I encountered the following in the introduction of the paper "Duality for Koszul Homology over Gorenstein Rings":
I assume it's easy, but why is this fact true? I played with modding out by regular sequences to try to get to a "codimension 1 reduction" or do something something with interpreting $S_1$ and $S_2$ in terms of duality (e.g., reflexive modules)... But nothing stuck. Any ideas?
Also, is there a more general phenomenon at play here or is this something special with $S_2$ and $S_1$? Could these be replaced by $S_m$ and $S_n$ (for some $n$) to yield "$\varphi:M\to N$ is an isomorphism when localized at any prime ideal $\mathfrak{p}$ such that $\text{ht}(\mathfrak{p})\leq m$"?
Let $K=\ker\varphi$ and suppose that $\varphi$ is not injective, i.e., $K\neq 0$. Then $\operatorname{Ass} K$ is non-empty, and we can find some associated prime $\newcommand{\p}{\mathfrak p}\p$ of $K$ such that $K_\p\neq 0$. If $\mathfrak q$ is a prime ideal of height $\leq 1$, then we know from our assumption that $K_{\mathfrak q}=0$, so the height of $\p$ must be $\geq2$. Thus, as $M$ satisfies $S_2$, $\operatorname{depth} M_\p\geq 2$. But $\p R_\p\in\operatorname{Ass} K_\p\subseteq \operatorname{Ass} M_\p$, so there is no way $\p R_\p$ contains a nonzerodivisor on $M_\p$. This contradiction concludes the injectivity part.
The surjectivity part is similar, and you might find the depth lemma useful, as @metalspringpro suggests.