Some remarks on symplectic injective stability

Abstract

It is shown that if $ A$ is an affine algebra of odd dimension $ d$ over an infinite field of cohomological dimension at most one, with $ (d +1)! A = A$, and with $ 4\vert(d -1)$, then Um $ _{d+1}(A) = e_1\textrm{Sp}_{d+1}(A)$. As a consequence it is shown that if $ A$ is a non-singular affine algebra of dimension $ d$ over an infinite field of cohomological dimension at most one, and $ d!A = A$, and $ 4\vert d$, then $ \textrm{Sp}_d(A) \cap \textrm{ESp}_{d+2}(A) = \textrm{ESp}_d(A)$. This result is a partial analogue for even-dimensional algebras of the one obtained by Basu and Rao for odd-dimensional algebras earlier.