Characteristic-free Knörrer periodicity

We show that the Knörrer functor induces an equivalence $\underset{}{\mathrm{MCM}}(R)\simeq \underset{}{\mathrm{MCM}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R=S\mathrm{/}(f)$ is a complete hypersurface ring and $A=S[\text{\hspace{0.17em}⁣}[u,v]\text{\hspace{0.17em}⁣}]\mathrm{/}(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\mathrm{♯}\mathrm{♯}}=S[\text{\hspace{0.17em}⁣}[z,w]\text{\hspace{0.17em}⁣}]\mathrm{/}(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying ${\mathrm{syz}}_2^A(M)$ with the image of $M\oplus {\mathrm{syz}}_1^{R}M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\text{\hspace{0.17em}⁣}[z]\text{\hspace{0.17em}⁣}]\mathrm{/}(f+z^2)$ fails.