20 Result(s)
within
Shuji Saito
Algebra
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Chapter and Conference Paper
Additional Invariants in the Case e x(X) = 2
In order to show key Theorem 6.40 in Chap. 6, we recall further invariants for singularities, which were defined by Hironaka. The definition works for any dimension, as long as the directrix is 2-dimensiona...
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Chapter and Conference Paper
Proof in the Case \(e_x(X)=\overline {e}_x(X)=2\) , II: Separable Residue Extensions
In this chapter we prove Theorem 13.7 below, which implies Key Theorem 6.40 under the assumption that the residue fields of the initial points of X n \(\mathcal {X}_n\) are separably algebraic ov...
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Chapter and Conference Paper
Non-existence of Maximal Contact in Dimension 2
Let Z be an excellent regular scheme and let X ⊂ Z be a closed subscheme.
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Chapter and Conference Paper
Functoriality, Locally Noetherian Schemes, Algebraic Spaces and Stacks
In this chapter we reformulate the obtained functoriality for our resolution and apply this to obtain resolution of singularities for (two-dimensional) locally noetherian excellent schemes, algebraic spaces an...
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Chapter and Conference Paper
Proof in the Case e x(X) = es x(X) = 2 , I: Some Key Lemmas
In this chapter we prepare some key lemmas for the proof of Theorem 6.40.
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Chapter and Conference Paper
An Alternative Proof of Theorem 6.17
We give another Proof of Theorem 6.17 which uses more classical tools in algebraic geometry.
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Chapter and Conference Paper
Basic Invariants for Singularities
In this chapter we introduce some basic invariants for singularities.
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Chapter and Conference Paper
\(\mathcal {B}\) -Permissible Blow-Ups: The Embedded Case
Let Z be a regular scheme and let 𝔹 ⊂ Z \(\mathbb {B} \subset Z\) be a simple normal crossing divisor on Z. For each x ∈ Z, let 𝔹 ( x ) \(\mathbb {B}(x)\) be the subdivisor of 𝔹 $$ \mathbb {...
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Chapter and Conference Paper
Main Theorems and Strategy for Their Proofs
We will treat the following two situations in a parallel way: (E)
(embedded case) X is an excellent noetherian scheme, i : X↪Z is a closed immersion into an excellent regular noetherian scheme Z, and ℬ $...
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Chapter and Conference Paper
Characteristic Polyhedra of J ⊂ R
In this chapter we are always in Setup A (beginning of Chap. 7). We introduce a polyhedron Δ(J, u) which plays a crucial role in this monograph. It will provide us with useful invariants of singularities of Sp...
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Chapter and Conference Paper
Introduction
Let X be an irreducible and reduced excellent noetherian scheme.
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Chapter and Conference Paper
Permissible Blow-Ups
We discuss some fundamental results on the behavior of the ν ∗-invariants, the e-invariants, and the Hilbert-Samuel functions under permissible blow-ups.
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Chapter and Conference Paper
Termination of the Fundamental Sequences of \( \mathcal {B}\) -Permissible Blow-Ups, and the Case e x(X) = 1
In this chapter we prove the Key Theorem 6.35 in Chap. 6, by deducing it from a stronger result, Theorem 10.2 below. Moreover we will give an explicit bound on the length of the fundamental sequence, by the...
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Chapter and Conference Paper
\( \mathcal {B}\) -Permissible Blow-Ups: The Non-embedded Case
Let X be a locally noetherian scheme. We start with the following definition.
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Chapter and Conference Paper
(u)-standard Bases
Let R be a regular noetherian local ring with maximal ideal 𝔪 \(\mathfrak {m}\) and residue field k = R ∕ 𝔪 \(k=R/\mathfrak {m}\) , and let J ⊆ 𝔪 \(J \subseteq \mathfrak {m}\) be an ideal.
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Chapter and Conference Paper
Proof in the Case e x(X) = e x(X) = 2 , III: Inseparable Residue Extensions
In this chapter we complete the proof of key Theorem 6.40 (see Theorem 14.4 below).
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Chapter and Conference Paper
Transformation of Standard Bases Under Blow-Ups
In this chapter we will study the transformation of a standard base under permissible blow-ups, in particular with respect to near points in the blow-up.
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Chapter and Conference Paper
Appendix. Hironaka’s Characteristic Polyhedron. Notes for Novices(B.Schober)
In this appendix, we discuss some of the ideas behind Hironaka’s characteristic polyhedron. In particular, we present pictures that are often hidden in the background.
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Article
Cohomological Hasse principle and motivic cohomology for arithmetic schemes
In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmet...
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Chapter
Recent Progress on the Kato Conjecture
This is a survey paper on recent works in progress by Jannsen, Kerz, and the author on the Kato conjecture on the cohomological Hasse principle.
