### 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__:
- R. Hartshorne, Algebraic Geometry, Springer Verlag, 1977.
Available here.
- R Vakil, Foundations of Algebraic Geometry, here.
- D. Arapura, Complex Algebraic Varieties and their Cohomology, here

**Tentative Course Plan**:

- 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.
- 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.
- Week 3: Simplicial spaces/schemes. Cech complex and Cech cohomology. Properties. Computation of the cohomology of line bundles on projective spaces.
- Week 4: Cohomology of Sheaves of
*k*-forms on projective space.
- Week 5: Hodge cohomology of projective space. Serre duality. Support of sheaves, algebraic description. Many examples. Torsion sheaves and torsion-free sheaves on varieties; ideal sheaves of closed subschemes.
- Week 6: Direct images and relationship to cohomology of fibers. Constant sheaves and similar statement for topological cohomology.
- Week 7:
- Week 8:

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:
- B+: attended class between 6 and 10 times and participated in activities during those class periods. (If you will attend fewer than 6 times, you should drop the course.)
- A-: attended class at least 11 times and participated in activities during those class periods.
- A: attended class at least 11 times and participated in activities during those class periods. Substantially completed at least 3 of the homework assignments.
- A+: attended class at least 11 times and participated in activities during those class periods. Substantially completed at least 6 of the homework assignments.

**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 ***P**^{n}.

**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 Ext*^{1}.
**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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