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$ \DeclareMathOperator{\init}{in} \DeclareMathOperator{\gr}{gr} \newcommand{\calO}{\mathcal{O}} $Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\gr_{\mathfrak{m}}(R)=\bigoplus_{i=0}^\infty\mathfrak{m}^i/\mathfrak{m}^{i+1}$ be the associated graded ring. For $f\in R$, the initial form of $f$ is $\init(f)=f+\mathfrak{m}^{i+1}\in\mathfrak{m}^i/\mathfrak{m}^{i+1}$ where $f\in\mathfrak{m}^i$ but $f\notin\mathfrak{m}^{i+1}$. For an ideal $I\subset R$, let $\init(I)$ be the ideal of $\gr_{\mathfrak{m}}(R)$ generated by the initial forms of elements of $I$. In general, if $f_1,\dots,f_k$ generate $I$, then $\init(I)$ is not necessarily generated by $\init(f_1),\dots,\init(f_k)$ (see for example here).

In Eisenbud's Commutative Algebra he gives an exercise (5.2) to find a local ring $R$ and $f\in R$ where $(\init(f))\subsetneq\init((f))$. However in the "Tangent Cones" section of Milne's notes on algebraic geometry, he states that in the setting where we view $f$ as an element of a polynomial ring, then $(\init(f))=\init((f))$.

I interpret Milne's comment to mean that at any regular point $P$ of a (say quasiprojective) scheme $X$, and for any $f\in\calO_P$, we have $(\init(f))=\init((f))$ since here $\gr_{\mathfrak{m}}(\calO_P)$ is a polynomial ring. (Edit: I now see this is Exercise 4.6.12 in Bruns and Herzog.) Is there a more general condition on a local ring $R$ for when $(\init(f))=\init((f))$? For example, I think this is true as long as $R$ is a domain, but even here bad things can happen with $\gr_{\mathfrak{m}}(R)$ (e.g. it may not be a domain) so I'm not sure.

mbert
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1 Answers1

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Let $\newcommand{\In}{\operatorname{in}}\In(f)=f+\newcommand{\m}{\mathfrak m}\m^{p+1}$. Then $\In(\langle f\rangle)=\langle \In(f)\rangle$ if and only if $\m^n\cap \langle f\rangle=\m^{n-p}\langle f\rangle$ for all $n\geq 1$.

This is a special case of a more general theorem (Theorem 1.1) proved by Valabrega and Valla in their paper Form Rings and Regular Sequences (Project Euclid link).

The proof of sufficiency is easy: all we have to do is notice that the $n$-th homogeneous component of $\In(\langle f\rangle)$ is $\m^n\cap \langle f\rangle+\m^{n+1}/\m^{n+1}$ and that of $\langle \In(f)\rangle$ is $\m^{n-p}f+\m^{n+1}/\m^{n+1}$.

KReiser
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cqfd
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