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 n → A m 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 ɛ ∈ En(A) and every v ∈ Umn(A) we have φ(vɛ) = φ(v
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