Exercise 8, § 3

Let A be a ring, u : A^p \to A^q an a-linear mapping. The rank of u is called, and is denoted by rg(u), the largest integer r such that \wedge^r u \neq 0. We denote by \partial(u) the ideal of A generated by the minors of order rg(u) of the matrix of u. a) Show that for \partial(u) to be equal to A, it is necessary and sufficient that the cokernel of u be projective of rank q – rg(u) (reduce to the case where A is local and write the matrix of u in a suitable basis). b) Let v : A^q \to A^s be an a-linear mapping; suppose that \partial(u) = \partial(v) = A. For Ker v = Im u to hold, it is necessary and sufficient that rg(u) + rg(v) = q (same method). c) If rg(u) = q, prove that \partial(u) annihilates the cokernel of u (reduce to the case p = q).