Abstract
The main result of this article is to prove that any Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen–Macaulay algebra. The proof of this result is based on the construction of almost Cohen–Macaulay algebras in mixed characteristic due to Yves André. Moreover, we prove that the absolute integral closure of a complete Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen–Macaulay algebra.
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Notes
“Tilt” is a standard name now. In [28], “Fontaine ring” was employed and \(\mathbf {E}(A)\) was used for the Fontaine ring. As this symbol is obsolete now, we decided to switch to a standard one.
It is probably better to write \(p^\flat :=(\overline{p},\overline{p^{\frac{1}{p}}},\ldots ) \in A^\flat \), but we make a simple choice of notation.
Fontaine, Gabber-Ramero and Kedlaya-Liu also considered versions of perfectoid algebras.
Precisely speaking, this is not necessary in the general theory of perfectoid geometry. But this extra assumption will be sufficient for the main results in this paper.
One uses the following fact: Let R[a] be a simple ring extension of R contained in \(R^+\). Then there is a unique monic polynomial \(f \in R[x]\) such that \(R[x]/(f) \cong R[a]\) by the normality of R.
It is shown in [10, Lemma 3.6] that the ideal-adic completion of a perfect \(\mathbb {F}_p\)-algebra remains perfect.
Quite recently, O. Gabber also announced a similar result, using his weak local uniformization theorem and ultra-products of rings.
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Acknowledgements
I am grateful to Bhargav Bhatt, Raymond Heitmann, Kiran Kedlaya, Linquan Ma, Kei Nakazato and Paul Roberts for useful comments. My special gratitude goes to Yves André for his numerous and kind comments and for his inspiration. Finally, I am grateful to the referee for pointing out errors and providing remarks.
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Communicated by Vasudevan Srinivas.
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Shimomoto, K. Integral perfectoid big Cohen–Macaulay algebras via André’s theorem. Math. Ann. 372, 1167–1188 (2018). https://doi.org/10.1007/s00208-018-1704-x
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DOI: https://doi.org/10.1007/s00208-018-1704-x