The Stacks project

Exercise 111.28.6. Let $(A, {\mathfrak m}, k)$ be a Noetherian local ring with associated graded $Gr_{\mathfrak m}(A)$.

1. Suppose that $x\in {\mathfrak m}^ d$ maps to a nonzerodivisor $\bar x \in {\mathfrak m}^ d/{\mathfrak m}^{d + 1}$ in degree $d$ of $Gr_{\mathfrak m}(A)$. Show that $x$ is a nonzerodivisor.

2. Suppose the depth of $A$ is at least $1$. Namely, suppose that there exists a nonzerodivisor $y \in {\mathfrak m}$. In this case we can do better: assume just that $x\in {\mathfrak m}^ d$ maps to the element $\bar x \in {\mathfrak m}^ d/{\mathfrak m}^{d + 1}$ in degree $d$ of $Gr_{\mathfrak m}(A)$ which is a nonzerodivisor on sufficiently high degrees: $\exists N$ such that for all $n \geq N$ the map of multiplication by $\bar x$

   \[ {\mathfrak m}^ n/{\mathfrak m}^{n + 1} \longrightarrow {\mathfrak m}^{n + d}/{\mathfrak m}^{n + d + 1} \]

   is injective. Then show that $x$ is a nonzerodivisor.