Introduction to Cohomology

Igor Kriz () and Sophie Kriz ()
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Igor Kriz: University of Michigan, Department of Mathematics
Sophie Kriz: University of Michigan, Department of Mathematics

Chapter 5 in Introduction to Algebraic Geometry, 2021, pp 271-365 from Springer

Abstract: Abstract It is now time to study the subject of cohomology in detail. In Chap. 4 , we already encountered its special cases in degrees 0 and 1. This motivates understanding the general machinery which lets us set up cohomology groups in any degree. Even more importantly, a careful reader noticed that there are very important facts about regular rings (for example the fact that a localization of a regular ring is regular) which we have not proved so far, and deferred to when we can characterize regular rings cohomologically. For this, we will certainly need cohomology in higher degrees. Without filling this gap, we would not even be able to use Weil divisors to calculate Picard groups rigorously in general examples such as those of Chap. 4 . We will complete those proofs in the present chapter.

Date: 2021
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DOI: 10.1007/978-3-030-62644-0_5

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