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 $4|(d -1)$, 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 $4|d$, 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.

Locations

• Proceedings of the American Mathematical Society - View - PDF