Intersection theory of algebraic obstructions

Abstract

Let A be a noetherian commutative ring of dimension d and L be a rank one projective A -module. For 1 ≤ r ≤ d , we define obstruction groups E r ( A , L ) . This extends the original definition due to Nori, in the case r = d . These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E r ( A , A ) × E s ( A , A ) → E r + s ( A , A ) . For a projective A -module Q of rank n ≤ d , with an orientation χ : L → ∼ ∧ n Q , we define a Chern class like homomorphism w ( Q , χ ) : E d − n ( A , L ′ ) → E d ( A , L L ′ ) , where L ′ is another rank one projective A -module.