Cancellation and Splitting of Symplectic Modules in the Critical Range and Euler Class Group

Abstract In this paper, we discuss the cancellation and splitting of the symplectic modules. The symplectic cancellation result presented here can be thought of as an analog of the projective module cancellation result of Fasel. The symplectic splitting is similar to Murthy’s splitting theorem. To prove the cancellation and splitting, we analyze the Postnikov towers in the $\mathbb{A}^{1}-$homotopy category. We study the top cohomology group with coefficients in certain $\mathbb A^{1}-$homotopy sheaf of even quadrics and show certain structural properties of these cohomology groups. As another application of this, we answer partially a question of Mrinal Das about the isomorphism of $(d-1)-$th Euler class group and $(d-1)-$th Chow group of a smooth affine scheme of dimension $d$.