The Stacks project

111.14 Singularities

Exercise 111.14.1. Let $k$ be any field. Suppose that $A = k[[x, y]]/(f)$ and $B = k[[u, v]]/(g)$, where $f = xy$ and $g = uv + \delta $ with $\delta \in (u, v)^3$. Show that $A$ and $B$ are isomorphic rings.

Remark 111.14.2. A singularity on a curve over a field $k$ is called an ordinary double point if the complete local ring of the curve at the point is of the form $k'[[x, y]]/(f)$, where (a) $k'$ is a finite separable extension of $k$, (b) the initial term of $f$ has degree two, i.e., it looks like $q = ax^2 + bxy + cy^2$ for some $a, b, c\in k'$ not all zero, and (c) $q$ is a nondegenerate quadratic form over $k'$ (in char 2 this means that $b$ is not zero). In general there is one isomorphism class of such rings for each isomorphism class of pairs $(k', q)$.

Exercise 111.14.3. Let $R$ be a ring. Let $n \geq 1$. Let $A$, $B$ be $n \times n$ matrices with coefficients in $R$ such that $AB = f 1_{n \times n}$ for some nonzerodivisor $f$ in $R$. Set $S = R/(f)$. Show that

\[ \ldots \to S^{\oplus n} \xrightarrow {B} S^{\oplus n} \xrightarrow {A} S^{\oplus n} \xrightarrow {B} S^{\oplus n} \to \ldots \]

is exact.

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