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Two comparison theorems for semiring schemes

Oliver Lorscheid

Abstract

In this note, we compare the two approaches to semiring schemes as topological spaces with a structure sheaf and as a functor of points. We explain and prove the following two results: (1) the topological space can be recovered from the functor of points; (2) the two notions of semiring schemes are canonically equivalent as categories.

Two comparison theorems for semiring schemes

Abstract

In this note, we compare the two approaches to semiring schemes as topological spaces with a structure sheaf and as a functor of points. We explain and prove the following two results: (1) the topological space can be recovered from the functor of points; (2) the two notions of semiring schemes are canonically equivalent as categories.

Paper Structure

This paper contains 24 sections, 12 theorems, 14 equations.

Table of Contents

  1. The two comparison theorems
  2. Prime spectra
  3. The Zariski site
  4. The first comparison theorem
  5. Semiring schemes
  6. Semiring schemes as semiringed spaces (GKMX)
  7. Semiring schemes as Zariski sheaves (BLMV)
  8. The second comparison theorem
  9. Proof of \ref{['thmA']}
  10. The monoid spectrum
  11. Proof strategy
  12. Technique #1: localizations
  13. Technique #2: covering conditions
  14. Technique #3: symmetric algebras
  15. The inclusion of the prime spectrum into the Stone dual
  16. ...and 9 more sections

Key Result

Theorem 1

There is a canonical homeomorphism $\varphi:\mathop{\mathrm{PSpec}}\nolimits R\to\Lambda_R^\textup{top}$ that is functorial in $R$ and that maps a principal open $U^P_h$ of $\mathop{\mathrm{PSpec}}\nolimits R$ (for $h\in R$) to the principal open $U^\Lambda_h$ of $\Lambda_R^\textup{top}$.

Theorems & Definitions (24)

  • Theorem 1: first comparison theorem
  • Theorem 2: second comparison theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 14 more