Generalized Matric Massey Products for Graded Modules

Abstract

The theory of generalized matric Massey products has been applied for some time to $A$-modules $M$, $A$ a $k$-algebra. The main application is to compute the local formal moduli ${\hat{H}}_M$, isomorphic to the local ring of the moduli of $A$-modules. This theory is also generalized to ${\mathcal{O}}_X$-modules $\mathcal{M}$, $X$ a $k$- scheme. In these notes we consider the definition of generalized Massey products and the relation algebra in any obstruction situation (a differential graded $k$-algebra with certain properties), and prove that this theory applies to the case of graded $R$-modules, $R$ a graded $k$-algebra, $k$ algebraically closed. When the relation algebra is algebraizable, that is the relations are polynomials rather than power series, this gives a combinatorial way to compute open (étale) subsets of the moduli of graded $R$-modules. This also gives a sufficient condition for the corresponding point in the moduli of $\mathcal{O}_{\Proj(R)}$-modules to be singular. The computations are straight forward, algorithmic, and an example on the postulation Hilbert scheme is given.