Exercises from The Rising Sea

Keywords: #algebraic-geometry
Table of Contents


During the summer before my first year of graduate school, I began reading Ravi Vakil’s wonderful textbook The Rising Sea: Foundations of Algebraic Geometry and solving every exercise along the way. The purpose of this blogpost is to publish my progress and host the completed solutions document someday in the future. Please note that I may have omitted solutions to straightforward exercises I have done previously or which are both tricky and not essential to the surrounding content, though this is rare. An exception to this is section 6.7 on coherent modules over non-Noetherian rings – I plan to return to this section at some point in the future. Since the textbook will likely be edited further and problem numbers may change as Ravi prepares it for publication, I have included the main portion of the problem statement with every solution to make clear which exercise is being solved. Many problems refer to other parts of the text. For my own sanity, I have not typeset the text of every such reference, so I recommend having a copy of the book on hand while reading these solutions.


In no particular order, I am indebted to Martin Olsson, Yunqing Tang, Owen Barrett, and Gabriel Dorfsman-Hopkins for many enlightening conversations about algebraic geometry and for their patience accepting my questions as a novice algebraic geometer. This project was, in large part, inspired by those interactions. I would also like to thank Dongryul Kim and his solutions to exercises from Hartshorne’s book which inspired me to format this project as a blogpost. Dongryul’s advice was also very helpful for actually setting up this post, in particular the “weak encryption” described below, which was directly copied from his website.

I would also like to thank Ravi for his many years of hard work cultivating The Rising Sea. I highly recommend his book to anyone wishing to learn algebraic geometry.

The solutions

Unless you wish for Ravi to come to your house and pummel you with EGA until you beg for mercy, I suggest you solve these exercises yourself. In the spirit of not tempting anyone out of a learning experience, the solutions have a sort of “weak encryption” – almost every problem/solution pair is on its own page, and the pages are randomly numbered 1 through 5000. You can choose a number in that range and enter it as a “Key” in the box below, and the “View file” button will show the corresponding page. The “Height” entry controls the height of the viewer which opens up below. Some solutions are longer than a page, so you may end up only seeing a portion of a solution! This is not by design, I just haven’t thought of a way to easily split up the large solutions document in a way which leaves these multipage solutions intact. As of May 17, 2024, there are solutions through Chapter 7.

If you find any errors, please let me know by email.

Key:   Height: px