无特征限制的 Knörrer 周期性
Characteristic-free Knörrer periodicity
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中文总结 AI 辅助
本文证明 Knörrer 函子对任意特征和奇点给出稳定范畴等价，并构造自由消解，对比双分支覆盖在特征 2 的失效。
中文摘要 AI 辅助
我们证明 Knörrer 函子诱导极大 Cohen-Macaulay 模的稳定范畴之间的等价 ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$，其中 $R = S/(f)$ 是完全交超曲面环，$A = S[\\![u,v]\\!]/(f+uv)$ 是 $R$ 的双曲扩张。对剩余域或其特征不加任何假设，$f$ 不必定义孤立奇点，$S$ 也不必包含域。这与迭代双分支覆盖 $R^{\sharp\sharp} = S[\\![z,w]\\!]/(f+z^2+w^2)$ 形成对比，后者仅在特征不等于 2 时才与 $R$ 稳定等价。我们还给出了一个自由消解的直接构造，将 $\operatorname{syz}_2^A(M)$ 与 $M \oplus \operatorname{syz}_1^R M$ 在函子下的像等同，并给出了特征为 2 时双分支覆盖 $S[\\![z]\\!]/(f+z^2)$ 的相应陈述失败的例子。
英文摘要
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.
发表机构
- Syracuse University（雪城大学）
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