree (see Blanc and J. (r4 BLANC J AND DESERTI J. IN PRESS. Degree Growth of Birational Maps of the Plane.), Friedland and Milnor ( 1989r7 FRIEDLAND S AND MILNOR J. 1989. Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn Syst 9: 67-99.), Silverman ( 2012r19 SILVERMAN JH. 2012. Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space. Ergodic Theory and Dynamical Systems 34(2): 647-678.)) of as
Corollary 9 If and is a simple derivation of , then .
Proof Suppose . By (Friedland and Milnor 1989r7 FRIEDLAND S AND MILNOR J. 1989. Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn Syst 9: 67-99., Theorem 3.1.), has exactly fixed points counted with multiplicities. Then, by Proposition 7, , which shows that the dynamical degree of is 1.
ACKNOWLEDGMENTS
I would like to thank Ivan Pan for his comments and suggestions. Research of R. Baltazar was partially supported by Coordenação de Aperfeiçoamento de Pessoal de NÃvel Superior (CAPES).
REFERENCES
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r1ATIYAH MF AND MACDONALD IG. 1969. Introduction to Commutative Algebra. Massachusetts: Addison-Wesley Publishing Company.
-
r2BALTAZAR R. 2014. Sobre soluções de derivações em k-algebras Noetherianas e simplicidade. Tese de Doutorado, Universidade Federal do Rio Grande do Sul.
-
r3BALTAZAR R AND PAN I. 2015. On solutions for derivations of a Noetherian k-algebra and local simplicity. Commun Algebra 43(7): 2739-2747.
-
r4BLANC J AND DESERTI J. IN PRESS. Degree Growth of Birational Maps of the Plane.
-
r5BRUMATTI P, LEQUAIN Y AND LEVCOVITZ D. 2003. Differential simplicity in Polynomial Rings and Algebraic Independence of Power Series. J London Math Soc 68(3): 615-630.
-
r6FARKAS HM AND KRA I. 1992. Riemann Surfaces. 2nd edition. Graduate Texts in Mathematics, Springer.
-
r7FRIEDLAND S AND MILNOR J. 1989. Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn Syst 9: 67-99.
-
r8KAPLANSKY I. 1974. Commutative Rings. Chicago, (2nd edition).
-
r9KOUR S AND MALOO AK. 2013. Simplicity of Some Derivations of k[x, y]. Commun Algebra 41(4): 1417-1431.
-
r10LANE DR. 1975. Fixed points of affine Cremona transformations of the plane over an algebraically closed field. Amer J Math 97(3): 707-732.
-
r12LEQUAIN Y. 2008. Simple Shamsuddin derivations of K[X; Y1; :::; Yn]: An algorithmic characterizarion. J Pure Appl Algebra 212(4): 801-807.
-
r11LEQUAIN Y. 2011. Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization. J Pure Appl Algebra 215(4): 531-545.
-
r14NOWICKI A. 1994. Polynomial derivations and their rings of constants. TORUN. at http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf
» http://www-users.mat.umk.pl/anow/psdvi/pol-der.pdf -
r13NOWICKI A. 2008. An Example of a Simple Derivation in Two Variables. Colloq Math 113(1): 25-31.
-
r15SARAIVA C. 2012. Sobre Derivações Simples e Folheações holomorfas sem Solução Algébrica, Tese de Doutorado.
-
r17SHAMSUDDIN A. 1977. Automorphisms and Skew Polynomial Rings. Ph.D. thesis, Univesity of Leeds.
-
r18SHAMSUDDIN A. 1982. Rings with automorphisms leaving no nontrivial proper ideals invariant. Canadian Math Bull 25: 478-486.
-
r19SILVERMAN JH. 2012. Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space. Ergodic Theory and Dynamical Systems 34(2): 647-678.
Publication Dates
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Publication in this collection
Dec 2016
History
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Received
26 Jan 2015 -
Accepted
29 Sept 2015