Session 1B — Lectures by Fellows/Associates

Umesh Waghmare, JNCASR, Bengaluru

On Separable 𝔸² and 𝔸³-form View Presentation

Let k be a field and F be its algebraic closure. A k-algebra B is said to be an 𝔸ⁿ-form over k if B⊗_k F is isomorphic to the polynomial ring F[Y₁,...,Y_n].

It is well-known that separable 𝔸¹-forms over k are isomorphic to the polynomial ring k[Y] and that there exist non-trivial purely inseparable 𝔸¹- forms over fields of positive characteristic. A nontrivial result of T. Kam-bayashi establishes that separable 𝔸²-forms over k are also isomorphic to the polynomial ring k[Y₁, Y₂]. However, for n > 2, it is not known whether every separable 𝔸ⁿ-form is necessarily isomorphic to the polynomial ring k[Y₁, Y_n].

In this talk, we shall discuss a partial solution to this problem for the case n = 3. We shall also discuss 𝔸²-forms over commutative rings.