🌿Algebraic Geometry Unit 3 – Sheaves and Schemes

Key Concepts and Definitions

• Sheaves generalize the notion of functions on a topological space by assigning data to open sets in a way that is compatible with restrictions
• Presheaves consist of a contravariant functor from the category of open sets of a topological space $X$ to a target category (often the category of sets, abelian groups, or rings)
• Sheaf axioms ensure local data can be glued together consistently
  • Identity axiom: The identity morphism on an open set induces the identity morphism on the corresponding sheaf
  • Gluing axiom: If $\{U_i\}$ is an open cover of an open set $U$ and $s_i \in \mathcal{F}(U_i)$ agree on overlaps, then there exists a unique $s \in \mathcal{F}(U)$ restricting to each $s_i$
• Stalks capture the local behavior of a sheaf at a point by considering the direct limit of sections over all open neighborhoods of the point
• Ringed spaces combine a topological space with a sheaf of rings, providing a foundation for studying schemes
• Locally ringed spaces require the stalks of the structure sheaf to be local rings, a key property for defining schemes

Sheaf Theory Foundations

• Sheaves can be defined on various categories, such as topological spaces, sites, or topoi, allowing for a wide range of applications
• Morphisms of sheaves are natural transformations between the corresponding functors, preserving the sheaf structure
• Operations on sheaves include direct sums, tensor products, and sheaf $\mathcal{Hom}$, enabling the study of sheaf cohomology and other constructions
• Exact sequences of sheaves play a crucial role in understanding the relationships between different sheaves and computing cohomology
• The sheafification process turns a presheaf into a sheaf by enforcing the gluing axiom, providing a universal way to associate a sheaf to a presheaf
• The étalé space construction provides a geometric perspective on sheaves by representing them as continuous maps to the base space
• Sheaf cohomology extends the notion of cohomology to sheaves, capturing global information from local data
  • Derived functors of the global sections functor $\Gamma$ yield sheaf cohomology groups $H^i(X, \mathcal{F})$

From Ringed Spaces to Schemes

• Affine schemes are the building blocks of scheme theory, defined as the spectrum of a commutative ring $A$ with the Zariski topology and the structure sheaf $\mathcal{O}_{\text{Spec}(A)}$
  • The Zariski topology on $\text{Spec}(A)$ is generated by basic open sets of the form $D(f) = \{p \in \text{Spec}(A) \mid f \notin p\}$ for $f \in A$
  • The structure sheaf $\mathcal{O}_{\text{Spec}(A)}$ assigns to each open set $U$ the ring of regular functions on $U$, given by the localization $A_f$ for $f \in A$ such that $D(f) \subseteq U$
• Schemes are obtained by gluing affine schemes along open subsets, generalizing the notion of algebraic varieties
  • A scheme $(X, \mathcal{O}_X)$ is a locally ringed space that is locally isomorphic to an affine scheme
• Morphisms of schemes are morphisms of locally ringed spaces, compatible with the structure sheaves
• Fiber products of schemes allow for the construction of new schemes from existing ones, enabling the study of geometric properties and moduli problems

Properties of Schemes

• Reducedness and irreducibility are important geometric properties of schemes
  • A scheme is reduced if its structure sheaf has no nilpotent elements in its stalks
  • A scheme is irreducible if it is not the union of two proper closed subschemes
• Separation axioms, such as the Hausdorff property and separatedness, control the global behavior of schemes
  • A scheme is separated if the diagonal morphism $\Delta: X \to X \times X$ is a closed immersion
• Properness is a stronger condition than separatedness, requiring the diagonal morphism to be proper (universally closed and separated)
• Dimension theory for schemes extends the notion of dimension from algebraic varieties
  • The dimension of a scheme at a point is the Krull dimension of the local ring at that point
  • The dimension of a scheme is the supremum of the dimensions at its points
• Smoothness and regularity characterize the local geometry of schemes
  • A scheme is regular at a point if the local ring at that point is a regular local ring
  • A scheme is smooth over a base scheme if it is locally of finite presentation and the fibers are geometrically regular

Morphisms of Schemes

• Morphisms of schemes are locally determined by morphisms of affine schemes, which correspond to ring homomorphisms in the opposite direction
• Properties of morphisms, such as being affine, finite, proper, smooth, or étale, reflect geometric and algebraic properties of the schemes involved
  • Affine morphisms are characterized by the preimage of an affine open set being affine
  • Finite morphisms have finite fibers and are affine
  • Proper morphisms are universally closed and separated
  • Smooth morphisms are locally of finite presentation with geometrically regular fibers
  • Étale morphisms are flat and unramified, providing a notion of local isomorphism
• Fiber products and base change allow for the study of morphisms and their behavior under pullbacks
• Valuative criteria for properness and separatedness provide a way to check these properties using valuation rings
• Flatness is a crucial property for families of schemes, ensuring that fibers vary continuously and have compatible dimensions

Cohomology of Sheaves

• Sheaf cohomology is a powerful tool for studying global properties of schemes and sheaves
• Čech cohomology provides a concrete way to compute sheaf cohomology using open covers and Čech cochains
  • The Čech complex associated to an open cover $\mathfrak{U} = \{U_i\}$ of a scheme $X$ and a sheaf $\mathcal{F}$ yields Čech cohomology groups $\check{H}^i(\mathfrak{U}, \mathcal{F})$
  • Refining the open cover leads to isomorphic Čech cohomology groups, allowing for the definition of the Čech cohomology groups $\check{H}^i(X, \mathcal{F})$
• Derived functor cohomology extends the notion of sheaf cohomology to derived categories, providing a more abstract and functorial approach
• Serre duality relates the cohomology of a coherent sheaf on a proper smooth scheme to the cohomology of its dual sheaf, twisted by the canonical bundle
• The Riemann-Roch theorem is a powerful tool for computing the Euler characteristic of a coherent sheaf on a proper smooth scheme, generalizing the classical theorem for curves and surfaces

Applications in Algebraic Geometry

• Schemes provide a unified framework for studying algebraic varieties, arithmetic geometry, and complex analytic spaces
  • Algebraic varieties can be studied as schemes over algebraically closed fields, allowing for the use of sheaf-theoretic techniques
  • Arithmetic geometry studies schemes over rings of integers or finite fields, leading to important questions in number theory (Fermat's Last Theorem)
  • Complex analytic spaces can be viewed as schemes over the complex numbers, connecting algebraic geometry to complex analysis and topology (GAGA theorems)
• Moduli spaces parametrize geometric objects, such as curves, surfaces, or vector bundles, and can often be constructed as schemes or stacks
  • The moduli space of elliptic curves $\mathcal{M}_{1,1}$ is a scheme classifying elliptic curves up to isomorphism
  • The moduli space of stable curves $\overline{\mathcal{M}}_g$ is a proper smooth Deligne-Mumford stack parametrizing stable curves of genus $g$
• Intersection theory on schemes extends the classical notion of intersecting subvarieties to the setting of schemes, allowing for the computation of intersection numbers and the study of enumerative geometry
• Grothendieck's theory of motives aims to unify various cohomology theories and provide a universal cohomology theory for schemes, leading to deep conjectures and connections to other areas of mathematics

Advanced Topics and Open Problems

• Derived algebraic geometry extends the theory of schemes to derived schemes, incorporating homotopical and higher categorical techniques
  • Derived schemes are locally modeled on the spectrum of a simplicial commutative ring, allowing for the study of derived intersections and virtual fundamental classes
• Algebraic stacks generalize schemes by allowing for quotients by group actions and more general moduli problems
  • Artin stacks are algebraic stacks that admit a smooth cover by a scheme, providing a framework for studying moduli spaces with non-trivial automorphisms (moduli of curves, vector bundles)
• The Weil conjectures, proved by Deligne using étale cohomology, relate the zeta function of a variety over a finite field to its topological properties, connecting arithmetic and geometry
• The Hodge conjecture predicts that every Hodge class on a projective complex manifold is a linear combination of classes of algebraic cycles, linking complex geometry to algebraic cycles
• The Tate conjecture suggests that the Galois invariants of the étale cohomology of a variety over a finitely generated field are spanned by classes of algebraic cycles, relating arithmetic to algebraic cycles
• Motive theory and the standard conjectures aim to provide a unified framework for understanding cohomology theories and cycles on varieties, with connections to arithmetic, geometry, and representation theory
• The Langlands program seeks to unify various areas of mathematics, including representation theory, number theory, and algebraic geometry, by relating Galois representations to automorphic forms and motives