54 research outputs found
Relative vanishing theorems for -schemes
We prove the relative Grauert-Riemenschneider vanishing, Kawamata-Viehweg
vanishing, and Koll\'ar injectivity theorems for -schemes, solving
conjectures of Boutot and Kawakita. Our proof uses Grothendieck's limit theorem
for sheaf cohomology and Zariski-Riemann spaces. As an application, we extend
Boutot's theorem to the case of locally quasi-excellent -algebras
by showing that if is a cyclically pure homomorphism of locally
quasi-excellent -algebras, and has rational singularities,
then has rational singularities. This solves a conjecture of Boutot and
answers a question of Schoutens in the locally quasi-excellent case.Comment: 31 pages. Comments welcome! v2: Removed excellence hypotheses, added
references, other change
Dilated Floor Functions That Commute
We determine all pairs of real numbers such that the
dilated floor functions and
commute under composition, i.e., such that holds for all
real .Comment: 6 pages, to appear in Amer. Math. Monthl
Pure subrings of Du Bois singularities are Du Bois singularities
Let be a cyclically pure map of rings essentially of finite type
over the complex numbers . In this paper, we show that if has
Du Bois singularities, then has Du Bois singularities. Our result is new
even when is faithfully flat. As a consequence, we show that under
the same hypotheses on , if has log canonical type singularities
and is Cartier, then has log canonical singularities.Comment: 7 pages; comments welcome. v2: Added part (ii) of Corollary B, other
small change
Permanence properties of -injectivity
We prove that -injectivity localizes, descends under faithfully flat
homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and
geometrically -injective fibers, all for arbitrary Noetherian rings of prime
characteristic. As a consequence, we show that the -injective locus is open
on most rings arising in arithmetic and geometry. Furthermore, we prove that
over an algebraically closed field of positive characteristic, generic
projection hypersurfaces associated to normal projective varieties are weakly
normal, and generic projection hypersurfaces associated to suitably embedded
smooth projective varieties of low dimension are even -pure, and hence
-injective. The former result proves a conjecture of Bombieri and
Andreotti-Holm, and the latter result is the positive characteristic analogue
of a theorem of Doherty.Comment: 38 pages; comments welcome! v2: added Theorem 6.6, fixed Lemma A.2,
more transparent proof of Lemma 4.5, other small additions and change
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero
We establish the relative minimal model program with scaling for projective
morphisms of quasi-excellent algebraic spaces admitting dualizing complexes,
formal schemes admitting dualizing complexes, semianalytic germs of complex
analytic spaces, rigid analytic spaces, and Berkovich spaces, all in equal
characteristic zero. To do so, we prove finite generation of relative adjoint
rings associated to projective morphisms of such spaces using the strategy of
Cascini and Lazi\'c and the generalization of the Kawamata-Viehweg vanishing
theorem to the scheme setting recently established by the second author. To
prove these results uniformly, we prove GAGA theorems for Grothendieck duality
and dualizing complexes to reduce to the algebraic case.Comment: 96 pages. Comments welcom
- β¦