非诺特环中的上同调Cohen--Macaulay性
Cohomological Cohen--Macaulayness in Non-Noetherian Rings
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中文总结 AI 辅助
本文研究非诺特环的上同调Cohen--Macaulay性，证明局部有限维CCM环必为局部HMCM环，给出满足条件的环类实例，并建立特征p>0诺特环与其完备化的相关等价关系。
中文摘要 AI 辅助
我们研究由大Cohen--Macaulay代数导出的非诺特环的Hamilton--Marley意义下的Cohen--Macaulay性。受Bhatt的上同调Cohen--Macaulay性概念启发，若局部有限维环的结构层满足该条件，则称其为CCM。我们的主要比较定理表明，每个局部有限维CCM环都是局部HMCM环。利用该定理，我们证明：若A是诺特环，R是A的整代数且在A上是局部平衡大Cohen--Macaulay代数，则R是CCM环，因此也是局部HMCM环。特别地，若A是优秀诺特整环，p是素数，n≥1且A/pA≠0，则A⁺/pⁿA⁺是CCM环且为局部HMCM环。我们还利用有限维赋值环证明，CCM性严格强于局部HMCM性。最后，对于特征p>0的诺特环A及其完备化A_perf，我们证明以下条件等价：A是局部弱F-幂零的；A_perf是局部平衡大Cohen--Macaulay A-代数；A_perf是CCM环。在这些等价条件下，A_perf是局部HMCM环。
英文摘要
We study Cohen--Macaulayness, in the sense of Hamilton--Marley, of non-Noetherian rings arising as big Cohen--Macaulay algebras. Motivated by Bhatt's notion of cohomological Cohen--Macaulayness, we call a locally finite-dimensional ring CCM if its structure sheaf satisfies this condition. Our main comparison theorem shows that every locally finite-dimensional CCM ring is locally HMCM. Using this theorem, we prove that if $A$ is Noetherian and $R$ is an integral $A$-algebra that is locally balanced big Cohen--Macaulay over $A$, then $R$ is CCM and hence locally HMCM. In particular, if $A$ is an excellent Noetherian domain, $p$ is a prime, $n\geq1$, and $A/pA\neq0$, then $A^+/p^nA^+$ is CCM and locally HMCM. We also show, using finite-dimensional valuation domains, that CCM is strictly stronger than locally HMCM. Finally, for a Noetherian ring $A$ of characteristic $p>0$ and its perfection $A_{\mathrm{perf}}$, we prove that the following conditions are equivalent: $A$ is locally weakly $F$-nilpotent; $A_{\mathrm{perf}}$ is a locally balanced big Cohen--Macaulay $A$-algebra; and $A_{\mathrm{perf}}$ is CCM. Under these equivalent conditions, $A_{\mathrm{perf}}$ is locally HMCM.
