MATH 632: ALGEBRAIC GEOMETRY II COHOMOLOGY ON ALGEBRAIC VARIETIES LECTURES BY PROF. MIRCEA MUSTA¸TA;˘ NOTES BY ALEKSANDER HORAWA These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸t˘a in Winter 2018, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from May 24, 2018. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at
[email protected]. The problem sets, homeworks, and official notes can be found on the course website: http://www-personal.umich.edu/~mmustata/632-2018.html This course is a continuation of Math 631: Algebraic Geometry I. We will assume the material of that course and use the results without specific references. For notes from the classes (similar to these), see: http://www-personal.umich.edu/~ahorawa/math_631.pdf and for the official lecture notes, see: http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf The focus of the previous part of the course was on algebraic varieties and it will continue this course. Algebraic varieties are closer to geometric intuition than schemes and understanding them well should make learning schemes later easy. The focus will be placed on sheaves, technical tools such as cohomology, and their applications. Date: May 24, 2018. 1 2 MIRCEA MUSTA¸TA˘ Contents 1. Sheaves3 1.1. Quasicoherent and coherent sheaves on algebraic varieties3 1.2. Locally free sheaves8 1.3.