Class and rank of differential modules

Abstract. A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class—a substitute for the length of a free complex—and on the rank of a differential module in terms of invariants of its homology. These re-sults specialize to basic theorems in commutative algebra and algebraic topol-ogy. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over commutative noetherian rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings