18.726: List of Topics

Here is the list of topics to be covered in 18.726. Topics already presented (or about to be presented) include lecture notes and dates of presentation. (References to Hartshorne, EGA, etc. appear in the notes.) In many cases, I made changes after the notes were handed out in class, to correct errors or add discussion that happened in class but was not originally reflected in the notes.

Topics not yet presented are subject to change, as I make decisions about how things will be organized.

• Introduction and overview [tex] (Feb 4)
• Basics of category theory [tex] (Feb 6; updated Feb 8): Categories, morphisms, functors, Yoneda's lemma, universal properties, adjoint pairs.
• Sheaves [tex] (Feb 9, 11, 13; updated Feb 12): Presheaves on topological spaces. Sheaf axiom, sheaves, examples. Sheaves specified on a basis, glueing. Stalks, morphisms of sheaves. Sheafification. Direct and inverse images.
• Abelian sheaves [tex](Feb 13): Facts about abelian groups. Exact sequences, five and snake lemmas, exactness of functors. Sheaves of abelian groups, kernel, cokernel, image. The global sections functor. Abelian categories.
• Schemes [tex] (Feb 17, 18): Ringed and locally ringed spaces. Prime spectrum, Zariski topology. Existence of the structure sheaf. Glueing, examples (projective space, line with a doubled point).
• Morphisms of schemes [tex] (Feb 18, 20, 23; updated Feb 20): Morphisms of locally ringed spaces. Adjointness property of affine schemes. Fibre products. Functor of points. Base change.
• Sheaves of modules [tex] (Feb 23, 25; updated Feb 27): Sheaves of modules on ringed spaces. Direct and inverse images. Quasicoherent sheaves. Ideal sheaves. Closed immersions. Separated morphisms.
• More properties of morphisms [tex] (Feb 27, Mar 2; updated Mar 5): Morphisms (locally) of finite type/presentation. Proper morphisms.
• Projective morphisms, part 1 [tex] (Mar 2, 4; updated Mar 3): Graded rings and Proj. Projective morphisms and why they are proper.
• Projective morphisms, part 2 [tex] (Mar 4, 6; updated Mar 7): Description using relative Proj. Blowing up. Chow's lemma.
• More properties of schemes [tex](Mar 9; updated Mar 9): Reduced. Connected. Irreducible. Integral. Normal. Dimension and codimension. Regular. (Warning: the lemma in section 4 appears to be incorrect.)
• Flat morphisms and descent [tex] (Mar 11, 13; updated Mar 13): Flat modules and flat morphisms. Faithfully flat morphisms. Descent for quasicoherent sheaves. Descent for (a few) properties of morphisms.
• Differentials [tex] (Mar 13, 16): Modules of Kähler differentials. Sheaves of differentials. Unramified, smooth, étale morphisms.
• Divisors [tex] (Mar 16, 18; updated Apr 1): Weil divisors. Cartier divisors. Linear systems and maps to projective space.
• Divisors on curves [tex] (Mar 18, 20; updated Mar 31): Riemann-Roch theorem (without proof yet). Riemann-Hurwitz formula.
• Homological algebra [tex] (Mar 30, Apr 1, 3; updated Apr 8): Abelian categories. Complexes and their cohomology. Cohomological functors (delta-functors). Universality of cohomological functors, effaceability. Acyclic objects, computation using acyclic resolutions. Injective objets, construction of derived functors. Examples: Tor, Ext, group (co)homology.
• Sheaf cohomology [tex] (Apr 6, 8, 10; updated Apr 13): Construction of injective objects in categories of abelian sheaves. Grothendieck's criterion for having enough injectives. Sheaf cohomology. Properties of flasque sheaves. Relationship with singular cohomology. Cech cohomology and when it computes sheaf cohomology.
• Cohomology of quasicoherent sheaves [tex] (April 13; updated Apr 25): Acyclicity of quasicoherent sheaves on affine schemes.
• Cohomology of projective spaces [tex] (April 13, 15, 17; updated Apr 14): Cohomology of twisting sheaves. Serre's vanishing theorem.
• Hilbert polynomials [tex] (April 17, 22): Existence of the Hilbert polynomial. Flatness of projective morphisms and Hilbert polynomials. Hilbert schemes. Leading term of the Hilbert polynomial.
• Spectral sequences and Cech cohomology [tex] (not to be covered in class?): Exact couples. Filtered complexes and double complexes. Spectral sequence of a filtered complex. Spectral sequence of a double complex. Application to Cech cohomology.
• GAGA [tex] (April 22, 24, 27, 29; updated April 30): Coherent sheaves on locally ringed spaces. Analytification of coherent sheaves. Computation of analytic cohomology of algebraic coherent sheaves. Comparison of algebraic and analytic morphisms of algebraic coherent sheaves. Descent of analytic coherent sheaves to algebraic coherent sheaves. Extension to proper and projective schemes. Applications. Variants.
• Serre duality for projective space [tex] (April 29, May 1): Ext and sheaf Ext. Duality for projective space. Interpretation using the canonical sheaf.
• Dualizing sheaves and Riemann-Roch [tex] (May 4, 6; updated May 6): Statement of results. Application to Riemann-Roch for curves. Construction of dualizing sheaves. Comparison to the canonical sheaf for a smooth scheme.
• Cohen-Macaulay schemes and Serre duality [tex] (May 6, 8): Statement of the duality theorem. Reduction to a local property. The Cohen-Macaulay condition.
• Higher Riemann-Roch [tex] (May 11): Riemann-Roch for surfaces. Hirzebruch-Riemann-Roch theorem. Grothendieck-Riemann-Roch theorem.
• Étale cohomology [tex] (May 13; updated May 13): Weil conjectures. Grothendieck topologies. Étale cohomology.