# 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

**Enrolling for the oral exam:**I checked back and received the information that the course should be present in the list of the master’s program as Algebraic Geometry 3 (or maybe Algebraische Geometrie 3, a

*Vertiefungsmodul*; probably with a fictitious date of March 31 — we will fix the actual dates later on an individual basis), and that all master’s students, in particular the ALGANT students, should register until November 22. We can discuss again on Wednesday. If all else fails, you could email Ms. Czogala at the central examination office of the university,

`dominika.czogala@uni-due.de`(with your name, student ID number, and the name of this course).

**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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## 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

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

**III. Finiteness results**

**IV. Grothendieck duality**

## Literature

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

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

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