Today, I finished (for the most part) discussing sheaves — in particular, we discussed inverse image sheaves. We’ve talked more about the underlying set of an affine variety or scheme (mSpec and Spec). In particular, we’ve begun to flesh out a dictionary between algebra and geometry. We saw why maps of rings give maps of Spec’s (or mSpec’s, if appropriate) in the other direction.

Things to think about in the next couple of weeks

You’re undoubtedly still thinking about things from the previous week, so here are some more things to ponder. If you are relatively new to commutative algebra, and you find that you can do exercises, then declare victory — you can learn commutative algebra as you need it. It is worth thinking through the properties we need on quotient rings, localization of rings, and Noetherian rings.

Things to read this week

On top of the things you read as of last week, you should now read the final section 2.7 on sheaves, and basically all of Chapter 3. We haven’t really discussed the topology on (m)Specs (the Zariski topology), but you have already told me how it should work — sets cut out by a bunch of functions should be declared to be closed sets, and nothing else. So you can now read all of Chapter 3 (and also think through what things mean even in the case of mSpec).

Problems to think about this week and next

For everyone: please do the same three meta-problems of what was interesting, and what was challenging, and what was confusing.
 You may have noticed that your answers have a big effect on what I choose to say in the pseudolectures.

If you are new to commutative algebra: There are a bunch of things in commutative algebra that are now coming up — you may be able to understand them all, by judiciously working through exercises. I hesitate to suggest any in particular — just pick several that you think are at the border of your understanding. I am hoping that a number of you will think about the same problems, and discuss them (and call in me or some shepherds if you have questions or things to talk about). For example, you might be able to learn all you need (for now) about Noetherian rings by doing a few exercises in section 3.6.

Do what you kind to understand the inverse image sheaf. For example, 2.7.B.

Exercise 3.4.E and 3.4.F will help see what nilpotents do (or more precisely don’t do) geometrically. We’ve seen that maps of rings induce maps of (m)Spec’s in the opposite direction; 3.4.H will show you that this is a continuous map, and 3.4.I will give a bit more insight. Definitely do 3.4.J.

When you read 3.5 on distinguished open subsets, definitely solve 3.5.E if you can.

Section 3.6 has a lot of words in it, but the important concepts have already happened by 3.5. 3.6.A and Remark 3.6.3 relate to Taylor’s comments on idempotents. 3.6.B and 3.6.C give examples that help you see the weirdness of Zariski topologies. 3.6.E and 3.6.F are concrete problems that might test your understanding; perhaps 3.7.G and 3.7.H too.

In section 3.7, do 3.7.E and 3.7.F.

And next Saturday, I’ll quickly review things you have read on the Zariski topology, and we will define schemes (and, almost, varieties)!

Bonus links

There were a couple of links on zulip to help make things in tikz, and I thought they were worth highlighting: