Deveney, James K.

Finston, David R.

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Additive group actions with finitely generated invariants

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Deveney, J.K. and Finston, D.R.
Osaka J. Math.
44 (2007), 91–98
ADDITIVE GROUP ACTIONS
WITH FINITELY GENERATED INVARIANTS
JAMES K. DEVENEY and DAVID R. FINSTON
(Received October 31, 2005, revised March 27, 2006)
Abstract
Every locally trivial action of the additive group of complex numbers on a
factorial affine variety has finitely generated ring of invariants. A criterion is
given for such an action on complex four space to be conjugate to a translation.
Restrictions on the nature of the singularities of the variety defined by the ring of
invariants of triangular actions are noted.
1. Introduction
Let Ga denote the additive group of complex numbers, and X a complex affine va-
riety. By an action of Ga on X we will mean an algebraic action. It is well known that
every such action can be realized as the exponential of some locally nilpotent derivation
D of the coordinate ring C[X ] and that every locally nilpotent derivation gives rise to
an action. The ring C0 of Ga invariants in C[X ] is equal to the ring of constants of the
generating derivation. While it is known that C0 need not be finitely generated, even
for actions on C5, the question of finite generation is interesting for special kinds of
actions. Indeed, while we have finite generation for all actions on normal varieties of
dimension  3, the known actions on Cn with nonfinitely generated invariants all are
(quasi)homogeneous and therefore have fixed points. It is unknown for n > 3 whether
every fixed point free action on Cn has finitely generated invariants, but the ring of
Ga invariants is finitely generated for all actions on C4 whose generating derivation is
triangulable (triangulable actions) [2].
An action is said to be equivariantly trivial if there is a variety Y for which X is
Ga equivariantly isomorphic to YGa , the action on YGa being given by g(y,h) =
(y, g + h). Equivariant triviality of an action on X is equivalent with the existence of a
regular function s 2 C[X ] for which Ds = 1. Such a function is called a slice and, if
one exists, C[X ] = C0[s]. In this case Y is a geometric quotient and C0 = C[Y ]. The
action is locally trivial if there are affine varieties Yi and a cover of X by Ga stable
affine open subsets X i on which the action is equivariantly trivial.
There are locally trivial actions on normal affine varieties with nonfinitely gener-
ated invariants [3]. Throughout this paper, the term “factorial affine variety” means an
2000 Mathematics Subject Classification. Primary 13A50; Secondary 14L24.
92 JAMES K. DEVENEY AND DAVID R. FINSTON
affine variety with trivial divisor class group, i.e. one who coordinate ring is a unique
factorization domain. The authors asserted in [4, Prop. 2.9 and Cor. 2.10] the existence
of locally trivial actions on factorial affine varieties. The argument was incorrect and
it is shown in Section 2 that in fact the ring of invariants for a locally trivial action
on a factorial affine variety is finitely generated.
It was shown in [9] that a locally trivial triangular action on C4 is equivariantly
trivial with quotient isomorphic to C3. In Section 4 a topological criterion for equi-
variant triviality of a locally trivial action is given.
Given an action  : Ga  X ! X , let ¯ : Ga  X ! X  X denote the graph
morphism and ˆ : C[X ] ! C[X , t] (resp. ˜ : C[X  X ] ! C[X , t]) denote the induced
maps on coordinate rings. The action is said to be proper if ¯ is a proper morphism
(i.e. if C[X ,t] is integral over the image of ˜ ). A proper action on X = Cn is known to
be locally trivial if C[X ] is a flat ring extension of C0 or if C0 defines a smooth variety
[6, Theorem 2.8]. While a finitely generated C0 need not define a smooth variety in
general, for triangular actions on C4 isolated singularities of this variety can be only of
a restricted type, namely canonical singularities. An example is given of a fixed point
free but nonproper triangular action on C4 with nonisolated singularities. The authors
know of no example of a proper triangular action on C4 with nonregular C0, i.e. all
known proper actions are equivariantly trivial.
2. Finite generation for locally trivial actions
From [8] we know that the quotient of a locally trivial action on an affine factorial
variety X exists as a quasiaffine variety Y 0  Spec R0, where R0 is the subring of C0
constructed as follows: Let Æ(a1), : : : , Æ(an) 2 C0 generate the unit ideal in C[X ], and
set Ri = C[X , 1=Æ(ai )]Ga . Note that C[X , 1=Æ(ai )] = Ri [ai=Æ(ai )] so that Ri is a finitely
generated C algebra, say Ri = C[bi1, : : : , bim , 1=Æ(ai )], with bi j 2 C0. It is easy to
check that Ri = C0[1=Æ(ai )]. The ring
R0 = C[bi j , Æ(ai ) j 1  i  n, 1  j  m]
is the required subring of C0 and, with Yi  Spec Ri , we have Y 0 =
S
i Yi . Note that
Ri is a unique factorization domain, so that Yi and therefore Y 0 are normal.
Denote by R the integral closure of R0, which is a finitely generated R0 module.
Factorial closedness of C0 in C[X ] implies that R is a subring of C0, in fact C0 is
the factorial closure of R and therefore the morphism X ! Spec R0 factors through
Spec R. We can therefore replace R0 by R and Y 0 by the image Y of X in Spec R.
Thus X ! Y is a geometric quotient and X ! Spec R an open morphism. Moreover,
C0 is the ring of global sections of the structure sheaf of Y , and isomorphic to TI (R),
the ideal transform of R with respect to the radical ideal defining the (Zariski closed)
complement of Y in Spec R.
ADDITIVE GROUP ACTIONS 93
REMARK 1. If R is regular then Spec R   V (J ) is affine variety for any height
one ideal J e.g. [12]. When R is regular it follows that C0 is affine and regular, and
Y is smooth as well. Since X is the total space of a principal Ga bundle over the
smooth quasiaffine Y , X is smooth.
More generally, we have the
Theorem 2.1. Let Ga act locally trivially on the factorial affine variety X . Then
the ring of Ga invariants in C[X ] is finitely generated.
Proof. With R, Y , I as above, note that if the height of I is at least 2, i.e. prime
ideals minimal over I all have height  2 so that
codimSpec R(Spec R   Y ) > 1
then C0 = R. Assume then that ht(I ) = 1 and write I = J \ K where J is the
intersection of the height one prime ideals minimal over I and K the intersection of
the minimal prime ideals of I of height greater than one. Then C0 = TI (R) = TJ (R)
since R is integrally closed [13, p.41 Corollary]. We claim that for each maximal ideal
m of R, C0m is flat over Rm and therefore that C0 is flat over R. The assertion then
follows from [15, Corollary 3.5].
Denote by Z the subscheme of Spec R defined by J and by W its complement.
In the terminology of [1],
Naf (Z )  fx 2 Spec R : W \ Spec Rx is not affineg
is a closed subset of Z , since R is noetherian, and empty if and only if W is affine.
We claim that Naf (Z ) = ;. Let m be the maximal ideal of R defining a closed point
z of Z , and set S = R   m  C0. Since Z is a component of the complement of
the image of f : X ! Spec R, mC[X ] = C[X ]. We claim that q f (R)  S 1C[X ].
Indeed, if 0 6= r 2 R is a nonunit in S 1C[X ], let M be a maximal ideal of S 1C[X ]
containng r S 1C[X ]. Then C = C[X ]=M \C[X ] is a field which is finitely generated
as a ring over R=M \ R. Thus R=M \ R is a field, and M \ R = m, contradicting the
assumption that z =2 im( f ).
Since q f (R) = q f (C0), we obtain S 1C[X ] = q f (C0)[s] where s is transcendental
over q f (C0) and Æ(s) 2 C0   f0g. Since S  C0,
(S 1C[X ])Ga = q f (C0)
= S 1C0
= S 1TJ (R)
= TJ Rm (Rm)
= J Rm TJ Rm (Rm).
94 JAMES K. DEVENEY AND DAVID R. FINSTON
The last equality shows that W \Spec Rm is affine for every maximal ideal of R, from
which it follows that Naf (Z ) = ;, i.e. that W is affine.
From affineness of W we conclude that TJ (R) is flat over R and therefore finitely
generated over R and over C [15, Corollary 3].
3. A slice criterion
Following Miyanishi [11], a morphism f : Z ! W of complex algebraic schemes
is said to be geometrically irreducible in codimension one (GICO) if for any irreducible
subvariety T of Z of codimension one the field extension C(T )=C  f (T ) is regular.
Here f (T ) denotes the closure of f (T ) in W . An action of Ga on a complex affine
variety X is said to be GICO if C[X ]Ga  C0 is affine and the induced morphism
Spec C[X ]Ga ! Spec C0 is GICO. It was shown in [6] that proper actions on X = Cn
with finitely generated invariants are GICO and that the GICO condition is equivalent
with the intersection of the kernel and image of the generating derivation not lying in
any height one ideal of C[X ] (equivalently of C0). A GICO action on Cn with C0
finitely generated and regular is locally trivial [5].
An example of a locally trivial Ga action on C5 with finitely generated regular
invariants but no slice was given in [18]. The next result suggests that such an example
might not occur for actions on C4.
Theorem 3.1. Consider a GICO (e.g. proper) Ga action on X = C4 and assume
that C0 is finitely generated and regular. Set W  Spec C0,  : X ! W the morphism
induced by the ring inclusion C0  C[X ], and V = W   im(). If V is smooth
and distinct irreducible components of V are disjoint then the action is equivariantly
trivial.
Proof. It is shown in [6] that a proper action is GICO and a GICO action with
regular invariants is locally trivial. Thus U  im() is open in W and  : X ! U is
a geometric quotient. If the action doesn’t admit a slice, then [11, Theorem 2] yields
that V is of pure codimension 2 in W (the flatness of  , missing from the hypothe-
sis of that theorem, holds in the present context [4]). In fact, with V = Ski=1 Vi the
decomposition into irreducible components, we show that the Vi are all isomorphic to
the line C1. In the following, we use singular homology with integral coefficients.
From the Lefschetz theorem on the homology of complex affine varieties we obtain
H j (Vi ) = 0 = H j (V ) for j  2, and H j (W ) = 0 for j  5. The Thom isomorphism
yields H j (W , U ) = H j 4(V ) so that H j (W , U ) = 0 for j < 4 and j > 5. Also, the
local triviality realizes X as a principal Ga bundle over U , and therefore H j (U ) = 0
for j > 0.
The long exact sequence for the pair (W , U ) shows that H0(W ) = H0(U ) = Z,
H j (W ) = 0 for j = 1, 2, 3 and H5(W , U ) = 0. Thus H1(V ) = 0, from which we deduce
that all Vi = C1.
ADDITIVE GROUP ACTIONS 95
For each i , let Ti  W be a tubular neighborhood of Vi and set T =
Sk
i=1 Ti . From
the Mayer-Vietoris sequence for (U , T ) it follows that H j (U \ T ) = 0 for all j > 0.
But each U \ Ti is homotopic to R4   fpt.g, which in turn is homotopy equivalent to
S3. Since H3(S3) 6= 0 we obtain a contradiction unless V = ;.
It should be noted that an example of a proper but not locally trivial action on a
smooth factorial fourfold (not isomorphic to C4) with finitely generated but nonregular
invariants was given in [8].
4. Singularities
Recall the following result from [9]:
Theorem 4.1. Let X be a smooth factorial quasiaffine variety. Suppose that Ga
acts algebraically on X and that C0 is finitely generated over C. If dim X  5 then
C0 is Gorenstein.
This applies in particular to triangular Ga actions on C4 for which the ring of
invariants is known to be finitely generated. Since the ring of invariants is identical
with the kernel of the generating derivation, the following lemma is easily verified:
Lemma 2. Let Ga act on Cn via a nonzero triangular derivation
Æ =
n
X
i=2
pi (x1, : : : , xi 1) 
xi
.
Then:
(1) Æ commutes with =xn .
(2) =xn restricts to a locally nilpotent derivation on the kernel of Æ.
(3) The associated Ga action on ker(Æ) is trivial if and only if Æ = =xn .
Flenner and Zaidenberg [10, Corollary 1.13] have shown that an isolated Cohen-
Macaulay singularity of a complex affine variety admitting a nontrivial Ga action is a
rational singularity. As a consequence we obtain the
Corollary 4.2. Let Ga act on C4 via a triangular derivation Æ and let Y denote
the affine variety defined by C0. If y 2 Y is an isolated singularity then y is a rational
singularity.
REMARK 3. (1) The proper but not locally trivial action on C5 in [5] has finitely
generated ring of invariants defining a variety singular in codimension exactly 3.
96 JAMES K. DEVENEY AND DAVID R. FINSTON
(2) Fixed point free triangular actions on C4 for which Y has isolated compound
du Val singularities have been given in [18] and [7]. A typical example is the fol-
lowing:
EXAMPLE 1. Let Ga act on C4 via the triangular derivation
Æ = [(x22   2x1x3)  1]

x4
+ x2

x3
+ x1

x2
.
Setting c1 = x1, c2 = x22   2x1x3   1, c3 = x1x4   x2c2,
C0 = C[c1, c2, c3, c4] with the relation c1c4   c23   c22(c2 + 1) = 0.
Observe that the origin is the unique singular point. After a change of variables,
the completion of the local ring of the singular point is easily seen to be isomorphic
to C[[c1, c2, c3, c4]]=(c21 + c24 + c23 + c22).
REMARK 4. We know of no proper triangular action on C4 for which the variety
defined by the ring of invariants is singular (and therefore know of no proper action
on C4 which is not conjugate to a translation).
With the aid of Singular [16] Parag Mehta discovered the following example of a
fixed point free nonproper action with maximally singular invariants (i.e. the singular
locus has codimension 2, minimal for a factorial variety):
EXAMPLE 2. Let Ga act on C4 via the triangular derivation
Æ = [x3(x22   2x1x3) + 1]

x4
+ x2

x3
+ x1

x2
.
Set
c1 = x1
c2 = x
2
2   2x1x3
c3 = x
5
2   5x1x32 x3 + 6x21 x2x23 + 3x21 x4   3x1x2
c4 = x
6
2 x
2
3  
20
3
x1x
4
2 x
3
3 +
44
3
x21 x
2
2 x
4
3  
32
3
x31 x
5
3
  2x52 x4 + 10x1x32 x3x4   12x21 x2x23 x4 + 2x42 x3
  12x1x22 x
2
3 + 16x21 x33   3x21 x24 + 6x1x2x4   3x22 .
Then C0 = C[c1, c2, c3, c4] and the relation satisfied by the generators is
c23 + 6c1c32   c52 + 3c21c4 = 0.
ADDITIVE GROUP ACTIONS 97
With Y denoting as usual the affine variety with coordinate ring C0, the singu-
lar locus of Y is given by c1 = c2 = c3 = 0. The singularities are again seen to be
compound du Val by applying the quasihomogeneity criterion for du Val surface sin-
gularities [14, p.275] to the surface defined by:
c23 + 6c1c32   c52 = 0.
It is of interest to explore the kinds of isolated singularities that can arise as iso-
lated singularities of Y . A three dimensional rational Gorenstein singularity y is known
to be canonical. It follows that the singularity is either compound du Val or the gen-
eral hyperplane section through y is an elliptic surface singularity [14]. On the other
hand, the following proposition indicates that the class of singularities that can arise is
even more restricted. The argument is a slight modification of one given in [17].
Proposition 4.3. Let Ga act via a triangular derivation Æ on C4. If the variety
Y defined by the ring C0 of Ga invariants has only isolated singularities, then they are
not quotient singularities.
Proof. Let  : C4 ! Y denote the morphism induced by the ring inclusion. With
Z1 and Z2 denoting the zero loci of C0\ im(Æ) in C4 and Y respectively,  jC4 Z1: C4 
Z1 ! Y   Z2 is a geometric quotient. In particular  fibers are generically connected.
Note that for any point y,  1(y) has codimension at least 2. Indeed, as argued in
[17] a codimension 1 component would be the zero locus of an invariant polynomial
and therefore not the fiber of a single point.
Suppose that an isolated singularity y of Y is a quotient singularity. Let V be an
analytic neighborhood of y analytically isomorphic to C3=G for some nontrivial finite
subgroup of GL3(C). Note that the fundamental group 1(V   fyg) = G. Let B be
an open ball in  1(V ) and consider the morphism B   jB  1(y) : B    1(y) !
V   fyg. Since codimB  1(y)  2, B    1(y) is simply connected and therefore
B factors through the simply connected universal covering space U of V   fyg. This
contradicts the generic connectedness of  fibers.
We close with the following
Conjecture 1. A proper Ga action on C4 has regular ring of invariants. Thus a
proper action is locally trivial, and a proper triangular action is equivariantly trivial.
98 JAMES K. DEVENEY AND DAVID R. FINSTON
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James K. Deveney
Department of Mathematical Sciences
Virginia Commonwealth University
1015 W. Main St.
Richmond, Virginia 23284
USA
e-mail: jdeveney@mail2.vcu.edu
David R. Finston
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico 88003
USA
e-mail: dfinston@nmsu.edu


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Additive group actions with finitely generated invariants

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