The Stacks project

Lemma 10.117.1. Let $k$ be a field. Let $S$ be a graded $k$-algebra generated over $k$ by finitely many elements of degree $1$. Assume $S_0 = k$. Let $P(T) \in \mathbf{Q}[T]$ be the polynomial such that $\dim (S_ d) = P(d)$ for all $d \gg 0$. See Proposition 10.58.7. Then

  1. The irrelevant ideal $S_{+}$ is a maximal ideal $\mathfrak m$.

  2. Any minimal prime of $S$ is a homogeneous ideal and is contained in $S_{+} = \mathfrak m$.

  3. We have $\dim (S) = \deg (P) + 1 = \dim _ x\mathop{\mathrm{Spec}}(S)$ (with the convention that $\deg (0) = -1$) where $x$ is the point corresponding to the maximal ideal $S_{+} = \mathfrak m$.

  4. The Hilbert function of the local ring $R = S_{\mathfrak m}$ is equal to the Hilbert function of $S$.

Proof. The first statement is obvious. The second follows from Lemma 10.57.8. By (2) every irreducible component passes through $x$. Thus we have $\dim (S) = \dim _ x\mathop{\mathrm{Spec}}(S) = \dim (S_\mathfrak m)$ by Lemma 10.114.5. Since $\mathfrak m^ d/\mathfrak m^{d + 1} \cong \mathfrak m^ dS_\mathfrak m/\mathfrak m^{d + 1}S_\mathfrak m$ we see that the Hilbert function of the local ring $S_\mathfrak m$ is equal to the Hilbert function of $S$, which is (4). We conclude the last equality of (3) by Proposition 10.60.9. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 10.117: Dimension of graded algebras over a field

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00P6. Beware of the difference between the letter 'O' and the digit '0'.