# Lecture Course: Algebraic Geometry 3

We will continue our study of the cohomology of quasi-coherent $\mathcal O_X$-modules. Key topics which I intend to discuss are

- The derived category of an abelian category
- Finiteness of higher direct images under projective/proper morphisms
- Cohomology and base change
- Grothendieck—Serre duality

**Dates and times:** Mon, Wed, 10-12, S-U-3.01. First lecture: Mon, Oct. 14. **No lecture on Wed., Nov. 13.**

**Exercise group:** (by Heer Zhao) Wed, 2-4, S-U-3.02. First meeting: Oct. 23.

**ECTS points:** 9

**Prerequisites:** Algebraic Geometry 1, Algebraic Geometry 2 (a solid understanding of the notion of schemes and of basic properties of schemes and morphisms of schemes; $\mathcal O_X$-modules and the beginnings of the theory of cohomology of abelian sheaves and (quasi-)coherent $\mathcal O_X$-modules).

**Exercise group:** There will be a weekly problem sheet, and a weekly exercise session, to be held by Dr. Heer Zhao, where the solutions to the problems are discussed. While handing in the solutions to the problems is not mandatory, it is *strongly encouraged*.

## Problem sheets

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3 | (corrected Nov. 6) | |

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8 | In the printed version there is a mistake in Problem 30 (2). | |

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## Content of the course

**I. Derived categories**

References: [Kashiwara-Schapira], [Gelfand-Manin], [Weibel], [Krause], [Lipman]

- The category of complexes of an abelian category
- The homotopy category of complexes
- Colimits
- Localization of categories
- Triangulated categories
- The derived category of an abelian category
- Derived functors
- Spectral sequences

**II. The derived category of $\mathscr O_X$-modules**

- The category of quasi-coherent $\mathscr O_X$-modules and its derived category
- Comparison $D({\rm QuCoh}(X))$ versus $D_{\rm qc}(X)$
- Base change homomorphism, (derived) projection formula, Künneth morphism

**III. Finiteness results**

- Ample line bundles and cohomological characterization of ampleness
- Perfect complexes
- Cohomology and base change

**IV. Grothendieck duality**

- The Brown representability theorem
- A right adjoint functor to $Rf_*$
- The dualising complex for $f$ proper and smooth
- Application: The Theorem of Riemann-Roch

## Literature

Coherent cohomology , Seminar at Leiden University (2014)

B. Conrad, Grothendieck Duality and Base Change, Springer Lecture Notes in Math. 1750 (2000)

S. Gelfand, Y. Manin, Methods of Homological Algebra, Springer Monographs in Math.

R. Hartshorne, Residues and Duality, Springer Lecture Notes in Math. 20 (1966)

M. Kashiwara, P. Schapira, Categories and Sheaves, Springer

J. Franke, On the Brown representability theorem for triangulated categories, Topology 40 (2001), 667-680.

H. Krause, A Brown representability theorem via coherent functors, Topology 41 (2002), 853-861.

H. Krause, Derived categories, resolutions, and Brown representability

J. Lipman, Notes on derived functors and Grothendieck duality , in Springer Lecture Notes in Math. 1960

A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, JAMS 9, no. 1 (1996), 205-236.

A. Neeman, Grothendieck duality made simple, arxiv:1806.03293

C. Weibel, An introduction to homological algebra, Cambridge University Press