Strongly Cohen-Macaulay Schemes and Residual Intersections

Abstract

This paper studies the local properties of closed subschemes $Y$ in Cohen-Macaulay schemes $X$ such that locally the defining ideal of $Y$ in $X$ has the property that its Koszul homology is Cohen-Macaulay. Whenever this occurs $Y$ is said to be strongly Cohen-Macaulay in $X$. This paper proves several facts about such embeddings, chiefly with reference to the residual intersections of $Y$ in $X$. The main result states that any residual intersection of $Y$ in $X$ is again Cohen-Macaulay.