Math 595 Cohomology of Schemes, Spring 2019

Tuesday/Thursday, 11am-12:20pm (January 14 to March 8), 143 Henry Admin Building
My information:

Thomas Nevins
357 Altgeld Hall
217.265.6762
nevins AT illinois DOT edu
Office hours: To be announced; and by appointment.
Some Suggested Texts:

Tentative Course Plan:


Notes from weeks 1-4: [PDF]. Notes from week 5: [PDF]. Notes from week 6: [PDF]. Notes from week7: [PDF]. Notes from week 8: [PDF].
Grading: The class meets 16 times. The grading scale will be negotiated with the class on the first class day. Here is my opening proposal:

Weekly Summaries:

WEEK 1: I find out your background. Presheaves and sheaves, quasicoherent and coherent sheaves. A few examples. Exactness of global sections on affine varieties/schemes
Reading: Sections II.1 and II.5 of Hartshorne (and/or Chapters 2 and 13 of Vakil).
Homework #1 due Thursday, January 24: Hartshorne II.1 Problems 1 through 8; II.5 Problems 1, 3, 7.

WEEK 2: Grothendieck topologies. Brief introduction to the etale topology. Global sections functor and its failure of exactness. Derived functors. Cohomology as a derived functor. Vanishing of higher cohomology of quasicoherent sheaves on affine schemes.
Reading: Sections III.1 to III.3 of Hartshorne (and/or Chapter 18 of Vakil).
Homework #2 due Thursday, January 31: Hartshorne Problem III.2.7, III.3.2, III.3.6.

WEEK 3: Simplicial spaces/schemes. Cech complex and Cech cohomology. Properties. Computation of the cohomology of line bundles on projective spaces.
Reading:
Homework #3 due Thursday, February 7: Hartshorne Problem III.4.1, III.4.2, III.4.3, III.4.6, III.4.7.

WEEK 4: Computation of the cohomology of line bundles on projective spaces. Tangent sheaf, sheaf of Kahler differentials, Euler sequence on Pn.
Reading: Section II.8 of Hartshorne.
Homework: none.

WEEK 5: Computation of Hodge cohomology of projective space. Dualizing sheaf. Serre duality. Reading: Hartshorne, Sections III.6, III.7.
Homework #4: Fill in details for yourself of the sheets [Day 6], [Day 7].

WEEK 6: More on Serre duality. Ext-groups and Ext-sheaves and the meaning of Ext1. Reading: Hartshorne, Sections III.6, III.7.

WEEK 7: Classification of vector bundles on the projective line. No class Thursday! Reading: This paper and the references in it. Homework #5: [PDF]

WEEK 8: Higher direct images. How to calculate using the Cech complex. Theorems about them. Applications. Reading: Hartshorne, Chapter III Sections 8, 11, 12. Homework #6: TBA




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