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Some Applications of -actions on Affine Varieties

Neena Gupta

Communicated by Notices Associate Editor Steven Sam

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1. Introduction

Over a field , an affine -space simply refers to with certain mathematical (topological and sheaf) structures, and affine varieties refer to irreducible closed subspaces of defined by the zero locus of polynomials. Over an algebraically closed field , affine spaces correspond to polynomial rings and the affine varieties to affine domains, i.e., finitely generated -algebras which are integral domains.

In affine algebraic geometry (AAG), researchers are mainly interested in the study of certain affine varieties, especially the affine spaces, equivalently, the polynomial rings. There are many fascinating and fundamental problems on polynomial rings which can be formulated in an elementary mathematical language but whose solutions remain elusive. Any significant progress requires development of new and powerful methods and their ingenious applications. The most celebrated problems on polynomial rings include the Jacobian problem (first asked by Ott-Heinrich Keller in 1939), the Zariski cancellation problem (ZCP), the epimorphism or embedding problem of Abhyankar–Sathaye, the affine fibration problem of Dolgačev–Veǐsfeǐler, the linearization problem of Kambayashi, the problem of characterization of polynomial rings by a few chosen properties, and the study of the automorphism groups of polynomial rings. In this article, we will discuss how -actions and their invariants have been employed in recent decades to achieve breakthroughs on some of the above challenging problems on polynomial rings.

The concept of a -action and its equivalent formulations exponential map and locally nilpotent derivation will be defined in Section 2 along with a brief overview on -actions.

In Section 3, we recall Miyanishi’s algebraic characterization of the affine plane obtained in the 1970s using -action and then Fujita–Miyanishi–Sugie’s theorem on the ZCP for the affine plane using this characterization. We shall also discuss a more recent elementary solution of ZCP for the affine plane by Crachiola and Makar-Limanov using tools of -action.

In Section 4, we revisit an invariant of -actions introduced by Makar-Limanov in the 1990s, now known as the Makar-Limanov invariant. We discuss how Makar-Limanov used it to distinguish a well-known threefold from the affine three space. This result led to the solution of the linearization conjecture of Kambayashi for .

Finally in Section 5, we see how the tools of -action were used in the last decade to establish counterexamples to the ZCP in positive characteristic in higher dimensions.

Due to the restriction on the size of the bibliography, many important results and references on -actions and allied topics had to be omitted. Interested readers may refer to the monograph of Freudenburg (7) and the survey articles 18, 6 and 13.

Throughout the article, will denote a field. By a ring, we mean a commutative ring with unity. For a ring , denotes the group of all units of . For an algebra over a ring , the notation will mean that is isomorphic to a polynomial ring in -variables over , and for a prime ideal of , denotes , where . Capital letters like , etc., will mean indeterminates over respective rings or fields. For a ring and -algebras and , the notation means that is isomorphic to as -algebras. We shall denote the set of all maximal ideals of a ring by .

We recall below the concept of morphisms between affine varieties over and their relationships with certain -algebra homomorphisms.

Remark 1.1.

Recall that for two affine algebraic varieties and , a morphism or a polynomial function is simply a function defined by polynomials

such that for all . In particular, any polynomial gives rise to a polynomial function . The coordinate ring refers to the ring of polynomial functions on . Any morphism induces a -algebra homomorphism defined by for all .

Remark 1.2.

For an affine variety over an algebraically closed field , by the celebrated Hilbert Nullstellensatz (1893), the maximal ideals of the coordinate ring are in one to one correspondence with the set of points of . Hence is identified with itself. Moreover, as a consequence of Hilbert Nullstellensatz, for any -algebra homomorphism and any maximal ideal of , is a maximal ideal of . Thus any -algebra homomorphism induces a map and hence induces a map which can be shown to be a morphism of varieties. Further, any morphism can be recovered from its induced -algebra homomorphism , i.e., and conversely, for any -algebra homomorphism , .

2. -action, Exponential Map, and LND

In this section we shall recall the concept of -actions on affine algebraic varieties and its connection with the concept of exponential maps on affine domains (finitely generated -algebras which are integral domains) and the concept of locally nilpotent derivations in characteristic zero. We first recall the concept of the group .

Recall that an affine algebraic group over a field is an affine variety over such that is a group compatible with the underlying variety structure. This means that the binary group operation and the inverse group operation are morphisms of affine varieties. For example, is an affine algebraic group with as the group operation and is denoted by . More precisely:

Definition.

The group over a field is the affine algebraic group comprising as an affine variety together with the additive group structure ’, i.e., the binary group operation

and the inverse operation

are morphisms of affine varieties. When the underlying field is understood, the simplified notation is used in place of . We note that is isomorphic to the affine algebraic group

and hence is a unipotent group (that is, a matrix group whose eigenvalues are all ).

Definition.

A -action on an affine variety over an algebraically closed field is a morphism of affine varieties satisfying

(i)

, for all , and

(ii)

for all , .

Thus, for each , the -action induces an isomorphism given by for all . Let denote the coordinate ring of . Then, we get a -algebra automorphism of the ring defined by for .

Let denote the subset of comprising elements which are fixed by these induced automorphisms , i.e.,

It is easy to see that the set is a -subalgebra of . The ring is called the ring of invariants of the -action .

Definition.

Let be an integral domain containing a field and be a -algebra homomorphism. For an indeterminate over , let denote the map . Then is said to be an exponential map on , if the following conditions are satisfied:

(i)

, where is the evaluation map at .

(ii)

, where is extended to a -algebra homomorphism , by setting .

The ring of invariants of the exponential map is defined to be the subring of defined as

An exponential map is said to be nontrivial if .

Let EXP denote the set of all -linear exponential maps on . The Makar-Limanov invariant of , denoted by ML, is a subring of defined as

If admits no nontrivial exponential map, then . The invariant ML was introduced by L. Makar-Limanov and, as we shall see, it has turned out to be a powerful tool in solving certain problems on polynomial rings.

There is a connection between -action and exponential map as stated below.

Theorem 2.1.

When is algebraically closed, is an affine domain over and , then any -action on gives rise to an exponential map on and conversely. Further, .

Proof.

Any morphism of algebraic varieties induces a -algebra homomorphism of their corresponding coordinate rings by Remark 1.1. Further, it is easy to see that the conditions (i) and (ii) of the morphism correspond algebraically to the conditions (i) and (ii) of the -algebra homomorphism . The converse also follows similarly (cf. Remark 1.2).

For illustration we consider an example.

Example 2.2.

Let be the affine surface in defined by and be the coordinate ring of . Consider the morphism

Then,

(i)

for all and

(ii)

for all and .

Hence is a -action on .

Let , , and denote the images of , , and in . Now induces the ring homomorphism defined by

One can see that is an exponential map on and .

We now define locally nilpotent derivations.

Definition.

Let be an integral domain containing a field of characteristic zero. A -linear derivation on is said to be a locally nilpotent derivation (or LND) if, for any there exists an integer (depending on ) satisfying . Thus a -linear map is said to be an LND if

(i)

and

(ii)

for some (depending on ) for each .

For an LND , let Ker denote the kernel of the derivation , i.e.,

Then Ker is a subring of .

For example, the partial derivations , , and on the ring are LNDs with kernels , , and respectively. Thus an LND may be thought of as a generalization of partial derivation on a polynomial ring . The famous Jacobian conjecture can also be formulated as a problem in LND (7, Chapter 3).

We now see that, over an algebraically closed field of characteristic zero, the study of exponential maps (equivalently the study of -actions on an affine variety) is equivalent to the study of locally nilpotent derivations.

Theorem 2.3.

Let be an algebraically closed field of characteristic zero and be a -domain. An exponential map induces a locally nilpotent derivation on and conversely.

Proof.

Let be an exponential map on . For any , let

Note that since is a -algebra homomorphism, we have

(I)

are -linear maps on .

(II)

for all and

(III)

For each , there exists such that for all .

Further, as is an exponential map, we have

(IV)

is the identity map on , by property (i) of exponential maps.

(V)

for all , by property (ii) of exponential maps.

Therefore, by (V), in characteristic zero, we have

Let . Then is a -linear map satisfying by property (II) and (IV). Further, by equation 1, , and hence for some by (III). Thus is a locally nilpotent derivation on . Further, it follows that , i.e., .

Conversely, let be an LND. Then it is easy to see that the map defined by

induces an exponential map on . We note that the image of is actually a polynomial in since for some . Also, for any .

As an example we see below the LND induced by the exponential map defined in Example 2.2.

Example 2.4.

Let the notation and hypothesis be as in Example 2.2. The exponential map induces the locally nilptent derivation on defined by

Thus by Theorems 2.1 and 2.3, over an algebraically closed field of characteristic , we have

and

Hence, going forward we shall be using -action, exponential map, and LND interchangeably.

M. Nagata (in 1959) had constructed an example of a unipotent group action over a polynomial ring whose ring of invariants is not finitely generated. Thus the concept of studying unipotent group actions came to be perceived as pathological, not of interest to many geometers.

It is now known that the ring of invariants of even a -action on a polynomial ring need not be finitely generated. In 1990, P. Roberts constructed a nonfinitely -algebra over a field of characteristic zero, as the symbolic blow-up of a prime ideal of . Later, A. A’Campo-Neuen realized the ring in Roberts’s example as the ring of invariants of an LND on . This was the first example of a nonfinitely generated ring of invariants of a -action over a polynomial ring . For subsequent examples, one can see 7, Chapter 7.

In spite of the apparently pathological nature of unipotent group actions, a few mathematicians like P. Gabriel, Y. Nouaźe, R. Rentschler, and J. Dixmier proved some early fundamental results in the 1970s on -actions. M. Miyanishi began to systematically investigate -actions and LND. He highlighted the concepts, proved important theorems on them and applied them to mainstream problems in AAG like classification of surfaces, algebraic characterization of affine plane, etc. His algebraic characterization of the affine plane (1975) eventually led to the solution of the ZCP for the affine plane by Miyanishi–Sugie and Fujita in 1980. In 2008, L. Makar-Limanov and A. Crachiola (3) gave an elementary proof of the cancellation theorem for the affine plane using exponential maps. The proof is discussed in Section 3.

Since the 1990s, many algebraists including L. Makar-Limanov, J. Deveney, D. Finston, A. van den Essen, S. Kaliman, D. Daigle, G. Freudenburg, H. Derksen, A. Dubouloz, S.M. Bhatwadekar, A.K. Dutta, and subsequent researchers have been contributing regularly in the study of -actions on affine varieties. With passage of time, there has been an increasing recognition of the importance of the concept of -actions in its own right as well as for application to problems in AAG, some of them longstanding.

A major breakthrough in AAG was obtained during the 1990s when, using LND, L. Makar-Limanov distinguished the famous Koras–Russell threefold from the affine three space by showing that the Makar-Limanov invariant (named after him) of this threefold does not coincide with the affine three space. This led to the solution of the linearization conjecture of Kambayashi that “Every faithful algebraic -action on is linearizable”. Thus, even for the study of a “good” action (-action), one had to study a so called “bad” -action (see Section 4).

Another breakthrough was obtained by the author in 2010s when she used exponential maps to obtain counterexamples to the ZCP in positive characteristic in higher dimensions (see Section 5).

In the above two cases, the tools and techniques of -actions were the cornerstones in the solutions of the respective problems. Certain invariants of -actions could distinguish between two rings, when all hitherto known methods had failed. These episodes illustrate that -action” can be a part of the general armoury of algebraists and geometers, and not confined to specialists. More details are given in 7.

3. Algebraic Characterization of the Plane and ZCP

The ZCP for polynomial rings can be posed as follows (cf. 8, 7, Chapter 10, 13, Section 2):

Question 1.

Let be an affine -algebra. Suppose that . Does it follow that ? In other words, is the polynomial ring cancellative?

For an algebraically closed field , Question 1 is equivalent to the following geometric version:

Question .

Let be an algebraically closed field and let be an affine -variety such that . Does it follow that ? In other words, is the affine -space cancellative?

Question 1 has inspired many fruitful explorations over the past 50 years. Some of the major research accomplishments during the 1970s, like the characterization of the affine plane, originated from the efforts to investigate the question. It is not very difficult to show that the polynomial ring is cancellative over a field of any characteristic. A more general result was shown by S. S. Abhyankar, P. Eakin, and W. J. Heinzer (1972). But for the polynomial ring , the problem is much more intricate.

In an attempt to solve the cancellation problem for , C.P. Ramanujam established in 1971 his celebrated topological characterization of the affine plane over . In 1975, M. Miyanishi gave a characterization of the polynomial ring using -action (quoted below as Theorem 3.1). This characterization was used by T. Fujita, M. Miyanishi, and T. Sugie (8, 17) to prove the cancellation property of over fields of characteristic zero and by P. Russell (19) over perfect fields of arbitrary characteristic. Later (in 2002), using methods of D. Mumford and C.P. Ramanujam, R.V. Gurjar gave a topological proof of the cancellation property of . More recently, a simplified proof of the cancellation property of for an algebraically closed field was given by A. Crachiola and L. Makar-Limanov in 3 using tools of exponential maps. The arguments in this paper were used by S.M. Bhatwadekar and the author to establish the cancellation property of over any arbitrary field . For more details one can see 13, Sections 2 and 3.

In 1987, T. Asanuma constructed a three-dimensional affine ring over a field of positive characteristic (see Example 5.3) as a counterexample to the -fibration problem (defined in Section 5) over a PID not containing . Later, in 1994, this ring was envisaged as a possible candidate for a counter-example to either the ZCP or the linearization problem (discussed in Section 4) for the affine -space in positive characteristic. In 2014, the author showed that Asanuma’s ring is indeed a counter-example to the cancellation problem (see Section 5). Thus, when ch. , the affine -space is not cancellative. Subsequently, the author showed that when ch. , the affine -space is not cancellative for any . Thus, over a field of positive characteristic, the ZCP has been completely answered in all dimensions (12).

Question 1 is still open in characteristic zero for and is of great interest in the area of AAG.

We now return to the two-dimensional case. We first state Miyanishi’s algebraic characterization of the affine plane (18, Theorem 2.2.3).

Theorem 3.1.

Let be an algebraically closed field and be a finitely generated -algebra of dimension such that

(i)

is a UFD.

(ii)

.

(iii)

There exists a nontrivial exponential map on .

Then .

Theorem 3.1 had led to following fundamental result of T. Fujita, M. Miyanishi, and T. Sugie (17 and 8) proving the cancellation property of over any field of characteristic zero and the result of P. Russell (19) proving it over any perfect field of arbitrary characteristic.

Theorem 3.2.

Let be a field and be a -domain such that

Then .

The original proof uses many ideas making them not quite self-contained for a large class of mathematicians. A simplified proof of Theorem 3.2 for the case is algebraically closed was given by L. Makar-Limanov and A. Crachiola (3) using exponential maps (equivalently -action). Their proof merely uses the following lemma on exponential maps whose proof is elementary. The simplification indicates the power of the tools introduced by Makar-Limanov comprising the concept of -invariant of -action and elementary results on the invariant.

Lemma 3.3.

Let be a finitely generated -algebra such that there exists an exponential map of with . Then there exists an exponential map of such that , i.e., if then .

We note that the ring admits several exponential maps. More specifically, for each , , , given by

are exponential maps on . We see that