Characteristic-free Knörrer periodicity

Abstract

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ 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 $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.