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On some properties of birational derived splinters
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abstract
A Noetherian reduced ring $A$ is called a birational derived splinter if for all proper birational maps $X\to\operatorname{Spec}(A)$, the canonical map $A\to Rf_*\mathcal{O}_X$ splits. In equal characteristic zero this property characterizes rational singularities, but much less can be said in positive or mixed characteristics. In this paper, we prove some fundamental properties of this notion, including the behavior under localization, taking a pure subring, taking direct limit, and along an \'etale extension. In particular, direct limit of rational singularities in characteristic zero has rational singularities. Then, we study residue extensions (in arbitrary characteristic), and openness and regular extensions in positive characteristic, parallel to Datta-Tucker and the author's previous works on splinters.
Forward citations
Cited by 3 Pith papers
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The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
For pseudo-rational and many Du Bois singularities, the full Briançon–Skoda containment J^{n+k-1} ⊆ J^k holds, and quasi-excellent finite-dimensional rings satisfy uniform Briançon–Skoda and uniform Artin–Rees.
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Factoring maps to big Cohen-Macaulay algebras through blowups
Functorial balanced big Cohen-Macaulay algebra assignments factor through RΓ(Y, O_Y) for every proper birational map Y → Spec R.
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Measuring birational derived splinters
The paper defines μ_bds, a level-based invariant of the derived category that measures the failure of a scheme to be a birational derived splinter.
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