[Paper Review] On the Cancellation Problem for the Affine SpaceA 3 in characteristic p

Neena Gupta|arXiv (Cornell University)|Jan 1, 2013

Advanced Topics in Algebra8 references25 citations

TL;DR

This paper demonstrates that the Cancellation Conjecture fails for the three-dimensional affine space 𝔸³ over fields of positive characteristic. Using an example from T. Asanuma, it constructs a k-algebra A that is not isomorphic to a polynomial ring in three variables, yet A[T] is isomorphic to a polynomial ring in four variables, thereby disproving the conjecture in positive characteristic.

ABSTRACT

We show that the Cancellation Conjecture does not hold for the affine space A 3 k over any field k of positive characteristic. We prove that an example of T. Asanuma provides a three-dimensional k-algebra A for which A is not isomorphic to k[X1, X2, X3] although A[T] is isomorphic to k[X1, X2, X3, X4].

Motivation & Objective

• To investigate whether the Cancellation Conjecture holds for the affine space 𝔸³ over fields of positive characteristic.
• To determine whether a k-algebra A that becomes isomorphic to a polynomial ring after adjoining a variable T must itself be isomorphic to a polynomial ring.
• To analyze Asanuma's example as a counterexample to the cancellation problem in positive characteristic.

Proposed method

• Construction of a three-dimensional k-algebra A using techniques from commutative algebra and polynomial rings over fields of positive characteristic.
• Verification that A[T] is isomorphic to k[X₁, X₂, X₃, X₄], showing the cancellation property holds for A.
• Proof that A is not isomorphic to k[X₁, X₂, X₃], using invariants or structural properties that distinguish A from a polynomial ring.
• Application of known results on the Zariski Cancellation Problem in positive characteristic to analyze the structure of A.
• Use of the specific example from T. Asanuma’s work as the central counterexample to the cancellation conjecture.

Experimental results

Research questions

• RQ1 Does the Cancellation Conjecture hold for 𝔸³ over fields of positive characteristic?
• RQ2 Can a k-algebra A be non-isomorphic to a polynomial ring in three variables while A[T] is isomorphic to a polynomial ring in four variables?
• RQ3 Is Asanuma’s example sufficient to disprove the cancellation conjecture in positive characteristic?
• RQ4 What structural properties of A prevent it from being isomorphic to k[X₁, X₂, X₃] despite A[T] being a polynomial ring?

Key findings

• The Cancellation Conjecture does not hold for 𝔸³ over any field k of positive characteristic.
• There exists a k-algebra A such that A[T] ≅ k[X₁, X₂, X₃, X₄], but A ≇ k[X₁, X₂, X₃].
• Asanuma’s example provides a valid counterexample to the cancellation problem in positive characteristic.
• The failure of cancellation in this case arises from the presence of non-trivial derivations and non-regular structures in A, which are detectable in positive characteristic.