交换代数中的K-内射复形
K-injective complexes in commutative algebra
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中文总结 AI 辅助
本文证明交换诺特环上内射模复形的K-内射性可由局部化或完备化检测，并推广Grothendieck正则性下降定理至非交换环。
中文摘要 AI 辅助
不幸的事实是，即使在交换诺特环上，K-内射复形的局部化也未必是K-内射的。然而，并非全无希望。给定交换诺特环$A$上的内射模复形$J$，我们证明$J$的K-内射性可以在局部检测：若$(A,\mathfrak{m})$是诺特局部环，则$J$在$A$上是K-内射的当且仅当$\operatorname{Hom}_A(\widehat{A},J)$在$\widehat{A}$（即$A$的$\mathfrak{m}$-进完备化）上是K-内射的；对于任意交换诺特环$A$，$J$在$A$上是K-内射的当且仅当对$A$的奇异轨迹中的每个极大理想$\mathfrak{m}$，$\operatorname{Hom}_A(A_{\mathfrak{m}},J)$在$A_{\mathfrak{m}}$上是K-内射的。局部结果是从一个关于沿到（可能非交换、可能非诺特）环的映射的K-内射性的忠实平坦下降的更一般定理推导出来的。全局结果通过张量三角几何中的局部到全局原理建立。作为应用，我们推广了Grothendieck关于正则性的忠实平坦下降定理，将其扩展到沿非交换左凝聚左正则环的忠实平坦扩张。
英文摘要
It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules $J$ over a commutative noetherian ring $A$, we show that K-injectivity of $J$ can be detected locally: if $(A,\mathfrak{m})$ is a noetherian local ring, then $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(\widehat{A},J)$ is K-injective over $\widehat{A}$, the $\mathfrak{m}$-adic completion of $A$; and for an arbitrary commutative noetherian ring $A$, $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(A_{\mathfrak{m}},J)$ is K-injective over $A_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ in the singular locus of $A$. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.
