Deveney, James K.

Finston, David R.

Osaka City University Repository

Deveney, J.K. and Finston, D.R.
Osaka J. Math.
39 (2002), 275–282
REGULAR Ga INVARIANTS
JAMES K. DEVENEY and DAVID R. FINSTON
(Received July 10, 2000)
1. Introduction
Let denote the additive group of complex numbers, and a complex affine
variety. By an action of on we will mean an algebraic action. It is well known
(e.g. [4]) that every such action can be realized as the exponential of some locally
nilpotent derivation of the coodinate ring C[ ] and that every locally nilpotent
derivation gives rise to an action. The ring 0 of invariants in C[ ] is equal to
the ring of constants of the generating derivation.
Given an action σ : × → , let σ¯ : × → × denote the graph
morphism and σˆ : C[ ] → C[ ] (resp. σ˜ : C[ × ] → C[ ]) denote the induced
maps on coordinate rings.
The action is said to be proper if σ¯ is a proper morphism (i.e. if C[ ] is in-
tegral over the image of σ˜). The action is said to be equivariantly trivial if there is
a variety for which is a equivariantly isomorphic to × , the action on
× being given by ∗ ( ) = ( + ). The action is locally trivial if there
are affine varieties and a cover of by stable affine open subsets on which
the action is equivariantly trivial. Equivariant triviality of an action on is equivalent
with the existence of a regular function ∈ C[ ] for which = 1. Such a function
is called a slice and, if one exists, C[ ] = 0[ ]. Local triviality is equivalent with the
intersection of the kernel and the image of generating the unit ideal in C[ ].
Locally trivial actions are proper, and proper actions on C are locally trivial pro-
vided C[ ] is a flat ring extension of 0 [4, Theorem 2.8]. It was also shown there,
for = C , that properness is equivalent with surjectivity of σ˜. It had been believed
e.g. [13] that proper actions on normal varieties are locally trivial, until an exam-
ple of a proper action on C5 which is not locally trivial was produced [5]. In that ex-
ample, 0 is affine, but the associated variety has a line of singularities. The fibers
of the morphism C5 → over the singular points are all two dimensional. The first
example of a locally trivial but not equivariantly trivial action on complex affine
space was discovered by Winkelmann [18]. In that example, 0 is affine and regular
and, at this writing, no example of a locally trivial action on C with non regular ring
of invariants is known.
1991 Mathematics Subject Classification : Primary 14L30, Secondary 20G20.
276 J.K. DEVENEY AND D.R. FINSTON
As a consequence of the main result of the paper, Theorem 2.1, singularities in
the variety associated to 0 are shown to be the only obstruction to local triviality of
a proper action. Moreover, the structure of the morphism C → is elucidated in
some cases where is singular. Finally, the main result leads to a generalization from
= 3 to arbitrary of an algorithm given in [12] to determine whether a set of − 1
elements of C[ 1 . . . ] is part of a set of variables.
2. Smooth points of Y
Denote the polynomial ring C[ 1 . . . ] by C[ ]. Suppose that is an affine
subring of C[ ] whose quotient field has transcendence degree − 1 over C, and
let be a multiplivatively closed subset of . If, for some { 1 . . . −1} ⊂ ,
{ 1 . . . −1} generates the module of differentials −1 /C, then −1 [ ]/ −1 is
the quotient of the free module generated by { | 1 ≤ ≤ } by the submodule
generated by { | 1 ≤ ≤ −1}. Thus the first Fitting ideal of −1 [ ]/ −1 is gen-
erated by the ( −1)×( −1) minors of the Jacobian matrix [∂ /∂ ] [10, Sec. 20.2].
In particular, −1 [ ]/ −1 is free of rank one if and only if the first Fitting ideal is
−1 [ ]
.
Theorem 2.1. Let be a locally nilpotent derivation of C[ ] whose associated
action is fixed point free. Suppose that the ring of invariants 0 is finitely gener-
ated and let denote the associated affine variety. If ∈ is a smooth point defined
by a maximal ideal of 0, and ≡ 0 − , then −1 [ ]/ −1 0 is free of rank one.
Proof. Since is a smooth point there are ∈ 0, 1 ≤ ≤ −1, which generate
the maximal ideal of 0 . These elements define a 0 derivation 1 of C[ ] by
1 ≡ det Jac( 1 . . . −1)
Since the action generated by is free, the C[ ] module of derivations of C[ ]
over 0 is free of rank one [9, Prop. 2.1]. Thus there are 0, 1 ∈ C[ ] with
0 = 1 1. The fixed point freeness of the action generated by is equivalent with
( 1 . . . ) = C[ ]. It follows that 1 divides 0 and thus 1 is a C[ ] multiple of
.
It is well known that ≡ 0 /C is generated by the . By the remarks above,
the 1( ), i.e. the ( −1)×( −1) minors of [∂ /∂ ], generate the first Fitting ideal
of ′, the module of differentials of −1C[ ] over 0 . According to [3, Cor. 3.9],
this ideal is contained in no height one prime ideal. Thus 1 is a multiple of by a
unit in −1 [ ]. Since the generate the unit ideal in C[ ], the 1 generate the
unit ideal in −1C[ ]. Thus the first Fitting ideal is −1C[ ] and ′ is free of rank 1.
REGULAR INVARIANTS 277
Corollary 2.2. With conditions as in the theorem, suppose in addition that is
smooth. Then the morphism C → induced by the ring inlusion 0 → C[ ] is
smooth of relative dimension 1.
Proof. Since C[ ]/ 0 is finitely presented, it suffices by the Quillen-Suslin theo-
rem to show that it is locally free as a C[ ] module. But this follows immediately from
the theorem.
It should be noted that the fixed point freeness assumption is essential. If is the
locally nilpotent derivation of C[ ] given by
: 7→ 7→ 7→ 0
then the ring of invariants for the associated action is well known to be C[ 2 − 2].
Since (2 − 2) = 2 − 2 + 2 and [ − ] is not a unimodular row over
C[ ], the module of differentials of C[ ] over C[ 2 − 2] is not free.
Corollary 2.3. If acts on C without fixed points and 0 is affine and regu-
lar, then the action is locally trivial if and only if it is proper.
Proof. By the previous corollary, C → is smooth and therefore flat. The result
then follows from [4, Theorem 2.8].
The next application of Theorem 2.1 generalizes a criterion for locally triviality
in [8]. The notion of GICO morphism was introduced by Miyanishi in [14].
DEFINITION 1. Let φ : → be a morphism of affine varieties. Then φ is GICO
over provided that for any height one prime ideal of C[ ] and prime ideal of
C[ ] minimal over C[ ], defining a codimension one subvariety of , the field
C(φ ) is algebraically closed in C( ).
Suppose that a action on the affine variety has finitely generated ring of
invariants 0. With denoting the affine variety with coordinate ring 0, the action
is said to be GICO if the morphism → induced by the inclusion 0 ⊂ C[ ]
is GICO. It should be noted that if is factorial, i.e. C[ ] is a ufd, then C[ ] is a
factorially closed subring of C[ ], hence also a unique factorization domain. Thus we
are concerned with the extension of the quotient field of 0/( ) to the quotient field
of C[ ]/ C[ ] for all principal prime ideals ( ) of 0.
For a action on a factorial affine variety, the GICO condition is easily seen
to be equivalent to the condition that the intersection of 0 and image of the generat-
ing derivation, which is an ideal of 0, is contained in no height one prime ideal of
0. In [8], it was shown that GICO actions on C with regular invariants are locally
278 J.K. DEVENEY AND D.R. FINSTON
trivial, with the added hypothesis that the morphism C → has open image. It was
also shown there that proper actions on factorial affine varieties are GICO. Testing the
GICO condition seems to be difficult, while properness is very easy to check. On the
other hand, no actions which are fixed point free but not GICO are known to the au-
thors.
In light of Cor. 2.2, the hypothesis that the image of C → is open can be
dropped:
Corollary 2.4. A GICO action on C is locally trivial provided that 0 is
finitely generated and regular.
3. Nonregular invariants
Consider a GICO action on = C generated by the locally nilpotent derivation
with finitely generated invariant ring 0. Let π : C → as above be the morphism
induced by the ring inclusion 0 ⊂ C[ ], and let denote the ideal 0 ∩ im . Denote
by the closed subset of defined by , observing that every irreducible component
of has codimension at least two and that π| −π−1 : − π−1 → − is a prin-
cipal bundle.
Recall the following lemma of Miyanishi [14, Section 2].
Lemma 3.1. Let ( ) e a regular local ring of dimension ≥ 2 and let
be a factorial, finitely generated domain with → . Let : → be the
morphism induced by the ring inclusion, where = Spec and = Spec . Let
= −{ }. Assume that : −1( ) → is an A1 bundle. Then either ∼= A1×
or −1({ }) = ∅ (the latter is only possible if = 2 ).
This lemma applies to the investigation of the dimensions of fibers of C →
over singular points when is not regular but the action is geometrically irreducible
in codimension one (GICO). All of the pathological examples known to the authors,
in particular the proper but not locally trivial action in [5] and the nonproper twin tri-
angular actions investigated in [7], satisfy the hypothesis of the following theorem.
Theorem 3.2. Consider a GICO action of on = C and assume that 0 is
affine and Cohen Macauley defining the affine variety . Assume also that the singular
locus of has dimension strictly less than the minimum of the dimensions of the
irredicible components of . Then either the action is locally trivial (i.e. π−1( ) = ∅ )
or π(π−1( )) ⊂ . In the latter case, fibers over points in are either empty or
have dimension strictly greater than 1.
Proof. Assume that the action is not locally trivial, so that the image of π has
nonempty intersection with some irreducible component 1 of . Let be a prime
REGULAR INVARIANTS 279
ideal of 0 defining 1, and note that by assumption 0 is a regular local ring. Set
= 0− . By Miyanishi’s lemma, applied to ( ) = ( 0 0 ) and = −1C[ ],
we see that either has a slice or the height of is equal to 2 and = .
The first case leads to a contradiction: If has a slice, write it as / with ,
∈ C[ ], ∈ . From 1 = ( / ) one readily concludes that and ( ) have a
common factor, which is impossible for a locally nilpotent derivation, unless ( ) = 0.
It follows that = ( ) ∈ 0 ∩ im( ) ⊂ , a contradiction.
As a consequence of the second case, π(π−1( 1)) is not dense in 1. Since the
dimension of is one more than the dimension of , and is normal, a theorem
of Chevalley [1] implies that nonempty fibers over points of 1 must have dimension
strictly greater than 1.
If is smooth at ∈ 1 ∩ im(π) then Theorem 2.1 shows that there are open
neighborhoods of in and of π−1( ) so that π| : → is smooth. But a
smooth morphism is open and therefore has dense image in 1. Since the image of π
is not dense in 1 no such exists.
4. Extendibility to a coordinate system
In [16], Rabier gives a simple algorithm to determine if ∈ C[ ] is a variable,
i.e. if there is an with C[ ] = C[ ]. In [17, Cor. p. 160], a criterion for an
element to be a variable in C[ ] is given, and in [12], van den Essen gives an
algorithm to determine if two elements , are part of a coordinate system for C3.
We say that a set { 1 . . . −1} ⊂ C[ ] is part of a coordinate system for C if
exists so that C[ 1 . . . −1 ] = C[ ]. We extend the method in [12] to give an
algorithm to decide whether −1 polynomials are part of a coordinate system for C .
The algorithm is based on the following theorem:
Theorem 4.1. A set of polynomials { 1 . . . −1} is part of a coordinate sys-
tem for C if and only if C[ 1 . . . −1] is the ring of invariants for a proper
action on C . In this case, the action is generated by the derivation
: 7→ λ det Jac( 1 . . . −1 )
for some λ ∈ C∗.
Proof. If C[ ] = C[ 1 . . . −1 ] then the derivation : 7→ 0, 7→ 1 gener-
ates the desired action. It is straightforward to verify that ( ) = λ∂ /∂ where
λ = det Jac( 1 . . . −1 ) ∈ C∗.
Conversely, a proper action is fixed point free [4, Theorem 2.3], so that
Corollary 2.3 shows that the action is locally trivial. But since the ring of invariants
is a polynomial ring, [6, Theorem 3.3] shows that the action is conjugate to a trans-
lation. Thus, with denoting the derivation generating the action, there is an ele-
280 J.K. DEVENEY AND D.R. FINSTON
ment ∈ C[ ] with ( ) = 1. From [19, Proposition 2.1], it follows that C[ ] =
C[ 1 . . . −1 ].
This theorem yields an extension of the algorithm in [12] to decide if
{ 1 . . . −1} is part of a coordinate system for C . Given { 1 . . . −1},
1. Define a derivation on C[ ] by
(θ) = det Jac(θ 1 . . . −1)
If ( ( 1) . . . ( )) is not the unit ideal of C[ ], then cannot generate a fixed
point free action, and therefore { 1 . . . −1} is not part of a coordinate system.
2. Check whether is locally nilpotent. As in [12], calculate
= max
1≤ ≤
{[C( 1 . . . ) : C( 1 . . . −1 )]}
using the algorithm in [15, Lemma 2.3]. is locally nilpotent if and only if = 0
for each . If is not locally nilpotent, then { 1 . . . −1} is not a part of a coordi-
nate system.
3. Check that C[ 1 . . . −1] = 0, the ring of invariants. While it is not known
a priori that the ring of invariants for a fixed point free action on C is finitely
generated for > 3, the algorithm in [11] can be modified to determine whether this
ring is C[ 1 . . . −1]. By steps 1 and 2, 0 6= ∈ 0 for some , . If is
not in C[ 1 . . . −1], then C[ 1 . . . −1] 6= 0 and { 1 . . . −1} is not part of a
coordinate system.
If is in C[ 1 . . . −1], set = −1 / , noting that = 1. Follow-
ing algorithm in [11] calculate = exp(− ) | = for each . Set = [ ( )]
where is the least exponent for which [ ( )] ∈ 0. Use the subalgebra mem-
bership algorithm [2] to determine whether the ∈ C[ 1 . . . −1]. If any does
not lie in C[ 1 . . . −1], then { 1 . . . −1} is not part of a coordinate system. If
∈ C[ 1 . . . −1] for each , then
C[ 1 . . . −1] ⊂ 0 ⊂ C
[
1 . . . −1
1
( )
]
Assuming these inclusions, the algorithm in [11] constructs an increasing chain
of subrings of 0 beginning with C[ 1 . . . −1] which eventually reaches 0 if this
ring is finitely generated. For our purposes, it suffices to construct the first such sub-
ring [11]. If it is properly larger than C[ 1 . . . −1] then { 1 . . . −1} is not part
of a coordinate system. Otherwise, C[ 1 . . . −1] = 0.
4. Check that the action is proper. In view of [4, Theorem 2.8] this is equivalent
to
∈ C[ 1 . . . exp( ) 1 . . . exp( ) ]
REGULAR INVARIANTS 281
and requires one application of the subalgebra membership algorithm to decide.
If the action is not proper, then { 1 . . . −1} is not part of a coordinate system.
Otherwise, Theorem 4.1 shows that { 1 . . . −1} is part of a coordinate system.
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J.K. Deveney
Department of Mathematical Sciences,
Virginia Commonwealth University,
1015 W. Main St., Richmond, Virginia 23284
e-mail: jdeveney@atlas.vcu.edu
282 J.K. DEVENEY AND D.R. FINSTON
D.R. Finston
Department of Mathematical Sciences,
New Mexico State University,
Las Cruces, New Mexico 88003
e-mail: dfinston@nmsu.edu


Regular G_{a} invariants

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Regular G_{a} invariants

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