School of Mathematics Seminars and Lectures
A Counterexample to the Cancellation Problem for Affine Spaces
by Dr. Neena Gupta (Visiting Fellow, TIFR)
Wednesday, September 26, 2012
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-77 )
at Colaba Campus ( AG-77 )
Description |
The Cancellation Problem for Affine Spaces (also known as Zariski Problem) asks: if $V$ is an affine variety over an algebraically closed field $k$ such that $V \times \A^1_k \cong \A^{n+1}_k$, does it follow that $V \cong \A_k^n$? Equivalently, if $A$ is an affine $k$-algebra such that $A[X]$ is isomorphic to the polynomial ring $k[X_1, \dots, X_{n+1}]$, does it follow that $A$ is isomorphic to $k[X_1, \dots, X_n]$?For $n=1$, an affirmative solution to the problem was given by S. Abhyankar, P. Eakin and W. Heinzer (1972). For $n=2$, an affirmative solution to the problem was given by T. Fujita (1979), M. Miyanishi and T. Sugie (1980) in characteristic zero and by P. Russell (1981) in positive characteristic. Over a field $k$ of positive characteristic, T. Asanuma (1987) constructed a three-dimensional $k$-algebra $A$ such that $A[T]$ is isomorphic to $k[X_1,X_2,X_3, X_4]$. The example gave rise to (in the words of P. Russell) ``Asanuma's Dilemma''. For, if $A$ were isomorphic to $k[X_1, X_2, X_3]$, then this would give a counterexample to the Linearisation Problem over $\A_k^3$ in positive characteristic; if not, then a counterexample to the Cancellation Problem. In this talk we will show that Asanuma's example $A$ is not isomorphic to $k[X_1, X_2, X_3]$. |