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.
111.14 Singularities
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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