Cancellation and splitting of Symplectic modules in the critical range and Euler class group

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 carefully analyze the Postnikov towers in the ${\mathbb{A}}^1$-homotopy category. Then we prove the vanishing of top cohomology with coefficients in some homotopy sheaf. As another application of the vanishing results, we answer partially a question of Mrinal Das about the isomorphism of $(d-1)$-th Euler class group and $(d-1)$-th Chow group, where $d$ is the dimension of the underlying smooth affine variety.