The Stacks project

Lemma 10.103.13. Let $R$ be a Noetherian ring. Let $M$ be a Cohen-Macaulay module over $R$. Then $M \otimes _ R R[x_1, \ldots , x_ n]$ is a Cohen-Macaulay module over $R[x_1, \ldots , x_ n]$.

Proof. By induction on the number of variables it suffices to prove this for $M[x] = M \otimes _ R R[x]$ over $R[x]$. Let $\mathfrak m \subset R[x]$ be a maximal ideal, and let $\mathfrak p = R \cap \mathfrak m$. Let $f_1, \ldots , f_ d$ be a $M_\mathfrak p$-regular sequence in the maximal ideal of $R_{\mathfrak p}$ of length $d = \dim (\text{Supp}(M_{\mathfrak p}))$. Note that since $R[x]$ is flat over $R$ the localization $R[x]_{\mathfrak m}$ is flat over $R_{\mathfrak p}$. Hence, by Lemma 10.68.5, the sequence $f_1, \ldots , f_ d$ is a $M[x]_{\mathfrak m}$-regular sequence of length $d$ in $R[x]_{\mathfrak m}$. The quotient

\[ Q = M[x]_{\mathfrak m}/(f_1, \ldots , f_ d)M[x]_{\mathfrak m} = M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \otimes _{R_\mathfrak p} R[x]_{\mathfrak m} \]

has support equal to the primes lying over $\mathfrak p$ because $R_\mathfrak p \to R[x]_\mathfrak m$ is flat and the support of $M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p}$ is equal to $\{ \mathfrak p\} $ (details omitted; hint: follows from Lemmas 10.40.4 and 10.40.5). Hence the dimension is $1$. To finish the proof it suffices to find an $f \in \mathfrak m$ which is a nonzerodivisor on $Q$. Since $\mathfrak m$ is a maximal ideal, the field extension $\kappa (\mathfrak m)/\kappa (\mathfrak p)$ is finite (Theorem 10.34.1). Hence we can find $f \in \mathfrak m$ which viewed as a polynomial in $x$ has leading coefficient not in $\mathfrak p$. Such an $f$ acts as a nonzerodivisor on

\[ M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \otimes _ R R[x] = \bigoplus \nolimits _{n \geq 0} M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \cdot x^ n \]

and hence acts as a nonzerodivisor on $Q$. $\square$

Comments (0)

There are also:

  • 6 comment(s) on Section 10.103: Cohen-Macaulay modules

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Your email address will not be published.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 0AAI. Beware of the difference between the letter 'O' and the digit '0'.