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• Tags
  • #MOC #AG
• Refs:
  • Eisenbud and Harris, Geometry of Schemes #resources/books
  • Szamuely, Galois groups and Fundamental Groups #resources/books
  • Contemplating spec: http://www.neverendingbooks.org/mumfords-treasure-map #resources/notes
  • Galois theory for schemes: https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf#page=1
• Links:
  • Study project: Algebraic Geometry
  • Study project: Commutative Algebra
  • properties of morphisms
  • variety
  • Unsorted/reduction mod p
  • GAGA
  • Zariski tangent space
  • schemes are functors
  • Unsorted/diagonal morphism
  • fiber of a vector bundle over a scheme
  • linear system
  • Common properties:
    • reduced
    • quasicompact
    • quasiseparated
    • geometrically connected

scheme

Definitions

An affine scheme \(X\in {\mathsf{Aff}}{\mathsf{Sch}}\) is a locally ringed space \((X, {\mathcal{O}}_X)\underset{\mathsf{RingSp}}{\cong}(\operatorname{Spec}R, {\mathcal{O}}_{\operatorname{Spec}R})\) for some \(R\in \mathsf{CRing}\).

Schemes as Zariski sheaves

Can generalize to define sheaves over an arbitrary symmetric monoidal category \(\mathsf{C}\). Define affine objects \({\mathsf{Aff}}(\mathsf{C}) \coloneqq{ {\mathsf{Comm}\mathsf{Mon}(\mathsf{C})}^{\operatorname{op}}}\) as model categories.

Motivations

Commutative Algebra prereqs

• regular ring
• local ring
• Noetherian ring
• completion
• localization of rings
• integral closure
• reduced ring
• Dedekind domain
• DVR

Topics

• smooth scheme
• finite type
• sheaf
• proper morphism
• separated
• geometric point
• closed point
• reduced scheme
• regular scheme
• geometric fiber
• group scheme
• geometrically connected scheme
• quasi-affine
• locally quasi-finite
• singular support
• characteristic cycle

Some other important ideas:

• derivations and cotangent complexes representing them,
• formally étale morphisms,
• flat morphisms,
• (Zariski) open immersions,
• formally unramified morphisms,
• finitely presented morphisms of commutative monoids and modules,
• projective modules and flat modules,
• Hochschild cohomology

Notes

• Gabriel-Rosenberg reconstruction theorem: \(X\) can be recovered from \({\mathsf{QCoh}}(X)\), the category of quasicoherent sheaves.on \(X\).

Analytic space

Morphisms of schemes