Intersection theory of algebraic obstructions

Let AA be a noetherian commutative ring of dimension dd and LL be a rank one projectiveAA-module. For 1≤r≤d1≤r≤d, we define obstruction groups Er(A,L)Er(A,L). This extends the original definition due to Nori, in the case r=dr=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) Er(A,A)×Es(A,A)→Er+s(A,A)Er(A,A)×Es(A,A)→Er+s(A,A). For a projective AA-module QQ of rank n≤dn≤d, with an orientation χ:L→∼∧nQ, we define a Chern class like homomorphism w(Q,χ):Ed−n(A,L′)→Ed(A,LL′),w(Q,χ):Ed−n(A,L′)→Ed(A,LL′), where L′L′ is another rank one projective AA-module.