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!

Solution 1:

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.