Some remarks on symplectic injective stability

Abstract

It is shown that if A A is an affine algebra of odd dimension d d over an infinite field of cohomological dimension at most one, with ( d + 1 ) ! A = A (d +1)! A = A , and with 4 | ( d − 1 ) 4|(d -1) , then Um d + 1 ( A ) = e 1 Sp d + 1 ( A ) _{d+1}(A) = e_1\textrm {Sp}_{d+1}(A) . As a consequence it is shown that if A A is a non-singular affine algebra of dimension d d over an infinite field of cohomological dimension at most one, and d ! A = A d!A = A , and 4 | d 4|d , then Sp d ( A ) ∩ ESp d + 2 ( A ) = ESp d ( A ) \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.