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 (53.16.0.1) 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 $A \to \mathop{\mathrm{Hom}}\nolimits _ A(\mathfrak m_ A, A)$.) $\square$

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