In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformatio...

We explore the lifting question in the context of cut-generating functions. Most of the prior literature on lifting for cut-generating functions focuses on which cut-generating functions have the unique lifting property. Here we develop a general theory for understanding how to do lifting for cut-generating functions which do not have the unique lifting property.

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.