# Solutions to "Vakil - Foundations of Algebraic Geometry" exercises [closed]

I think the notes of Professor Ravi Vakil are a great source to learn algebraic geometry. The exposition is very clear, and much effort was put to give an intuitive picture together with a flawless formal precision.

Of course throughout the book there are hundreds of exercises, some of them are easy, some others are more involved, depending also on the reader's background in the field.

I think it would be very nice and useful to start an archive with solutions to the exercises, for at least three reasons:

• It would be very useful for people trying to learn the subject. It's always a good idea to try to solve exercises by yourself, but sometimes you just can't do it, or you simply want to check your solution to make sure you were correct.

• Once a good number solutions is collected, it would be a great gift to Ravi. Probably, its notes will be published as a book eventually and he could decide to publish our archive in the form of a Solution book.

• It would be a great stimulus for learners to solve the exercises and write the solution carefully and precisely. Everybody wants to gain points here on StackExchange ;-)

I propose to post the solution of every single exercise as one separate answer, with the following format:

1. Exercise number in bold, together with the version of the notes from which it was taken. Specify if it is a starred or double-starred exercise.

2. Body of the exercise inside a quote block. I'd say this is not mandatory, but very appreciated.

3. "Solution:" in bold, followed by the solution to the exercise.

Here's one example of the formatting:

Exercise 1.3.A - June 11, 2013 version

Show that any two initial objects are uniquely isomorphic. Show that any two final objects are uniquely isomorphic.

Solution:

Let's assume $A$ and $A'$ are two initial objects $\dots$

The fact we put each solution in a different answer gives another good reason to make such an archive here on StackExchange: the solutions to the most interesting exercises will hopefully be upvoted and appear first on the list of answers, thus giving them the emphasis they deserve.

I hope you like this idea, and of course any suggestions to improve it is welcome!

• I had thought about posting solutions on my website, but contacted Vakil before proceeding to ask whether the Creative Commons license CC BY-NC-ND 3.0 on the book prevented this (a solution manual could be considered a derivative work). Vakil, in personal correspondence to me (Aug 11, 2013): The license, and my wishes, are intended to be very open, so there's no obstruction there. Dec 8, 2013 at 11:51
• But I'd still ask that you to do some variant of this instead, for different reasons. If you post your answers on the web, it is inevitable that others who will be learning the same material in the future will flip too quickly to your solutions, and not struggle themselves. And when learning mathematics, the struggle gives much more insight than the answers. (There will be a minority who, when given homework problems from the notes, will just go straight to your solutions; but I'm worried about the much larger number who have good intentions.) Dec 8, 2013 at 11:52
• So, there you have it: Vakil would prefer if people's collected solutions are kept private. Dec 8, 2013 at 11:54
• As Ravi has communicated to Zev, there are numerous reasons one doesn't want solutions for this text easily available. Algebraic Geometry in particular is a subject where many students come up against a kind of wall they haven't come across before, and torturing yourself with the exercises is the only want to really 'grok' it. However I still think your three reasons for wanting to do this are valid points -- how about organizing a learning group of 3-8 people and create a shared latex document (on Google Drive say) to begin this project more privately? Dec 8, 2013 at 12:26
• I have begun a meta-thread on this topic here. Dec 8, 2013 at 13:29

Exercise 9.2.B - June 11, 2013 version

Let $\phi: B\to A$ be a ring homomorphism and $I\subset B$ and ideal. Let $I^e := \langle\phi(i)\rangle_{i\in I} \subset A$ be the extension of $I$ ot $A$. Describe a natural isomorphism $A/I^e \cong A\otimes_B (B/I)$.

Solution:

First of all notice that we have a canonical identification $I^e \cong I\otimes_B A$, which follows immediately from the definition of the $B$-module structure induced by $\phi$ on $A$.

Next, apply the right exact functor $\square\otimes_B A$ to the exact sequence $$I\to B \to B/I \to 0$$ to obtain the exact $$I\otimes_B A\to B \otimes_B A \to (B/I) \otimes_B A \to 0$$ and use the above identification and $B \otimes_B A \cong A$ to conclude.

• You have an extra ">" in the statement of the exercise. Be aware that a > followed by a block without a double break line is a perfectly good quote (you don't have to put a > in every single line of the quote).
– Pece
Dec 8, 2013 at 11:50