Vanishing of the homology modules of a Koszul complex

AbstractGiven a commutative Noetherian ring R, there is associated with each homomorphism φ: F → E between finitely generated free R modules, a Koszul complex K(φ) over the symmetric algebra of E. As a complex of R modules, K(φ) splits into direct summands of complexes Kμ(φ) for each integer μ. For rank F ≥ rank E, we obtain an upper bound for the degrees of the non-vanishing homology modules of Kμ(φ) in terms of the grades of the Fitting ideals of Coker φ, provided the grade of the 0th Fitting ideal is at least (rank F − rank E + 1)