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Non-reduced valuation rings and descent for smooth blowup squares

Shane Kelly

Abstract

We consider a class of non-reduced valuation rings, known in the literature as chain rings. We observe that the Grothendieck topology generated by the Zariski topology and smooth blowup squares is exactly the topology which has chain rings for its local rings, and that sheaves for this topology are \emph{not} characterised by excision for smooth blowup squares.

Non-reduced valuation rings and descent for smooth blowup squares

Abstract

We consider a class of non-reduced valuation rings, known in the literature as chain rings. We observe that the Grothendieck topology generated by the Zariski topology and smooth blowup squares is exactly the topology which has chain rings for its local rings, and that sheaves for this topology are \emph{not} characterised by excision for smooth blowup squares.
Paper Structure (5 sections, 15 theorems, 31 equations)

This paper contains 5 sections, 15 theorems, 31 equations.

Table of Contents

  1. Summary
  2. Some first order logic
  3. Non-reduced valuation rings; definition
  4. Chain rings as local rings
  5. Chain rings as quotients of valuation rings

Key Result

Proposition 1.2

Let $\mathcal{U} = \{Y_i \to X\}_{i \in I}$ be a family of morphisms in $\mathrm{Sch}_\mathbb{Z}$. The following are equivalent. In other words, the chain topology is generated by the Zariski topology, and blowup of the affine plane in the origin.

Theorems & Definitions (47)

  • Definition 1.1
  • Proposition 1.2: Proposition \ref{['prop:equivChain']}
  • Proposition 1.3: Corollary \ref{['coro:autoRH']}
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • ...and 37 more