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Valuation rings are derived splinters

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Abstract

We give three proofs that valuation rings are derived splinters: a geometric proof using absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the second author’s PhD thesis and which we reprise here), and a proof by approximation which reduces the problem to Bhatt’s proof of the derived direct summand conjecture. The approximation property also shows that smooth algebras over valuation rings are splinters.

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Notes

  1. The degree of a finite algebra over a field is its vector space dimension over the field.

  2. This is not standard terminology, but it is the notion we need. More common, for example as in [33, Sec. VI.1], is the notion of p-closed field where it is required that K have no Galois extensions of degree p (or equivalently of degree \(p^n\) for \(n>0\)). A p-separably closed field is p-closed since non-trivial field extensions do not admit sections, but the converse is not generally true. For example, let \(\mathbb {F}_\ell \) be a finite field and fix two distinct primes \(q_1\) and \(q_2\) which are both different from p. We let \(G\cong \prod _{r}\mathbb {Z}_r\) be the absolute Galois group of \(\mathbb {F}_\ell \), where the product ranges over all primes r, and we let \(G'\subseteq G\) be the subgroup \(\prod _{r\ne q_1,q_2}\mathbb {Z}_r\). Set \(K={\overline{\mathbb {F}}}_\ell ^{G'}\). Then, K is evidently p-closed since its Galois group is \(\mathbb {Z}_{q_1}\times \mathbb {Z}_{q_2}\), but it is not p-separably closed since we can solve the equation \(mq_1+nq_2=p^a\) for some \(m,n,a>0\).

  3. We thank Linquan Ma for pointing out to us that the property of local cohomology having finitely many associated primes descends under pure maps.

  4. Universal cohesiveness of a Prüfer domain does not follow from universal cohesiveness of all its localizations by Harris’s example of a ring which is not coherent even though its local rings are all Noetherian [16, Thm. 3].

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Acknowledgements

We thank Bhargav Bhatt, Linquan Ma, Akhil Mathew, Matthew Morrow, Takumi Murayama, Emanuel Reinecke and Kevin Tucker for helpful conversations. The first author is indebted to Elden Elmanto who pointed out [43] and suggested that Proposition 4.2.1 should be true. The second author is especially grateful to Emanuel Reinecke for numerous illuminating discussions about derived splinters and for making us aware of Fujiwara and Kato’s finiteness result (Theorem 2.3.3). Additionally, we thank Takumi and the referee for their comments. In particular, Remark 3.1.3 was added at the referee’s suggestion. The first author was supported by NSF Grant DMS-1552766.

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  1. Northwestern University, Evanston, Illinois, USA

    Benjamin Antieau

  2. University of Illinois at Chicago, Chicago, USA

    Rankeya Datta

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Antieau, B., Datta, R. Valuation rings are derived splinters. Math. Z. 299, 827–851 (2021). https://doi.org/10.1007/s00209-020-02683-6

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