## A particular Isomorphism of graded algebras over a regular local ring

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:

Proposition. Let $R$ be a regular local ring and $I=(x_1,\ldots,x_r)\subset R$ an ideal generated by a regular sequence. Let $A:=R/I$. Then, ```$$\begin{eqnarray*}
  \phi: A[T_1,\ldots,T_r] &\overset{_\sim}{\longrightarrow} & \mathrm{gr}_I(R) = \bigoplus\nolimits_{d\ge 0} {I^d}/{I^{d+1}} \\
   T_i & \longmapsto & x_i
  \end{eqnarray*}$$``` is an isomorphism of graded $A$-algebras.

In Hartshorne, the condition of being regular and local is strengthened to Cohen-Macaulay. However, I only need the above. I tried to look up the proof for the general statement in Matsumuura's book, but it seems rather involved (and honestly, a bit convoluted). I would like to use results about regular local rings and in turn, avoid introducing terminology like Hilbert-Samuel polynomials.

So, I guess I am asking for an "easy" proof of the above proposition. It seems rather easy for $r=1$, but I am somehow stuck trying to prove it by induction.

-

1
One reference is Bruns-Herzog, Theorem 1.1.8. You don't really need any assumption on $R$. – Hailong Dao Nov 2 2011 at 19:24
That's just perfect. Make it an answer and I'll accept. – Jesko Hüttenhain Nov 2 2011 at 23:13

## 1 Answer

One reference is Bruns-Herzog, Theorem 1.1.8. You don't really need any assumption on $R$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8860535025596619, "perplexity_flag": "middle"}