Openness of local properties in flat families of superschemes

Abstract

Recently, Jankowski introduced the Serre criteria for superschemes to define the super versions of normality and reducedness (called generic fermionic regularity and reducedness). We show that for a flat, proper morphism $f:X\rightarrow Y$, where $X$ is a superscheme and $Y$ is a Dedekind scheme of finite type over a field of characteristic $0$, the set of $y\in Y$ for which the fibers $X_y$ are normal, GFRR, regular, and Cohen-Macaulay is open in $Y$. In particular, if one has a degeneration over $\mathbb{A}_{\mathbb{C}}^{1|0}$, this shows that these properties can be lifted from the special fiber to the generic one.