Some Applications of ${\mathbb{G}}_a$-actions on Affine Varieties

1. Introduction

Over a field $k$, an affine $n$-space $\mathbb{A}^n_k$ simply refers to $k^n$ with certain mathematical (topological and sheaf) structures, and affine varieties refer to irreducible closed subspaces of $\mathbb{A}^n_k$ defined by the zero locus of polynomials. Over an algebraically closed field $k$, affine spaces correspond to polynomial rings and the affine varieties to affine domains, i.e., finitely generated $k$-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 ${\mathbb{G}}_a$-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 ${\mathbb{G}}_a$-action and its equivalent formulations exponential map and locally nilpotent derivation will be defined in Section 2 along with a brief overview on ${\mathbb{G}}_a$-actions.

In Section 3, we recall Miyanishi’s algebraic characterization of the affine plane obtained in the 1970s using ${\mathbb{G}}_a$-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 ${\mathbb{G}}_a$-action.

In Section 4, we revisit an invariant of ${\mathbb{G}}_a$-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 $\mathbb{C}^3$.

Finally in Section 5, we see how the tools of ${\mathbb{G}}_a$-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 ${\mathbb{G}}_a$-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, $k$ will denote a field. By a ring, we mean a commutative ring with unity. For a ring $R$, $R^*$ denotes the group of all units of $R$. For an algebra $A$ over a ring $R$, the notation $A= R^{[n]}$ will mean that $A$ is isomorphic to a polynomial ring in $n$-variables over $R$, and for a prime ideal $P$ of $R$, $A_P$ denotes $S^{-1}A$, where $S=R\ \setminus P$. Capital letters like $X,Y,Z,W, T, X_1, \dots , X_n$, etc., will mean indeterminates over respective rings or fields. For a ring $R$ and $R$-algebras $A$ and $B$, the notation $A\cong _R B$ means that $A$ is isomorphic to $B$ as $R$-algebras. We shall denote the set of all maximal ideals of a ring $R$ by $\operatorname {Max}(R)$.

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

Remark 1.1.

Recall that for two affine algebraic varieties $V \subseteq \mathbb{A}^n_k$ and $W\subseteq \mathbb{A}^m_k$, a morphism or a polynomial function $\Phi : V \to W$ is simply a function defined by $m$ polynomials

$$\begin{equation*} \phi _1(X_1, \dots , X_n),\nobreakspace\nobreakspace \phi _2(X_1, \dots , X_n),\nobreakspace\nobreakspace \dots ,\nobreakspace\nobreakspace \phi _m(X_1, \dots , X_n) \end{equation*}$$

such that $(\phi _1(a_1, \dots , a_n), \phi _2(a_1, \dots , a_n), \dots , \phi _m(a_1, \dots , a_n)) \in W$ for all $(a_1, \dots , a_n) \in V$. In particular, any polynomial $\phi (X_1, \dots , X_n)$ gives rise to a polynomial function $\phi : V\to k$. The coordinate ring $k[V]$ refers to the ring of polynomial functions on $V$. Any morphism $\Phi : V \to W$ induces a $k$-algebra homomorphism $\widetilde{\Phi }: k[W] \to k[V]$ defined by $\widetilde{\Phi }(f)=f\circ \Phi$ for all $f \in k[W]$.

Remark 1.2.

For an affine variety $V$ over an algebraically closed field $k$, by the celebrated Hilbert Nullstellensatz (1893), the maximal ideals of the coordinate ring $k[V]$ are in one to one correspondence with the set of points of $V$. Hence $\operatorname {Max}(k[V])$ is identified with $V$ itself. Moreover, as a consequence of Hilbert Nullstellensatz, for any $k$-algebra homomorphism $\Psi : k[W]\to k[V]$ and any maximal ideal $\mathscr{m}$ of $k[V]$, $\Psi ^{-1}(\mathscr{m})$ is a maximal ideal of $k[W]$. Thus any $k$-algebra homomorphism $\Psi : k[W]\to k[V]$ induces a map $\operatorname {Max}(k[V]) \to \operatorname {Max}(k[W])$ and hence induces a map $\widehat{\Psi }: V \to W$ which can be shown to be a morphism of varieties. Further, any morphism $\Phi : V \to W$ can be recovered from its induced $k$-algebra homomorphism $\tilde{\Phi }$, i.e., $\widehat{\widetilde{\Phi }}= \Phi$ and conversely, for any $k$-algebra homomorphism $\Psi : k[W]\to k[V]$, $\widetilde{\widehat{\Psi }}=\Psi$.

2. ${\mathbb{G}}_a$-action, Exponential Map, and LND

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

Recall that an affine algebraic group $G$ over a field $k$ is an affine variety over $k$ such that $G$ is a group compatible with the underlying variety structure. This means that the binary group operation $\mu : G \times G \to G$ and the inverse group operation $\iota : G \to G$ are morphisms of affine varieties. For example, $k$ is an affine algebraic group with $+$ as the group operation and is denoted by ${\mathbb{G}}_a^k$. More precisely:

Definition.

The group ${\mathbb{G}}_a^k$ over a field $k$ is the affine algebraic group $(k, +)$ comprising $k$ as an affine variety together with the additive group structure ‘$+$’, i.e., the binary group operation

$$\begin{equation*} \mu : k \times k \to k,\nobreakspace\nobreakspace \text{defined by}\nobreakspace\nobreakspace \mu (a, b)= a+b \end{equation*}$$

and the inverse operation

$$\begin{equation*} \iota : k \to k,\nobreakspace\nobreakspace \text{defined by}\nobreakspace\nobreakspace \iota (a)= -a \end{equation*}$$

are morphisms of affine varieties. When the underlying field $k$ is understood, the simplified notation ${\mathbb{G}}_a$ is used in place of ${\mathbb{G}}_a^k$. We note that ${\mathbb{G}}_a$ is isomorphic to the affine algebraic group

$$\begin{equation*} \{ \begin{pmatrix} 1&a \\ 0&1 \end{pmatrix} \in \operatorname {SL}_2(k)\nobreakspace |\nobreakspace a \in k \}, \end{equation*}$$

and hence is a unipotent group (that is, a matrix group whose eigenvalues are all $1$).

Definition.

A ${\mathbb{G}}_a$-action on an affine variety $V$ over an algebraically closed field $k$ is a morphism of affine varieties $\theta : {\mathbb{G}}_a\times V \rightarrow V$ satisfying

(i)

$\theta (0, x) = x$, for all $x \in V$, and

(ii)

$\theta (\alpha , \theta (\beta , x))= \theta (\alpha +\beta , x)$ for all $x \in V$, $\alpha , \beta \in {\mathbb{G}}_a$.

Thus, for each $\lambda \in k$, the ${\mathbb{G}}_a$-action $\theta$ induces an isomorphism $\theta _{\lambda }: V\to V$ given by $\theta _{\lambda }(x)= \theta (\lambda , x)$ for all $x \in V$. Let $B$ denote the coordinate ring $k[V]$ of $V$. Then, we get a $k$-algebra automorphism $\widetilde{\theta _{\lambda }}$ of the ring $B$ defined by $\widetilde{\theta _{\lambda }} (f)=f\circ \theta _{\lambda }$ for $f \in B$.

Let $B^{\theta }$ denote the subset of $B$ comprising elements which are fixed by these induced automorphisms $\widetilde{\theta _{\lambda }}$, i.e.,

$$\begin{equation*} B^{\theta }\coloneq \left\{f\in B \,|\, \widetilde{\theta _{\lambda }}(f)=f\nobreakspace {\text{ for all }}\nobreakspace \lambda \in k\right\}. \end{equation*}$$

It is easy to see that the set $B^{\theta }$ is a $k$-subalgebra of $B$. The ring $B^{\theta }$ is called the ring of invariants of the ${\mathbb{G}}_a$-action $\theta$.

Definition.

Let $B$ be an integral domain containing a field $k$ and $\phi : B \rightarrow B^{[1]}$ be a $k$-algebra homomorphism. For an indeterminate $U$ over $B$, let $\phi _{U}$ denote the map $\phi : B \rightarrow B[U]$. Then $\phi$ is said to be an exponential map on $B$, if the following conditions are satisfied:

(i)

$\epsilon _{0} \phi _{U} =id_{B}$, where $\epsilon _{0}: B[U] \rightarrow B$ is the evaluation map at $U=0$.

(ii)

$\phi _{V} \phi _{U}=\phi _{U+V}$, where $\phi _{V}:B \rightarrow B[V]$ is extended to a $k$-algebra homomorphism $\phi _{V}:B[U] \rightarrow B[U,V]$, by setting $\phi _{V}(U)=U$.

The ring of invariants of the exponential map $\phi$ is defined to be the subring $B^{\phi }$ of $B$ defined as

$$\begin{equation*} B^{\phi }\coloneq \{f \in B\,|\, \phi (f)=f\}. \end{equation*}$$

An exponential map $\phi$ is said to be nontrivial if $B\neq B^{\phi }$.

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

$$\begin{equation*} \operatorname {ML}(B)\coloneq \bigcap _{\phi \in \operatorname {EXP}(B)} B^{\phi }. \end{equation*}$$

If $B$ admits no nontrivial exponential map, then $\operatorname {ML}(B)=B$. The invariant ML$(B)$ 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 ${\mathbb{G}}_a$-action and exponential map as stated below.

Theorem 2.1.

When $k$ is algebraically closed, $B$ is an affine domain over $k$ and ${V}\coloneq \operatorname {Max} (B)$, then any ${\mathbb{G}}_a$-action $\theta$ on $V$ gives rise to an exponential map $\phi$ on $B$ and conversely. Further, $B^{\theta }=B^{\phi }$.

Proof.

Any morphism $\theta : {\mathbb{G}}_a\times V \to V$ of algebraic varieties induces a $k$-algebra homomorphism $\phi : B \to B \otimes _k k[U]=B[U]$ of their corresponding coordinate rings by Remark 1.1. Further, it is easy to see that the conditions (i) and (ii) of the morphism $\theta$ correspond algebraically to the conditions (i) and (ii) of the $k$-algebra homomorphism $\phi$. The converse also follows similarly (cf. Remark 1.2).

For illustration we consider an example.

Example 2.2.

Let $V$ be the affine surface in $\mathbb{A}^3_k$ defined by $\{(a,b,c)\in \mathbb{A}^3_k\nobreakspace |\nobreakspace ac=b^2-1\}$ and $B= k[V]=k[X,Y,Z]/(XZ-Y^2+1)$ be the coordinate ring of $V$. Consider the morphism

$$\begin{multline*} \theta : {\mathbb{G}}_a\times V \rightarrow V\nobreakspace\nobreakspace \text{defined by}\nobreakspace\nobreakspace \theta (\lambda , (a,b,c))\\ = (a, b+ a\lambda , c+2b\lambda +a\lambda ^2). \end{multline*}$$

Then,

(i)

$\theta (0, (a,b,c))=(a,b,c)$ for all $(a,b,c) \in V$ and

(ii)

$\theta (\alpha , \theta (\beta , (a,b,c)))=\theta (\alpha +\beta , (a,b,c))$ for all $(a,b,c) \in V$ and $\alpha , \beta \in k$.

Hence $\theta$ is a ${\mathbb{G}}_a$-action on $V$.

Let $x$, $y$, and $z$ denote the images of $X$, $Y$, and $Z$ in $B$. Now $\theta$ induces the ring homomorphism $\phi : B \to B[U]$ defined by

$$\begin{equation*} \phi (x)=x,\nobreakspace\nobreakspace \phi (y)=y+xU\nobreakspace \text{ and }\nobreakspace \phi (z)=z+2yU+xU^2. \end{equation*}$$

One can see that $\phi$ is an exponential map on $B$ and $B^{\theta }=B^{\phi }=k[x]$.

We now define locally nilpotent derivations.

Definition.

Let $B$ be an integral domain containing a field $k$ of characteristic zero. A $k$-linear derivation $D$ on $B$ is said to be a locally nilpotent derivation (or LND) if, for any $a \in B$ there exists an integer $n$ (depending on $a$) satisfying $D^n (a)=0$. Thus a $k$-linear map $D: B \to B$ is said to be an LND if

(i)

$D(xy)=xD(y)+yD(x)$ $\forall$ $x,y \in B$ and

(ii)

$D^n(a)= 0$ for some $n \ge 1$ (depending on $a$) for each $a \in B$.

For an LND $D$, let Ker$(D)$ denote the kernel of the derivation $D$, i.e.,

$$\begin{equation*} \operatorname {Ker}(D)\coloneq \{f \in B\,|\, D(f)=0\}. \end{equation*}$$

Then Ker$(D)$ is a subring of $B$.

For example, the partial derivations $\partial /\partial (X)$, $\partial /\partial (Y)$, and $\partial /\partial (Z)$ on the ring $k[X,Y,Z]$ are LNDs with kernels $k[Y,Z]$, $k[X,Z]$, and $k[X,Y]$ respectively. Thus an LND may be thought of as a generalization of partial derivation on a polynomial ring $B$. 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 ${\mathbb{G}}_a$-actions on an affine variety) is equivalent to the study of locally nilpotent derivations.

Theorem 2.3.

Let $k$ be an algebraically closed field of characteristic zero and $B$ be a $k$-domain. An exponential map $\phi : B \to B[U]$ induces a locally nilpotent derivation $D$ on $B$ and conversely.

Proof.

Let $\phi : B \to B[U]$ be an exponential map on $B$. For any $b \in B$, let

$$\begin{equation*} \phi (b) = \phi _0(b)+\phi _1(b)U + \phi _2(b)U^2 + \dots + . \end{equation*}$$

Note that since $\phi$ is a $k$-algebra homomorphism, we have

(I)

$\{\phi _n\}$ are $k$-linear maps on $B$.

(II)

$\phi _n(ab)= \sum _{i+j=n}\phi _i(a)\phi _j(b)$ for all $a, b \in B$ and $n \ge 0$

(III)

For each $b \in B$, there exists $n \in \mathbb{N}$ such that $\phi _m(b)=0$ for all $m \geq n$.

Further, as $\phi$ is an exponential map, we have

(IV)

$\phi _0$ is the identity map on $B$, by property (i) of exponential maps.

(V)

$\phi _i\phi _j = {{i+j} \choose i}\phi _{i+j}$ for all $i, j \ge 0$, by property (ii) of exponential maps.

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

$$\begin{equation} \phi _2= \frac{1}{2!}\phi _1^2,\nobreakspace\nobreakspace \dots ,\nobreakspace\nobreakspace \phi _n= \frac{1}{n!}\phi _1^n\nobreakspace\nobreakspace \dots . {\tag{1}} \end{equation}$$

Let $D\coloneq \phi _1$. Then $D: B \to B$ is a $k$-linear map satisfying $D(ab)= aD(b)+bD(a)$ by property (II) and (IV). Further, by equation 1, $D^n(b)= n!\phi _n(b)$, and hence $D^n(b)=0$ for some $n \ge 1$ by (III). Thus $D$ is a locally nilpotent derivation on $B$. Further, it follows that $b \in \text{ Ker}(D)$ $\Longleftrightarrow$ $\phi (b)=b$, i.e., $b \in B^{\phi }$.

Conversely, let $D: B \to B$ be an LND. Then it is easy to see that the map $\phi : B \to B[U]$ defined by

$$\begin{equation*} b \longmapsto b+D(b)U + \frac{D^2(b)}{2!}U^2 +\frac{D^3(b)}{3!}U^3+ \dots \end{equation*}$$

induces an exponential map on $B$. We note that the image of $\phi$ is actually a polynomial in $U$ since $D^n(b)=0$ for some $n \ge 1$. Also, $D(b)=0$ $\Longleftrightarrow$ $\phi (b)=b$ for any $b \in B$.

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 $\phi$ induces the locally nilptent derivation $D$ on $B$ defined by

$$\begin{equation*} D(x)= 0,\nobreakspace\nobreakspace D(y)=x,\nobreakspace\nobreakspace \text{ and }\nobreakspace\nobreakspace D(z)= 2y. \end{equation*}$$

Thus by Theorems 2.1 and 2.3, over an algebraically closed field of characteristic $0$, we have

$$\begin{multline*} {\mathbb{G}}_a{\text{-action on }} V \Longleftrightarrow \text{ Exponential map on }B \\ \Longleftrightarrow \text{ LND of }B. \end{multline*}$$

and

$$\begin{align*} & \text{Ring of invariants of }{\mathbb{G}}_a\text{-action}\\ & \quad = \text{Ring of invariants of exponential maps} = \operatorname {Ker}(D). \end{align*}$$

Hence, going forward we shall be using ${\mathbb{G}}_a$-action, exponential map, and LND interchangeably.

M. Nagata (in 1959) had constructed an example of a unipotent group action over a polynomial ring $k^{[n]}$ 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 ${\mathbb{G}}_a$-action on a polynomial ring need not be finitely generated. In 1990, P. Roberts constructed a nonfinitely $k$-algebra $A$ over a field of characteristic zero, as the symbolic blow-up of a prime ideal $P$ of $k^{[3]}$. Later, A. A’Campo-Neuen realized the ring $A$ in Roberts’s example as the ring of invariants of an LND on $k^{[7]}$. This was the first example of a nonfinitely generated ring of invariants of a ${\mathbb{G}}_a$-action over a polynomial ring $k^{[7]}$. 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 ${\mathbb{G}}_a$-actions. M. Miyanishi began to systematically investigate ${\mathbb{G}}_a$-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 ${\mathbb{G}}_a$-actions on affine varieties. With passage of time, there has been an increasing recognition of the importance of the concept of ${\mathbb{G}}_a$-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 ${\mathbb{C}}^*$-action on ${\mathbb{C}}^3$ is linearizable”. Thus, even for the study of a “good” action (${\mathbb{C}}^*$-action), one had to study a so called “bad” $({\mathbb{C}}, +)$-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 ${\mathbb{G}}_a$-actions were the cornerstones in the solutions of the respective problems. Certain invariants of ${\mathbb{G}}_a$-actions could distinguish between two rings, when all hitherto known methods had failed. These episodes illustrate that “${\mathbb{G}}_a$-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 $A$ be an affine $k$-algebra. Suppose that $A[X]\cong _k k[X_1, \dots , X_{n+1}]$. Does it follow that $A\cong _k k[X_1, \dots , X_n]$? In other words, is the polynomial ring $k[X_1, \dots , X_{n}]$ cancellative?

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

Question $1^{\prime }$.

Let $k$ be an algebraically closed field and let $V$ be an affine $k$-variety such that $V \times \mathbb{A}^1_k \cong _k \mathbb{A}^{n+1}_k$. Does it follow that $V \cong _k \mathbb{A}_k^n$? In other words, is the affine $n$-space $\mathbb{A}^n_k$ 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 $k[X]$ is cancellative over a field $k$ 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 $k[X, Y]$, the problem is much more intricate.

In an attempt to solve the cancellation problem for $\mathbb{C}[X,Y]$, C.P. Ramanujam established in 1971 his celebrated topological characterization of the affine plane over $\mathbb{C}$. In 1975, M. Miyanishi gave a characterization of the polynomial ring $k[X,Y]$ using ${\mathbb{G}}_a$-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 $k[X,Y]$ 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 $\mathbb{C}[X,Y]$. More recently, a simplified proof of the cancellation property of $k[X, Y]$ for an algebraically closed field $k$ 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 $k[X, Y]$ over any arbitrary field $k$. 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 $\mathbb{A}^2$-fibration problem (defined in Section 5) over a PID not containing $\mathbb{Q}$. 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 $3$-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. $k>0$, the affine $3$-space $\mathbb{A}^3_k$ is not cancellative. Subsequently, the author showed that when ch. $k>0$, the affine $n$-space $\mathbb{A}^n_k$ is not cancellative for any $n \ge 3$. 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 $n \ge 3$ 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 $k$ be an algebraically closed field and $B$ be a finitely generated $k$-algebra of dimension $2$ such that

(i)

$B$ is a UFD.

(ii)

$B^*= k^*$.

(iii)

There exists a nontrivial exponential map on $B$.

Then $B= k^{[2]}$.

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 $k[X,Y]$ 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 $k$ be a field and $B$ be a $k$-domain such that

$$\begin{equation*} B^{[1]}=k^{[3]}. \end{equation*}$$

Then $B= k^{[2]}$.

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 $k$ is algebraically closed was given by L. Makar-Limanov and A. Crachiola (3) using exponential maps (equivalently ${\mathbb{G}}_a$-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 $\operatorname {ML}$-invariant of ${\mathbb{G}}_a$-action and elementary results on the invariant.

Lemma 3.3.

Let $B$ be a finitely generated $k$-algebra such that there exists an exponential map $\phi$ of $B[T]$ with $(B[T])^{\phi }\neq B$. Then there exists an exponential map $\psi$ of $B$ such that $B^{\psi }\neq B$, i.e., if $\operatorname {ML}(B)=B$ then $\operatorname {ML}(B[T])=B$.

We note that the ring $C=k[X_1,\dots ,X_n]$ admits several exponential maps. More specifically, for each $i$, $1\le i\le n$, $\psi _i: C \to C[U]$, given by

$$\begin{equation*} \psi _i(X_i)= X_i+ U \text{ and, for } i\neq j,\nobreakspace \psi _i(X_j)= X_j \end{equation*}$$

are exponential maps on $C$. We see that

$$\begin{equation*} C^{\psi _i}=k[X_1, \dots , X_{i-1}, X_{i+1}, \dots , X_n] \text{ and } \bigcap _{i}C^{\psi _i}=k. \end{equation*}$$

Therefore, $\operatorname {ML}(C)=k$.

We now show how Theorem 3.2 follows easily from Theorem 3.1 and Lemma 3.3.

Proof of Theorem 3.2.

Since $B^{[1]}=k^{[3]}$, it is easy to see that $B$ is a finitely generated $k$-algebra, $B$ is a UFD and $B^*=k^*$. Hence by Theorem 3.1, it is enough to show that $B$ admits a nontrivial exponential map. Again as $B^{[1]}= k^{[3]}$, the ring $B[T]$ admits an exponential map $\phi$ such that $(B[T])^{\phi }\neq B$. Hence the result follows from Lemma 3.3.

In view of the importance of Theorem 3.1, some characterizations of the affine three space have been obtained by Miyanishi (1984, 1987), Kaliman (2002) and the author with Nikhilesh Dasgupta (2021). So far, no suitable characterization of the affine $n$-space for $n \ge 4$ is known to the author. For more details on the characterization problem one can see 18, Section 2.2 for a detailed survey and 13, Section 3 for a more updated survey.

4. The Russell–Koras Threefold

A special case of the linearization conjecture of Kambayashi (15) asserts the following:

Conjecture.

Every algebraic action of $k^*$ on an affine $n$-space $\mathbb{A}^n_k$ is linearizable.

By the equivalence of $k^*$-action on affine varieties and $\mathbb{Z}$-graded structure on their respective coordinate ring, the conjecture asserts that given any $\mathbb{Z}$-graded structure of the polynomial ring $B=k^{[n]}$, there exists a homogeneous set of coordinates for $B$, i.e., $B=k[X_1, \dots , X_n]$, where $X_1, \dots , X_n$ are homogeneous elements with respect to the given $\mathbb{Z}$-grading.

By a result of Kambayashi (15), it follows that any $k^*$-action on $k^{[2]}$ is linearizable. However the case $n=3$ turned out to be highly nontrivial. An affirmative answer was finally obtained in characteristic zero as a culmination of efforts of several researchers over a period of nearly three decades.

We now discuss the solution of $\mathbb{C}^*$-action on $\mathbb{C}^{[3]}$ which had been eluding mathematicians for many years. While studying $\mathbb{C}^*$-action on $\mathbb{C}^{[3]}$, M. Koras and P. Russell encountered a family of contractible threefolds which shared several properties of $\mathbb{A}^3_{\mathbb{C}}$ and observed that if any such threefold was isomorphic to $A^3_{\mathbb{C}}$, then it would lead to a nonlinearisable $\mathbb{C}^*$-action on $\mathbb{C}^{[3]}$. Any member of this family was known as a Koras–Russell threefold. One such example (also known as Russell’s cubic) is the following ring:

$$\begin{equation*} A= {\mathbb{C}}[X,Y,Z,T]/(X^2Y+ X + Z^2+T^3). \end{equation*}$$

Let $V= \operatorname {Max}(A)$ and let $x$ denote the image of $X$ in $A$. The ring $A$ satisfies several properties of the polynomial ring $\mathbb{C}^{[3]}$, namely, it is a regular UFD with $A^*=\mathbb{C}^*$. Moreover, $V$ is diffeomorphic to $\mathbb{R}^6$ with respect to Euclidean topology and there exists a dominant morphism from $\mathbb{A}^3_{\mathbb{C}}$ to $V$. By C.P. Ramanujam’s or by M. Miyanishi’s characterization of the affine plane, the corresponding properties suffice for a two-dimensional surface to be the affine plane. There was an excitement, especially in the AAG community, regarding the possibility that $V$ is isomorphic to $\mathbb{A}^3_{\mathbb{C}}$. For, if $V=\mathbb{A}^3_{\mathbb{C}}$, i.e., $A=\mathbb{C}^{[3]}$, then the ring $A$ would have led to a counterexample to the Abhyankar–Sathaye epimorphism conjecture for $\mathbb{C}^{[3]}$ which asserts that if $F \in \mathbb{C}^{[3]}=B$ is such that $B/(F)= \mathbb{C}^{[2]}$, then $B= \mathbb{C}[F]^{[2]}$. Now note that $A/(x-\lambda )={\mathbb{C}}^{[2]}$ $\forall \lambda \in {\mathbb{C}}^*$ but $A/(x) \ne {\mathbb{C}}^{[2]}$. Thus, if $A=\mathbb{C}^{[3]}$, then $A \ne {{\mathbb{C}}[x-\lambda ]}^{[2]}$ for any $\lambda \in {\mathbb{C}}$, contradicting the Abhyankar–Sathaye conjecture.

In 1994, in a conference at McGill University, P. Russell discussed the open problem about the triviality of the above ring $A$ and the linearization conjecture. L. Makar-Limanov, who was attending the conference, announced in the meeting that the ring $A$ is not $\mathbb{C}^{[3]}$. His proof was elementary (though not easy) and involved an ingenious idea. He showed that $x \in \operatorname {ML}(A)$ (16), i.e., for every ${\mathbb{G}}_a$-action on $V$ (equivalently any LND $D$ on $A$), the regular function determined by $x$ on $V$ is fixed (i.e., $x\in$ Ker$(D)$). On the other hand it is easy to see that $\operatorname {ML}(\mathbb{C}[X,Y,Z])=\mathbb{C}$. As $x \notin A^*=\mathbb{C}^*$, it follows that $\operatorname {ML}(A) \neq {\mathbb{C}} (= \operatorname {ML}({\mathbb{C}}^{[3]}))$. Thus, $A \ncong \mathbb{C}^{[3]}$. Later, Kaliman and Makar-Limanov proved that none of the Russell–Koras threefolds is isomorphic to the affine $3$-space. This led to the solution of the linearization conjecture for $\mathbb{A}^3_{\mathbb{C}}$ by Kaliman–Koras–Makar-Limanov–Russell (14).

The ring $A$ is now a potential threat to the Zariski cancellation conjecture for the affine three space in characteristic zero. For if $A^{[1]}=\mathbb{C}^{[4]}$, then $A$ would be a counterexample to the ZCP in characteristic zero for $n=3$. A. Dubouloz has shown in 2009 (4) that $\operatorname {ML}(A^{[1]})=\mathbb{C}$ and A. Dubouloz and J. Fasel have shown in 2018 that $V$ is “$\mathbb{A}^1$-contractible” (5). The variety $V$ is in fact the first example of an $\mathbb{A}^1$-contractible threefold which is not algebraically isomorphic to $\mathbb{C}^3$.

5. The Asanuma Threefold

We first recall the affine fibration problem. For a ring $R$ and any prime ideal $P$ of $R$, the notation $k(P)$ denotes the residue field of the local ring $R_P$. It is also the same as the field of fractions of the integral domain $R/P$.

Definition.

Let $R$ be a ring. A finitely generated flat $R$-algebra $B$ is said to be an $\mathbb{A}^n$-fibration over $R$ if $B \otimes _R k(P) =k(P)^{[n]}$ for each prime ideal $P$ of $R$.

A major problem in the area of affine fibrations is the following question of Dolgačev and Veǐsfeǐler.

Question 2.

Let $R$ be a regular local ring of dimension $d$ and $A$ be an $\mathbb{A}^n$-fibration over $R$. Is $A$ necessarily a polynomial ring over $R$?

For a survey on the above problem one may see 6, Section 3.1. T. Kambayashi, M. Miyanishi, and David Wright have shown that Question 5 has an affirmative answer for $\mathbb{A}^1$-fibrations, i.e., any $\mathbb{A}^1$-fibration over a regular local ring is necessarily a polynomial ring. Their results were further refined by A.K. Dutta who showed that it is enough to assume the fiber conditions only on generic and co-dimension one fibers.

In 1983, A. Sathaye obtained the following major breakthrough on $\mathbb{A}^2$-fibrations (20):

Theorem 5.1.

Let $R$ be a PID containing $\mathbb{Q}$ and $A$ be an $\mathbb{A}^2$-fibration over $R$. Then $A= R^{[2]}$.

In 1987, T. Asanuma made the next major breakthrough on the affine fibration problem: a deep structure theorem on $\mathbb{A}^n$-fibrations (1, Theorem 3.4). From his main structure theorem, he deduced the following stable structure theorem for any affine fibration over a regular local ring (1, Corollary 3.5).

Theorem 5.2.

Let $R$ be a regular local ring of dimension $d\ge 1$ and $A$ be an $\mathbb{A}^n$-fibration over $R$. Then $A^{[m]}= R^{[m+n]}$ for some integer $m \ge 0$.

In the same paper, Asanuma also constructed the first counter-example to the $\mathbb{A}^2$-fibration problem over a PID not containing $\mathbb{Q}$ (1, Theorem 5.1). We present below a version of Asanuma’s example.

Example 5.3.

Let $k$ be a field of characteristic $p >0$ and let

$$\begin{equation} \begin{gathered} A= k[X, Y, Z, T]/(X^mY+Z^{p^e}+T+T^{sp}),\nobreakspace\nobreakspace m, e, s \in \mathbb{N}\\ \text{and }\nobreakspace p^e \nmid sp,\nobreakspace\nobreakspace sp \nmid p^e. \end{gathered} {\tag{2}} \end{equation}$$

Let $x$ denote the image of $X$ in $A$. Then $k[x] \subset A$ and the following properties are satisfied by $A$ (cf. 1, Theorem 5.1):

(1)

$A \otimes _{k[x]} k(P)= k(P)^{[2]}$ for every prime ideal $P$ of $k[x]$.

(2)

$A^{[1]} = k[x]^{[3]} = k^{[4]}$.

(3)

$A \neq k[x]^{[2]}$.

We note that (3) means that “$A$ is not a polynomial ring over its subring $k[x]$”. It does not imply that “$A$ is not a polynomial ring over $k$”, i.e., $A\neq k^{[3]}$.

(1) shows that $A$ is an $\mathbb{A}^2$-fibration over $k[x]$; (2) shows that $A$ is a stably polynomial ring over $k[x]$, in particular, a stably polynomial ring over $k$; and (3) shows that $A$ is not a polynomial ring over $k[x]$. Thus $A$ is a nontrivial $\mathbb{A}^2$-fibration over $k[x]$, apart from being a nontrivial stably polynomial ring over $k[x]$.

This ring $A$ was soon to acquire a wider significance. In a subsequent paper (2, Theorem 2.2), using the ring $A$, Asanuma constructed nonlinearizable $k^*$-actions on $\mathbb{A}^n_k$ over any infinite field $k$ of positive characteristic when $n \ge 4$. He then asked whether $A$ is a polynomial ring and explained the significance of his question as follows (2, Remark 2.3):

“If $A$ is a polynomial ring then it will give an example of a nonlinearizable torus action on $k^3$ in positive characteristic. On the other hand if $A$ is not a polynomial ring then it will clearly give a counter-example to the cancellation problem.”

Thus either way one would answer a major problem in AAG. This dichotomy has been popularized by P. Russell as “Asanuma’s Dilemma”.

In 10, using some basic results on exponential maps, the author has shown that the ring $A$ in Example 5.3 is not a polynomial ring when $m \ge 2$. Consequently, it follows that the ZCP does not have an affirmative answer in positive characteristic. In 11, the author made further investigations on the Asanuma ring discussed below.

Recall that a polynomial $g \in k[Z,T]$ is called a line if $k[Z,T]/(g) = k^{[1]}$ and a line $g$ is called a nontrivial line if $k[Z, T] \neq k[g]^{[1]}$. The famous epimorphism theorem of S.S. Abhyankar and T. Moh (also proved independently by M. Suzuki for $k=\mathbb{C}$) asserts that there does not exist any nontrivial line over any field of characteristic zero. However, as early as in 1957, B. Segre had exhibited an example of a nontrivial line over any field of positive characteristic. Later M. Nagata gave a family of examples of such nontrivial lines. For more details on the epimorphism problem, see 6, Section 2.

Asanuma’s three-dimensional ring in Example 5.3 can be considered as a special case of the general class of threefolds in $\mathbb{A}^4_k$ defined by the zero locus of a polynomial of the form $X^m Y-f(Z,T)$, where $f(Z,T)$ is a Segre–Nagata nontrivial line. This led us to a problem which was asked to the author independently by P. Russell.

Question 3.

Let $f$ be any nontrivial line in $k[Z,T]$ and let $A$ be a ring defined by the relation $x^ry=f(z,t)$. Is the ring $A$ necessarily not a polynomial ring over the field $k$?

As classification of nontrivial lines is still an open problem, Russell’s question has to be approached abstractly. In 11, the author has considered a more general threefold and answered Russell’s question affirmatively as a part of a general theory which is independent of the characteristic of the field. This generalization is more transparent and conceptual, and has simplified the earlier proof by the author in 10 of the noncancellative property of $k^{[3]}$ in positive characteristic. The precise statement proved by the author is (11, Theorem 3.11):

Theorem 5.4.

Let $k$ be a field of any characteristic and $F \in k[X,Z, T]$. Let $f(Z, T)= F(0, Z, T)$, $G= X^r Y - F(X, Z, T) \in k[X,Y,Z,T]$, where $r >1$ and $A= k[X,Y,Z,T]/(G)$. Then the following statements are equivalent:

(i)

$f(Z, T)$ is a variable in $k[Z, T]$, i.e., $k[Z,T]= k[f]^{[1]}$.

(ii)

$A = k[x]^{[2]}$, where $x$ denotes the image of $X$ in $A$.

(iii)

$A = k^{[3]}$.

(iv)

$G$ is a variable in $k[X, Y, Z, T]$, i.e., $k[X,Y,Z,T]=k[G]^{[3]}$.

(v)

$G$ is a variable in $k[X, Y, Z, T]$ along with $X$, i.e., $k[X,Y,Z,T]=k[X,G]^{[2]}$.

Theorem 5.4 answers in one statement several very different looking questions that had been of long interest in the field. The equivalence of (i) and (iii) answers Russell’s question affirmatively. It also explains the nontriviality of the Russell–Koras threefold defined in Section 4. On the other hand the following theorem of the author shows that if $f$ is a line, then $A$ is a stably polynomial ring. More precisely (11, Theorem 4.2):

Theorem 5.5.

Let $k$ be a field of any characteristic, $F \in k[X,Z, T]$ and let

$$\begin{equation*} A= k[X,Y,Z,T]/(X^rY - F(X, Z, T)), \text{ where } r \ge 1. \end{equation*}$$

Then $A$ is a stably polynomial ring if $F(0, Z,T)$ is a line in $k[Z,T]$. That is, $A^{[1]} =k[x]^{[3]}= k^{[4]}$ if $k[Z,T]/(F(0, Z,T))= k^{[1]}$.

By the equivalence of (i) and (iii) in Theorem 5.4, it follows that $A$ is not a polynomial ring if $k[Z,T]\neq k[F(0, Z,T)]^{[1]}$. Thus, if $F(0, Z,T)$ is a nontrivial line in $k[Z,T]$, then $A$ is a stably polynomial ring but not a polynomial ring over $k$. This gives a recipe for constructing counter-examples to the ZCP. We emphasize again that the proofs of Theorems 5.5 and 5.4 are independent of the characteristic of the field. However, we know by the Abhyankar–Moh–Suzuki theorem that a nontrivial line never exists in characteristic zero. Thus, for obtaining counter-examples to the cancellation problem for affine $3$-space, an application of Theorems 5.5 and 5.4 can be made only in positive characteristic.

Theorem 5.4 was achieved through the study of another invariant of a ring arising out of exponential maps that was introduced by H. Derksen, and is now known as the Derksen invariant. For an affine $k$-algebra $R$, the Derksen invariant of $R$ is the subring of $R$, denoted by $\operatorname {DK}(R)$ and defined by

$$\begin{equation*} \operatorname {DK}(R)\coloneq k[f \in R^{\phi }\,|\, \phi \in \operatorname {Exp}(R)\nobreakspace \text{and}\nobreakspace B^{\phi }\neq B]. \end{equation*}$$

A major result used in the proof of Theorem 5.4 is the following proposition.

Proposition 5.6.

Let $A$ be an integral domain defined by

$$\begin{equation*} A= k[X,Y,Z,T]/(X^m Y - F(X, Z, T)), {\text{ where }} m > 1. \end{equation*}$$

Set $f(Z, T)\coloneq F(0, Z, T)$. Let $x$, $y$, $z$, and $t$ denote, respectively, the images of $X$, $Y$, $Z$, and $T$ in $A$. Suppose that $\operatorname {DK}(A) \neq k[x,z,t]$. Then the following statements hold.

(i)

There exist $Z_1, T_1 \in k[Z, T]$ and $a_0, a_1 \in k^{[1]}$ such that $k[Z, T]=k[Z_1, T_1]$ and $f(Z, T) = a_0(Z_1) + a_1(Z_1)T_1$.

(ii)

If $k[Z, T]/(f)= k^{[1]}$, then $k[Z, T]=k[f]^{[1]}$.

From Proposition 5.6, it follows that in case $A$ is as in Example 5.3 with $m \ge 2$, then $\operatorname {DK}(A) =k[x,z,t] \subsetneqq A$. Thus $\operatorname {DK}(A)\neq A$ and hence $A \neq k^{[3]}$, as $\operatorname {DK}(k^{[3]})=k^{[3]}$.

Subsequently, the author generalized the Asanuma threefold to obtain counterexamples to the ZCP in positive characteristic in higher dimensions (12). Recently with Parnashree Ghosh, the author has used exponential maps to obtain the following generalization of Theorem 5.4 in higher dimensions (9).

Theorem 5.7.

Let $F= F(X_{1}, \ldots , X_{m}, Z,T)= f(Z,T)+ (X_{1} \cdots X_{m})g$, for some $g \in k[X_1,\ldots ,X_m,Z,T]$, $G\coloneq X_{1}^{r_{1}} \cdots X_{m}^{r_{m}}Y -F$ and

$$\begin{equation} \begin{gathered} A= \frac{k\left[ X_{1}, \ldots , X_{m}, Y,Z,T\right]}{(X_{1}^{r_{1}} \cdots X_{m}^{r_{m}}Y -F(X_{1}, \ldots , X_{m}, Z,T))},\\ r_i > 1 \text{ for all } i, 1 \leqslant i \leqslant m, \end{gathered} {\tag{3}} \end{equation}$$

Then the following statements are equivalent:

(i)

$k[X_{1}, \ldots , X_{m},Y,Z,T]=k[X_{1}, \ldots , X_{m},G]^{[2]}$.

(ii)

$k[X_{1}, \ldots , X_{m},Y,Z,T]=k[G]^{[m+2]}$.

(iii)

$A=k[x_{1}, \ldots , x_{m}]^{[2]}$.

(iv)

$A=k^{[m+2]}$.

(v)

$k[Z,T]=k[f(Z,T)]^{[1]}$.

Moreover, each of the above conditions is equivalent to each of the following four conditions involving the Derksen invariant; and each of the subsequent four conditions involving the Makar-Limanov invariant.

(vi)

$A^{[l]}=k^{[l+m+2]}$ for some $l \geqslant 0$ and $k[x_{1},\ldots ,x_{m},z,t] \subsetneqq \operatorname {DK}(A)$.

(vii)

$A$ is an $\mathbb{A}^{2}$-fibration over $k[x_{1}, \ldots , x_{m}]$ and $k[x_{1}, \ldots , x_{m},z,t] \subsetneqq \operatorname {DK}(A)$.

(viii)

$f(Z,T)$ is a line in $k[Z,T]$ and $k[x_{1}, \ldots , x_{m},z,t] \subsetneqq \operatorname {DK}(A)$.

(ix)

$A$ is a UFD, $k[x_{1},\ldots ,x_{m},z,t] \subsetneqq \operatorname {DK}(A)$ and $\left( \frac{A}{x_{i}A} \right)^{*}=k^{*}$, for every $i\in \{1,\ldots ,m\}$.

(x)

$f(Z,T)$ is a line in $k[Z,T]$ and $\operatorname {ML}(A)=k$.

(xi)

$A^{[l]}=k^{[l+m+2]}$ for $l \geqslant 0$ and $\operatorname {ML}(A)=k$.

(xii)

$A$ is an $\mathbb{A}^{2}$-fibration over $k[x_{1},\ldots ,x_{m}]$ and $\operatorname {ML}(A)=k$.

(xiii)

$A$ is a UFD, $\operatorname {ML}(A)=k$ and $\left( \frac{A}{x_{i}A} \right)^{*}=k^{*}$, for every $i \in \{1,\ldots ,m\}$.