From Mennicke symbols to Euler class groups

Bhatwadekar and Raja Sridharan have constructed a homomorphism from an orbit set of unimodular rows to an Euler class group. We show under weaker assumptions that a generalization of its kernel is a subgroup. Our tool is a partially defined operation on the set of unimodular matrices with two rows. 1 The exact sequence Consider a commutative noetherian Q-algebra A of dimension d. Let n = d + 1. We often assume n is odd. We try to understand the following exact sequence. 0 → MSn−1(A) → Um2,n(A)/En(A) → Um1,n(A)/En(A) → E(A). A good case to keep in mind is d = 6. 2 The terms in the sequence Let us recall the terms in the sequence. All matrices will have entries in A. An m by n matrix M with m ≤ n is called unimodular if it has a right inverse, which is thus an n by m matrix. In other words, M is called unimodular if the corresponding map A → A is surjective. Let Umn(A) = Um1,n(A) be the set of unimodular rows with n entries. Following Suslin [Su2], we say that a Mennicke symbol of order n on A is a map φ from Umn(A) to an abelian group G such that MS1 and MS2 hold: MS1 For every elementary matrix ǫ ∈ En(A) and every v ∈ Umn(A) we have φ(vǫ) = φ(v) MS2 φ(x, a2, . . . , an)φ(y, a2, . . . , an) = φ(xy, a2, . . . , an) Here we simplified notation from φ((x, a2, . . . , an)) to φ(x, a2, . . . , an). 2.1 Universal Mennicke symbols The group MSn−1(A) is by definition the universal target of order n − 1 Mennicke symbols on A. We have shown in [vdK2] that v, w ∈ Umn−1(A) have the same image in MSn−1(A) if and only if there is ǫ ∈ En(A) ∩ GLn−1(A) with vǫ = w. (This needs that n − 1 is even and d ≤ 2n− 7.) Moreover, the map ms : Umn−1(A) → MSn−1(A) is surjective, so we may think of MSn−1(A) as the orbit set Umn−1(A)/En(A) ∩ GLn−1(A) that has been provided with a group structure. (For n− 1 = 2, d = 1 this is exactly what Bass Milnor Serre did for ordinary Mennicke symbols.) 1 2.2 The term that we expect to be a group The term Um2,n(A)/En(A) is just an orbit set of unimodular two by n matrices. We expect it also carries a group structure. Indeed the analogy with algebraic topology, cf. section 6, predicts an abelian group structure for d ≤ 2n−6. (Such an analogy gave correct predictions in [vdK2].) But for now we have no such group structure, so our exact sequence will be one of pointed sets only! The map MSn−1(A) → Um2,n(A)/En(A) we define by sending ms(v) to the orbit of (