STUDENT-RUN SEMINAR ON MODERN ALGEBRAIC GEOMETRY

Waterloo, Spring 2013

Michael Baker • Erik Crevier • Robert Garbary • Yossef Musleh • Adam Venis • Eeshan Wagh • Shuntaro Yamagishi

May: sheaves

• Tue May 14: Michael Baker, "The affine ideal-variety correspondence"
  
  Affine $n$-space $\mathbf{A}^n$ and the polynomial ring $A=k[x_1,\ldots,x_n]$. Zero sets of polynomials. Hilbert Basis Theorem. Affine algebraic sets and examples. Zariski topology on $\mathbf{A}^n$. Irreducible algebraic sets (affine varieties). Irreducible decomposition. The ideal $I(X)$ of a set $X \subseteq \mathbf{A}^n$. Radical and prime ideals. Hilbert's Nullstellensatz. The correspondence between radical ideals and algebraic sets. The coordinate ring $A(X)=\Gamma(X)$ of a variety $X$.
  
  
• Thu May 16: Yossef Musleh, "Local rings, dimension, and projective varieties"
  
  Polynomial maps and induced homomorphisms between coordinate rings. Field of fractions and localization $S^{-1}R$ of integral domains. Local ring $\mathcal{O}_p(V)$ of a variety at a point. Krull dimension of rings and varieties. Equivalent definitions of dimension. Projective space and projective varieties. Homogeneous polynomials and the projective Nullstellensatz.
  
  
• Tue May 21: Eeshan Wagh, "Agriculture 101: Sheaves and things"
  
  Presheaves (on $X$). Sheaves. Examples of sheaves. Examples of presheaves that are not sheaves. Morphisms of presheaves. Germs and the stalk $\mathcal{F}_p$ of a sheaf at a point.
  
  
• Thu May 23: Michael Baker, "Grothendieck Agriculture: The method of stalking"
  
  Review of definitions. Categories, functors, natural transformations. Examples. Direct limits $\varinjlim X_i$ in a category, and further discussion of stalks. The "poset category" $\mathfrak{Top}(X)$ of open sets of $X$. Translation of definitions (presheaves, morphisms, stalks) into categorical language.
  
  
• Tue May 28: Michael Baker, "Algebraic aspects of sheaves" (tex source)
  
  Induced maps on stalks. Working "at the level of stalks". Sheafification $\mathcal{F}^+$ (existence and uniqueness). Kernels, cokernels, and images (of morphisms of presheaves and sheaves). Exact sequences of sheaves. Operations on sheaves (direct image and inverse image).
  
  
• Thu May 30: Yossef Musleh
  
  Review of operations on sheaves. Problems 1.2, 1.7, 1.13, 1.15, 1.16, 1.17 from Chapter 2 of Hartshorne were discussed.

June: schemes

• Tue Jun 4: Michael Baker, "Schemes on the horizon: Spectra of commutative rings" (tex source) (notes by Eeshan, Yossef, other)
  
  The spectrum $\text{Spec } A$ of a commutative ring $A$: its underlying set, topology, and sheaf of rings. Examples. Regular functions on $\text{Spec } A$. Ringed spaces $(X, \mathcal{O}_X)$ and their morphisms.
  
  
• Thu Jun 6: Eeshan Wagh, "The Joy of Schemes" (notes by Robert)
  
  Review of spectrum and ringed spaces. Locally ringed spaces. Affine schemes. Schemes. Gluing schemes together. Examples.
  
  
• Tue Jun 11: Yossef Musleh, "Introduction to Proj"
  
  Proposition 2.3 in Chapter 2 of Hartshorne and its proof. Introduction to projective schemes and their associated topology.
  
  
• Thu Jun 13: Yossef Musleh, "More on Proj"
  
  Review of Proposition 2.3 and projective schemes. Definition of the structure sheaf on a projective scheme. Projective schemes are schemes.
  
  
• Tue Jun 18: Eeshan Wagh, "Projective Schemes"
  
  Review of graded rings, the $\operatorname{Proj}$ construction, and the fact that it is a scheme. The fully faithful functor $V \mapsto t(V)$ from the category of varieties over $k$ to the category of schemes over $k$ (hence allowing us to view varieties as a special case of schemes).
  
  
• Thu Jun 20: Eeshan Wagh, "Properties of Schemes" (notes by Robert: here and here)
  
  Connected, irreducible, integral, and reduced schemes. A scheme is integral if and only if it is both irreducible and reduced. Locally noetherian and noetherian schemes. Morphisms of locally finite type and of finite type. Open immersions and open subschemes. Closed immersions and closed subschemes.
  
  
• Tue Jun 25: Michael Baker, "Properties of Schemes II"
  
  Locally noetherian schemes. Fibred products of schemes.
  
  
• Thu Jun 27: Yossef Musleh (notes by Robert)
  
  Conclusion of 2.3.

July: pre-cohomology

• Thu Jul 4: Eeshan Wagh, "Computing Fibered Products"
  
  Review and examples.
  
  
• Tue Jul 9: Stanley Xiao, "On the dimension growth conjecture I"
  
  
• Tue Jul 16: Stanley Xiao, "On the dimension growth conjecture II" (paper suggested by Stanley)
  
  
• Thu Jul 18: Robert Garbary, "What coherence means I"
  
  (Sheaves of) modules over ringed spaces. The structure sheaf of an affine scheme could be defined $\mathcal{O}_{\text{Spec } A}(U_f) = A_f$. The $\mathcal{O}_{\text{Spec } A}$-module $\widetilde{M}$ associated to an $A$-module $M$ ($\widetilde{M}(U_f) = M_f$). Quasi-coherent and coherent modules over affine schemes. The ideal sheaf and the short exact sequence $0 \to I_{Z \subset X} \to \mathcal{O}_X \to \iota_*(\mathcal{O}_Z) \to 0$.
  
  
• Tue Jul 23: Robert Garbary, "What coherence means II"
  
  The projective line, degree shifting, and Serre's twist sheaf $\mathcal{O}_{\mathbf{P}^n}(1)$ (turns out to be the dual bundle of the tautological line bundle on $\mathbf{P}^n$ and a generator of the Picard group, which consists of all isomorphism classes of invertible sheaves under the tensor product, or alternatively, divisors modulo linear equivalence).
  
  
• Thu Jul 25: Michael Baker, "The zeroth cohomology talk" (an old video where I define exact sequences in a group-theoretic context)
  
  Preparation for sheaf cohomology: abelian categories, complexes, homotopy operators, cohomology objects, additive and exact functors, injective and acyclic resolutions, derived functors.
  
  
• Tue Jul 30: Erik Crevier, "Non-sheaf cohomology/This is what you are already supposed to know about cohomology, before even picking up Hartshorne"
  
  $\Delta$-sets; geometric realization. The nerve of a category. Simplicial homology. Simplicial cohomology. Čech cohomology.

August: cohomology

• Thu Aug 1: Erik Crevier, "Sheaf cohomology"
  
  Review of derived functors. Left exactness of the global sections functor. Cohomology of sheaves. The exponential sequence. Review of Čech cohomology. Analytic continuation. Čech cohomology for sheaves.
  
  
• Tue Aug 6: Michael Baker, "An affinity for affineness"
  
  Review of sheaves of $\mathcal{O}_X$-modules, quasicoherence, and coherence. Proof that the category of sheaves of $\mathcal{O}_X$-modules has enough injectives. Injective sheaves are flasque, and flasque sheaves are ($\Gamma(X,-)$-)acyclic. $M \mapsto \widetilde{M}$ takes injective $A$-modules to flasque sheaves of $\mathcal{O}_{\text{Spec }A}$-modules (to be proved later). One part of Serre's affineness criterion: the cohomology of any quasicoherent sheaf on a (noetherian) affine scheme is trivial.
  
  
• Tue Aug 13: Michael Baker, "Cohomology of a noetherian affine scheme"
  
  Krull's theorem. Injectivity of $\Gamma_{\mathfrak{a}}(I)$ when $I$ is injective. The localisation map $I \to I_f$ is surjective for injective modules $I$. Proof that if $I$ is injective then $\widetilde{I}$ is flasque. Serre's affineness criterion: a (noetherian) scheme is affine if and only if all of its quasicoherent sheaves are acyclic (have no cohomology).