Happy New Year! The picture is of an amazing artwork by Gabriel Dorfsman-Hopkins and Daniel Rostamloo of 27 lines on a cubic surface. **Here** is a link to the webpage for the sculpture.

The latest (December 31, 2022 version) of the notes are posted here, at the **usual place**. In terms of content, it remains (and will remain) very similar to the previous version. But it has been substantially edited and polished.

In more detail: the first 18 chapters are much more polished than before, and are nearly in “potentially final form”. Most of the later chapters are still waiting for that level of tender loving care. Please continue to let me know a$bout any suggestions, corrections, typos, etc., in any chapter, no matter how small. The only exceptions are stylistic things (margins, fonts, low-quality figures).

Here are some changes I find interesting or worth mentioning.

Chapter 8: The Chevalley’s Theorem proof still had a small flaw, and is now fixed. (Thanks to Hikari Iwasaki for comments on this and many other things near there.) It is now one that I find so natural that I can’t forget it, and can easily explain while walking across the quad.

Chapter 10: The starred section, and doubled-starred proofs, on things like “geometrically reduced/integral/etc.” have been awaiting patching for a very long time. I finally did it. If you are curious, or want to solidify your understanding of those issues, please take a look. I hope that it has become very comprehensible. The main black box (that I punted to the chapter on flatness) is that if is any -scheme, then is universally open.

Chapter 12: The chapter on dimension is substantially rewritten, and I hope it is a cleaner way through the topic. There are some explanations directly or indirectly from Mel Hochster and Qing Liu. (I highly recommend Qing Liu’s book; I love how he thinks in a very “clean” way.)

Chapter 13: I had been meaning to rethink the older version of the chapter on regularity and smoothness for some time. I’ve now done so, and fixed a lot of problems. I know from experience that I have undoubtedly introduced new ones. From the new point of view, the central players here is that if you want to get at notions of regularity and smoothness, you are led to the notion of the Zariski tangent space and the Jacobian criterion, and everything revolves around them.

Chapter 15: If there is a single central fulcrum on which everything balances, it is at the almost literal center of the notes: the link between line bundles and divisors. This is a tricky topic the first time you see it, and section 15.2 has always been trickier for readers than I want it to be. I hope it is better now.

Also, Pol van Hoften told me the definition of abelian category which he prefers, and I was very surprised to find it much clearer and enlightening than anything I had known before. How could this be, given how old and important a notion it is? Pol’s definition is from Jacob Lurie’s Higher Algebra, although his description of it is his own. I have now included it in the notes. I’ll tell you it here, because it is something directly memorable enough that (unlike the previous definition) I can remember it without trying. Also, I learned something new: the notion of abelian category is intrinsic to the category itself — there is no additional structure required. That was so unclear from other definitions that I did not realize it despite using this notion for years! Here is Pol’s description. We are motivated by the abelian categories we already know and love. Suppose is a category. We give some axioms that determine whether it is an additive category, or an abelian category.

Axiom 0: has a zero object (an object that is both final and initial). Of course this axiom should be numbered zero. And of course abelian categories need a zero object, so this is easy to remember.

For the next axioms, we note that abelian categories have finite products and coproducts, and they are the same. Let’s axiomatize this.

Axiom 1: For any two objects and , the product and coproduct exists. (Hence has finite products and coproducts.)

Axiom 2: By the universal property of product and coproduct, we therefore have a map for all pairs of objects. We require that this map be an isomorphism.

Now here is the insight I didn’t realize: with this information, we can discover how to “add” two morphisms from to . And adding the zero morphism (the morphism that factors through the 0 object) does nothing under this operation. You can check that this gives intrinsically the structure of a commutative monoid (it is associative, has identity). And it even automatically distributes in the way we’d like (e.g., if we have two maps and from to , and one map , then ). It is fun seeing how some of these work out.

We don’t yet know that it is an abelian group, because we don’t know that + has an inverse. So that is our next axiom.

Axiom 3. For any two objects and , this operation + gives the structure of an abelian group.

At this point we have defined the notion of additive category. Note that we don’t *require* that have the structure of an abelian group (which I used to think of as an additional structure); we have the *property* that with its pre-existing operation + is such that + has an inverse.

Axiom 4 and 5 are just the two axioms making additive categories into abelian categories. Axiom 4: kernels and cokernels exist. Axiom 5: some property (you can have several choices here, leading to the same definition) that kernels and cokernels have to satisfy. Usually it is “some property and its dual”. Once again, there is no additional structure required; just a condition.

So if you have two abelian categories and , and you have a functor , how do you know if it is a map of abelian categories? All you need to do is to check that it preserves products and coproducts (of two elements) and kernels and cokernels. All other parts of the structure come at no extra charge.

I will teach the second and third quarters of “Foundations of Algebraic Geometry” this winter and spring quarters (Pol taught the first one and by all accounts did a remarkable job). I hope to continue to edit the file during this time, and clean up more chapters.