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# Some Applications of on Affine Varieties -actions

Communicated by *Notices* Associate Editor Steven Sam

## 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 , which are integral domains. -algebras

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 and their invariants have been employed in recent decades to achieve breakthroughs on some of the above challenging problems on polynomial rings. -actions

The concept of a and its equivalent formulations -action*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 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 -action.

In Section 4, we revisit an invariant of 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 -actions .

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

Due to the restriction on the size of the bibliography, many important results and references on and allied topics had to be omitted. Interested readers may refer to the monograph of Freudenburg ( -actions7) 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 over -variables 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 We shall denote the set of all maximal ideals of a ring -algebras. by .

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

## 2. Exponential Map, and LND -action,

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

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: .

Thus, for each the , -action induces an isomorphism given by for all Let . denote the coordinate ring of Then, we get a . automorphism -algebra 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 of -subalgebra The ring . is called the *ring of invariants* of the -action .

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 exponential maps on -linear 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 and exponential map as stated below. -action

For illustration we consider an example.

We now define locally nilpotent derivations.

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 on an affine variety) is equivalent to the study of locally nilpotent derivations. -actions

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

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 exponential map, and LND interchangeably. -action,

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 on a polynomial ring need not be finitely generated. In 1990, P. Roberts constructed a nonfinitely -action -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 . over a polynomial ring -action 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 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 ( -actions3) 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 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. -actions

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 on -action is linearizable”. Thus, even for the study of a “good” action ( one had to study a so called “bad” -action), (see Section -action4).

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 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 -actions“ can be a part of the general armoury of algebraists and geometers, and not confined to specialists. More details are given in -action”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):

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

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 (quoted below as Theorem -action3.1). 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 problem (defined in Section 5) over a PID not containing -fibration 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 in positive characteristic. In 2014, the author showed that Asanuma’s ring is indeed a counter-example to the cancellation problem (see Section -space5). 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 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.

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 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 -action). of -invariant and elementary results on the invariant. -action

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

are exponential maps on We see that .