The Stacks project

Lemma 53.16.4. Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \in X$ be a multicross singularity (Definition 53.16.2). If $X$ is Gorenstein, then $x$ is a node.

Proof. The map $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge $ is flat and unramified in the sense that $\kappa (x) = \mathcal{O}_{X, x}^\wedge /\mathfrak m_ x \mathcal{O}_{X, x}^\wedge $. (See More on Algebra, Section 15.43.) Thus $X$ is Gorenstein implies $\mathcal{O}_{X, x}$ is Gorenstein, implies $\mathcal{O}_{X, x}^\wedge $ is Gorenstein by Dualizing Complexes, Lemma 47.21.8. Thus it suffices to show that the ring $A$ in ( with $n \geq 2$ is Gorenstein if and only if $n = 2$.

If $n = 2$, then $A = k[[x, y]]/(xy)$ is a complete intersection and hence Gorenstein. For example this follows from Duality for Schemes, Lemma 48.24.5 applied to $k[[x, y]] \to A$ and the fact that the regular local ring $k[[x, y]]$ is Gorenstein by Dualizing Complexes, Lemma 47.21.3.

Assume $n > 2$. If $A$ where Gorenstein, then $A$ would be a dualizing complex over $A$ (Duality for Schemes, Definition 48.24.1). Then $R\mathop{\mathrm{Hom}}\nolimits (k, A)$ would be equal to $k[n]$ for some $n \in \mathbf{Z}$, see Dualizing Complexes, Lemma 47.15.12. It would follow that $\mathop{\mathrm{Ext}}\nolimits ^1_ A(k, A) \cong k$ or $\mathop{\mathrm{Ext}}\nolimits ^1_ A(k, A) = 0$ (depending on the value of $n$; in fact $n$ has to be $-1$ but it doesn't matter to us here). Using the exact sequence

\[ 0 \to \mathfrak m_ A \to A \to k \to 0 \]

we find that

\[ \mathop{\mathrm{Ext}}\nolimits ^1_ A(k, A) = \mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)/A \]

where $A \to \mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)$ is given by $a \mapsto (a' \mapsto aa')$. Let $e_ i \in \mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)$ be the element that sends $(f_1, \ldots , f_ n) \in \mathfrak m_ A$ to $(0, \ldots , 0, f_ i, 0, \ldots , 0)$. The reader verifies easily that $e_1, \ldots , e_{n - 1}$ are $k$-linearly independent in $\mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)/A$. Thus $\dim _ k \mathop{\mathrm{Ext}}\nolimits ^1_ A(k, A) \geq n - 1 \geq 2$ which finishes the proof. (Observe that $e_1 + \ldots + e_ n$ is the image of $1$ under the map $A \to \mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)$.) $\square$

Comments (0)

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Your email address will not be published.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