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Summary: CLASS AND RANK OF DIFFERENTIAL MODULES
LUCHEZAR L. AVRAMOV, RAGNAR-OLAF BUCHWEITZ, AND
SRIKANTH IYENGAR
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.
Contents
Introduction 1
1. Differential modules 4
2. Differential flags 9
3. Class inequality. I 14
4. Class inequality. II 18
5. Rank inequalities 20
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