Locally tame plane polynomial automorphisms
Joost Berson, Adrien Dubouloz, Jean-Philippe Furter, Stefan Maubach
aa r X i v : . [ m a t h . AG ] N ov LOCALLY TAME PLANE POLYNOMIAL AUTOMORPHISMS
JOOST BERSON, ADRIEN DUBOULOZ, JEAN-PHILIPPE FURTER, AND STEFAN MAUBACH
Abstract.
For automorphisms of a polynomial ring in two variables over a domain R , we show thatlocal tameness implies global tameness provided that every -generated locally free R -module of rank is free. We give many examples illustrating this property. Introduction
A natural problem in commutative algebra and algebraic geometry is to understand the group GA n ( R ) of automorphisms of a polynomial ring R [ X , . . . , X n ] over a ring R . Although much progress has beenmade in this direction during the last decades, one can state that only the case n = 2 and R is a field isfully understood. A central and fruitful notion in the study of polynomial automorphisms is the notion oftameness: an automorphism is called tame if it can be written as a composition of affine and triangularones, where by a triangular automorphism, we mean an automorphism F = ( F , . . . , F n ) ∈ GA n ( R ) such that F i ∈ R [ X i , . . . , X n ] for every i = 1 , . . . , n . Tame automorphisms form a subgroup TA n ( R ) of GA n ( R ) and a classical theorem due to Jung in characteristic zero [8] and van der Kulk in the generalcase [9] asserts that if R is a field k then GA ( k ) = TA ( k ) . The result is even more precise: GA ( k ) is the free product of the subgroups of affine and triangular automorphisms amalgamated over theirintersection. In contrast, even the equality TA ( R ) = GA ( R ) is no longer true for a general domain R ,as illustrated by a famous example due to Nagata : for an element z ∈ R \ { } the endomorphism F = (cid:0) X − Y ( zx + Y ) − z ( zX + Y ) , Y + z ( zX + Y ) (cid:1) of R [ X, Y ] is in GA ( R ) and can be decomposed as F = ( X − z − Y , Y )( X, z X + Y )( X + z − Y , Y ) in GA ( K ( R )) = TA ( K ( R )) . Such a decomposition being essentially unique, this implies in particularthat if z is not invertible in R , then F cannot be tame over R . Note that more generally, given a primeideal p ∈ Spec ( R ) , F ∈ TA ( R p ) if and only if z p .Automorphisms F ∈ GA n ( R ) such that F ∈ TA n ( R p ) for every p ∈ Spec ( R ) are said to be locallytame . Of course, every tame automorphism is locally tame. In contrast, the Nagata automorphism isneither tame nor locally tame. This could suggest that, at least for plane polynomial automorphisms,tameness is a property that can be checked locally on the base ring. In particular, one could hope that theonly reason why an automorphism F ∈ GA ( R ) is not tame is because there exists a prime p ∈ Spec ( R ) such that F is already non tame over R p . It turns out that this hope is too optimistic, and that ingeneral, some “global” properties of R have to be taken into account to be able to infer tameness directlyfrom local tameness. The main result of this article is the following characterization of rings for whichglobal tameness can be checked locally : Theorem.
For a domain R , the following assertions are equivalent :1) TA ( R ) = \ p ∈ Spec( R ) TA ( R p ) ,2) Every -generated locally free R -module of rank is free. In particular, it follows that over a unique factorization domain R , tameness is a local property ofautomorphisms.The article is organized as follows. Section one is devoted to the proof of the above characterization,that we essentially derive from the fact that tame automorphisms of a polynomial ring in two variablescan be recognized algorithmically. In section two, we consider many examples that illustrate condition2) in the Theorem above. Funded by a free competition grant of the Netherlands Organisation for scientific research (NWO).Partially supported by FABER grant 07-512-AA-010-S-179 and PHC Grant Van Gogh 18153NA.Partially supported by PHC Grant Van Gogh 18153NA.Funded by Veni-grant of council for the physical sciences, Netherlands Organisation for scientific research (NWO). From Local tameness to global tameness
In this section, we characterize domains R with the property that an automorphism F = ( F , F ) ∈ GA ( R ) is tame if and only if it is locally tame. For an automorphism F = ( F , F ) ∈ GA ( R ) , we let deg F = (deg F , deg F ) ∈ ( N ∗ ) considered as equipped with the product order. We denote by F i the homogeneous component of F i ofdegree deg F i , i = 1 , . An automorphism with deg F = (1 , is affine, and we denote by Aff ( R ) thecorresponding subgroup of GA ( R ) .1.1. Properties of automorphisms.
Even if the equality GA ( R ) = TA ( R ) is no longer true for a general domain R , tame automorphismsof a polynomial ring in two variables can be recognized algorithmically. Indeed, the following result quotedfrom [5, Prop. 1] (see also [4, Cor. 5.1.6]) says in essence that for every F ∈ TA ( R ) with deg F > (1 , there exists a linear or a triangular automorphism ϕ such that deg ϕF < deg F . Proposition 1.
Let F = ( F , F ) ∈ TA ( R ) and let ( d , d ) = deg F . Then the following holds:a) d | d or d | d .b) If max ( d , d ) > then we have:(i) If d < d then F = cF d /d for some c ∈ R ,(ii) If d < d then F = cF d /d for some c ∈ R ,(iii) If d = d then there exists ϕ ∈ Aff ( R ) such that ϕF = ( F ′ , F ′ ) satisfies deg F ′ = d and deg F ′ < d . In contrast to the tame case, for an arbitrary automorphism F = ( F , F ) ∈ GA ( R ) with deg F =deg F there is no guarantee in general that there exists ϕ ∈ Aff ( R ) such that deg ϕF < deg F . Indeed,such a ϕ exists if and only there exists a unimodular vector ( α , α ) ∈ R such that α F + α F = 0 ,which is the case if and only if the R -module RF + RF is free of rank . Combined with [4, Ex. 6 p.94], this observation leads to a natural procedure to construct families of locally tame but not (globally)tame automorphisms, namely: Proposition 2. If z, w ∈ R and q ( T ) ∈ R [ T ] is a polynomial of degree at least , then F := ( X + wq ( zX + wY ) , Y − zq ( zX + wY )) is an element of GA ( R ) . Furthermore F is tame if and only if ( z, w ) is a principal ideal of R .In particular, if ( z, w ) is a locally principal but not principal ideal, then F is a locally tame but notglobally tame automorphism.Proof. A straightforward verification shows that G = ( X − wq ( zX + wY ) , Y + zq ( zX + wY )) is aninverse for F . Suppose that ( z, w ) = aR for some a ∈ R . Replacing q ( T ) , z and w by aq ( aT ) , a − z and a − w respectively, we may assume that ( z, w ) = R . But then if we take any ϕ ∈ SL ( R ) having zX + wY as its first component, one checks that F = ϕ − ( X, Y − q ( X )) ϕ ∈ T A ( R ) . Conversely, if F ∈ TA ( R ) , then, since deg F = deg F = deg Q > , it follows from Proposition 1 and the abovediscussion that the R -module generated by F = wq ( zX + wY ) and F = − zq ( zX + wY ) is free of rank . Simplifying by q ( zX + wY ) , we get that the R -module generated by w and − z is free a rank , i.e., ( w, − z ) is a principal ideal. (cid:3) It follows that locally tame but not globally tame automorphisms abound : for instance, in theproposition above, one can take for R the coordinate ring of a smooth non rational affine curve C andfor z, w a pair of generators of the defining ideal of a non principal Weil divisor on C (see also section 2below for more examples).1.2. A criterion.
It turns out that the examples discussed above illustrate the only global obstruction to infer globaltameness from local tameness, namely, the existence of -generated locally free but not globally freemodules of rank . Indeed, we have the following criterion. Theorem 3.
For a domain R , the following assertions are equivalent:1) TA ( R ) = \ p ∈ Spec( R ) TA ( R p ) ,2) Every -generated locally free R -module of rank is free. OCALLY TAME PLANE POLYNOMIAL AUTOMORPHISMS 3
Proof. ⇒ R is a domain, every locally free R -module of rank is isomorphic to an R -submodule of the field of fractions K ( R ) of R (see e.g. [7, Prop. 6.15]). In turn, every such submoduleis isomorphic to an ideal of R . In particular, if there exists a locally free but non free -generated R -module of rank , then there exists locally principal but not principal ideal ( z, w ) of R . But then any F ∈ GA ( R ) as in Proposition 2 above is locally tame but not tame.2) ⇒ R , it is clear that TA ( R ) ⊆ \ p ∈ Spec( R ) TA ( R p ) ⊆ \ p ∈ Spec( R ) GA ( R p ) = GA ( R ) . Let F = ( F , F ) ∈ GA ( R ) be a locally tame automorphism and let d i = deg F i , i = 1 , . We mayassume that d ≤ d . If d = d = 1 then F is affine, whence tame. We now proceed by induction on ( d , d ) , assuming that every locally tame automorphism of degree ( d ′ , d ′ ) < ( d , d ) is globally tame. • Case : d < d . Since F ∈ TA (cid:0) R (0) (cid:1) = TA ( K ( R )) , it follows from Proposition 1 that e = d /d ∈ N ∗ and that there exists α ∈ K ( R ) such that F = αF e . But since F ∈ TA ( R p ) for every p ∈ Spec ( R ) , it follows that α ∈ \ p ∈ Spec( R ) R p = R. Now, the automorphism ( X, Y − αX e ) F satisfies the induction hypothesis and we are done with case. • Case : d = d . Since for any p ∈ Spec ( R ) , we have F ∈ TA ( R p ) , it follows from Proposition1 and the discussion 1.2 that for every p ∈ Spec ( R ) , the R p module generated by F and F is free ofrank . This means exactly that the R -module generated by F and F is locally free of rank . Ourassumption implies that it is globally free, and so, we deduce from 1.2 that there exist ϕ ∈ Aff ( R ) suchthat deg ϕF < deg F . (cid:3) Recall that the
Picard group of a ring R is the group Pic ( R ) of isomorphy classes of locally free R -modules of rank . In view of the above criterion, it is natural to introduce the subgroup Pic ( R ) of Pic ( R ) generated by isomorphy classes of locally free R -modules of rank that can be generated by elements. With this definition, property 2) in Theorem 3 is equivalent to the triviality of Pic ( R ) . Inparticular, we obtain: Corollary 4. If Pic ( R ) = { } and F belongs to GA ( R ) , then F is tame if and only if it is locallytame. Example 5.
The class of rings with
Pic ( R ) = { } contains in particular unique factorization domainssince for these domains the Picard group itself is trivial. This also holds for Bézout rings , that is, domainsin which every finitely generated ideal is principal (see e.g. [2]).1.3.
Minimal overring for tameness.
Recall that GA ( R ) = TA ( R ) if and only if R is a field [4, Proposition 5.1.9]. If F ∈ GA ( R ) ,then F is tame over the field of fractions K of R , but, in general, there does not exist a smallest ring S between R and K such that F is tame over S . Indeed, letting R = C [ z, w ] every automorphism F as inProposition 2 is tame over R (cid:2) z − (cid:3) and R (cid:2) w − (cid:3) but not over R = R (cid:2) z − (cid:3) ∩ R (cid:2) w − (cid:3) . However, if wefurther assume that R is a Bézout domain, we have the following result. Proposition 6.
Let R be a Bézout domain and let ( R j ) j ∈ J be a family of rings between R and K suchthat R = \ j ∈ J R j . Then TA ( R ) = \ j ∈ J TA ( R j ) .Proof. Similarly as in the proof of Theorem 3, we proceed by induction on the degree of F = ( F , F ) ∈ GA ( R ) ∩ \ j ∈ J TA ( R j ) , the case deg F = (1 , being obvious. Letting d i = deg F i , we may assume that d ≤ d . • Case : d < d . Then e = d /d ∈ N ∗ and there exists α ∈ K such that F = αF e . Since F ∈ T A ( R j ) , we have α ∈ R j for every j ∈ J , and so α ∈ R = \ j ∈ J R j . Now the automorphism ( X, Y − αX e ) F satisfies the induction hypothesis. • Case 2 : d = d . Since F and F are K -linearly dependent, the R -module RF + RF is isomorphicto a proper ideal of R . As R is a Bézout domain, the latter is free of rank , and so, we conclude from1.2 above that there exists ϕ ∈ Aff ( R ) such that deg ϕF < deg F . (cid:3) OCALLY TAME PLANE POLYNOMIAL AUTOMORPHISMS 4
Proposition 7. If R is a Bézout domain and F ∈ GA ( R ) then there exists a smallest ring S between R and K ( R ) such that F ∈ TA ( S ) . Furthermore, S is a finitely generated R -algebra.If we assume further that R is a principal ideal domain, then there exists r ∈ R \ { } such that S = R (cid:2) r − (cid:3) .Proof. Any ring between R and K ( R ) is again a Bézout domain [2, Theorem 1.3]. Therefore, theexistence of S is a consequence of the previous proposition. The fact that S is finitely generated fol-lows from Proposition 1 by easy induction. For the last assertion, since S is finitely generated over R , there exists a finitely generated ideal I ⊂ R and an element r ∈ R \ { } such that S = R [ I/r ] = (cid:8) a/r k ∈ K ( R ) , a ∈ I k , k = 0 , . . . (cid:9) . Since R is a p.i.d, I is a principal ideal, say generated by an element g ∈ R . After eliminating common factors if any, we may assume that r and g are relatively prime andthat S = R [ g/r ] ⊂ R (cid:2) r − (cid:3) . But by Bézout identity, there exists u, v ∈ R such that ur + vg = 1 and so, S = R (cid:2) r − (cid:3) . (cid:3) Example 8. If F ∈ GA ( C [ z ]) , then there exists a smallest ring S between C [ z ] and C ( z ) of the form C [ z ] (cid:2) r − (cid:3) such that F ∈ GA ( S ) .2. Examples and complements
Here we discuss examples of domains R which illustrate the property Pic ( R ) = { } .2.1. The condition
Pic ( R ) = { } for -dimensional noetherian domains. If R is a noetherian domain of Krull dimension , every locally free R -module of rank j is generatedby at most j + 1 elements (see e.g. [10, Th. 5.7]). In particular, we have Pic ( R ) = Pic ( R ) for everynoetherian domain of dimension . As a consequence, we get: Example 9. If R is a Dedekind domain, the following are equivalent :(1) Pic ( R ) = { } ; (2) Pic ( R ) = { } ; (3) R is a UFD; (4) R is a p.i.d.For the coordinate ring R of an affine curve C defined over an algebraically closed field, we have thefollowing classical result: Proposition 10.
The Picard group of R is trivial if and only if C is a nonsingular rational curve.Proof. Let ˜ C = Spec( ˜ R ) be the normalization of C . By virtue of [13, Theorem 3.2], the natural surjection Pic( C ) → Pic( ˜ C ) is an isomorphism if and only if R = ˜ R . Therefore, if Pic(C) is trivial, then C isnecessarily a nonsingular curve. Now it is well known that a nonsingular curve has trivial Picard groupif and only if it is rational (see e.g. [3, §11.4 p. 261]). (cid:3) Corollary 11.
Let R be the coordinate ring of a rational affine curve and let ˜ R be its integral closure in K ( R ) . If F ∈ GA ( R ) is locally tame, then F ∈ TA (cid:16) ˜ R (cid:17) .Proof. Indeed, with the notation of the previous proof, one has F ∈ TA ( O p ) for every p ∈ C = Spec ( R ) and so F ∈ TA (cid:16) ˜ O p (cid:17) for every p ∈ C . Since ˜ R = T p ∈ C ˜ O p , it follows that F is locally tame over ˜ R ,whence tame by virtue of Proposition 10. (cid:3) Example 12.
Let R = C [ u, v ] / (cid:0) v − u (cid:1) be the coordinate ring of a cuspidal rational curve C . Via thehomomorphism C [ u, v ] → C [ t ] , ( u, v ) (cid:0) t , t (cid:1) we may identify R with the subring C (cid:2) t , t (cid:3) of C [ t ] andthe integral closure ˜ R of R with C [ t ] . For every a ∈ C ∗ , we let I a = (cid:0) t − a , t − a (cid:1) be the maximalideal of the smooth point (cid:0) a , a (cid:1) of C . In particular, I a is locally principal but one checks easily thatit is not principal. So for ( z, w ) = (cid:0) t − a , t − a (cid:1) , any automorphism F as in Proposition 2 is locallytame but not tame. On the other hand, I a ˜ R is principal, generated by t − a , and so, F ∈ TA ( C [ t ]) .2.2. Examples of rings with
Pic ( R ) = { } but Pic ( R ) = { } . As observed above, for -dimensional domains R , the triviality of Pic ( R ) is equivalent to the one of Pic ( R ) . Here we give examples of domains with Pic ( R ) = { } and Pic ( R ) = { } which are coordinaterings of smooth affine algebraic varieties. Let Q be a smooth quadric in the complex projective space P n = P n C , n ≥ , and let U = P n \ Q . Asis well-known, U is smooth affine variety with Picard group isomorphic to Z , generated by the restrictionto U of the invertible sheaf O P n (1) on P n . Letting R n = Γ ( U, O U ) and M n = Γ ( U, O P (1)) , which is alocally free R n -module of rank , we have the following result. OCALLY TAME PLANE POLYNOMIAL AUTOMORPHISMS 5
Proposition 13.
The minimal number of generators of M n as an R n -module is [ n/
2] + 1 . In particular,if n ≥ then Pic ( R n ) = { } whereas Pic ( R n ) ≃ Z .Proof. Up to the action of
PGL n +1 ( C ) , we may assume that Q ⊂ P n = Proj ( C [ x , . . . , x n ]) is thehypersurface q = 0 , where q = x + · · · + x n ∈ C [ x , . . . , x n ] . Letting Q ⊂ A n +1 = Spec ( C [ x , . . . , x n ]) be the quadric defined by the equation q = 1 , the natural map A n +1 \ { } → P n restricts to an étaledouble cover Q → P n \ Q expressing the coordinate ring R n of P n \ Q as the ring of invariant functions of A = C [ x , . . . , x n ] / ( q − for the Z -action induced by − id on A n +1 . With this description, O P n (1) | U coincides with the trivial line bundle Q × A equipped with the nontrivial Z -linearization Q × A ∋ ( x, u ) ( − x, − u ) ∈ Q × A (see e.g. [11, §1.3]). It follows that we may identify regular functions on U and global sections of O P n (1) | U with cosets in A of even and odd polynomial functions on A n +1 respectively. • Case : n = 2 m is even. Clearly, the m + 1 odd polynomials p j = x j + ix j +1 for ≤ j ≤ m − and p m = x m have no common zero on Q . Therefore, the corresponding sections of O P n (1) | U generate M n as an R n -module. Let us show that M n cannot be generated by less than m + 1 elements. Otherwise,we could find in particular m odd polynomial functions s , · · · , s m on A n +1 with no common zero on Q .Writing s j = a j + ib j for suitable odd polynomials a j, b j ∈ R [ x , . . . , x n ] , this would imply in particularthat the n odd real polynomials a , . . . a m and b , . . . , b m have no common zero on Q ∩ R n +1 . This isimpossible. Indeed, since Q ∩ R n +1 is the real n -sphere S n , it follows from Borsuk-Ulam theorem that themap φ = ( a , . . . , a m , b , . . . , b m ) : S n → R n takes the same value on a pair of antipodal points, hence,being odd, vanishes on a pair of antipodal points. • Case : n = 2 m + 1 is odd. One checks in a similar way as above that the m + 1 global sectionsof O P (1) | U corresponding the odd polynomials p j = x j + ix j +1 , ≤ j ≤ m generate M n as an R n -module. Now if M n was generated by m elements, then there would exists m odd polynomials s j = a j + ib j as above for which the polynomials a j , b j ∈ R [ x , . . . , x n ] , j = 1 , . . . , m have no common zero on the real n -sphere S n . But then the continuous map φ = ( a , . . . , a m , b , . . . , b m ,
0) : S n = Q ∩ R n +1 → R n wouldcontradict the Borsuk-Ulam theorem. (cid:3) Remark . An argument very similar to the one used in the proof above shows that over the subring ˜ R n of R [ x , . . . , x n ] / (cid:0) x + · · · x n − (cid:1) consisting of cosets of even polynomials, the module ˜ M n consisting ofcosets of odd polynomials cannot be generated by less than n + 1 elements. This property seems to havebeen first observed by Chase (unpublished). Our proof is deeply inspired by an argument due to Gilmer[6] on a slightly different example.2.3. Further research.
One may wonder if there exists a complete characterization of obstructions to infer global tamenesfrom local tameness for higher dimensional polynomial automorphisms similar to Theorem 3. A goodstarting point would be to have in general an effective algorithmic way to recognize tame automorphisms.Unfortunately, at the present time, such an higher dimensional algorithm only exists for automorphismsof a polynomial ring in 3 variables over a field [12].
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OCALLY TAME PLANE POLYNOMIAL AUTOMORPHISMS 6
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