Lifting monomial ideals

We show how to lift any monomial ideal J in n variables to a saturated idea l I of the same codimension in n + t variables. We show that I has the same graded Betti numbers as J and we show holy to obtain the matrices for the resolution of I. The cohomology of I is described. Making general choices f or our lifting, we show that I is the ideal of a reduced union of linear va rieties with singularities that are "as small as possible" given the cohomo logical constraints. The case where J is Artinian is the nicest. III the ca se of curves we obtain stick figures for I, and in the case of points Ne ob tain certain k-configurations which we can describe in a very precise way.