A Theorem of Gruson

a presentation of M , where F is a free module of arbitrary rank. Let U denote the matrix (with n rows but possibly infinitely many columns) representing φ with respect to some fixed bases for F and An. For t ≥ 1 we let It(φ) denote the ideal of A generated by all t-sized minors U . (This ideal is independent of the choice of bases.) By convention, It(φ) = R for t ≤ 0 and It(φ) = 0 if t exceeds n or the rank of F . For an integer i, we let Fi(M) := In−i(φ). It is a result of Fitting that Fi(M) does not depend on the presentation of M (e.g., [3, Theorem 1, pg. 58]). If the ring A is not clear from the context, we write FA i (M) for Fi(M). A projective A-module is said to have constant rank r if Pq ∼= Aq for every prime ideal q.