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2014 version

Posted by ravivakil under

Actual notes
[6] Comments
The last version posted was in 2013. I wanted to get the next version out in 2014, and time is running out! So here it is, posted at **the usual place** (the Dec. 30, 2014 version). (There is also a version suited for an e-reader — thanks to Jack Sherk asking for it, and explaining to me how to make it.)

In this 2014 post, I can’t help mentioning the passing of Grothendieck. I was more moved than I expected to hear the news and feel the ripples in the mathematical community. It feels strange that he is now a historical figure, even though he had walked away from his unfinished cathedral long before I was even aware of its existence.

The notes have continued to evolve around the edges, although the material is stable. Please continue to give me corrections and suggestions! There are few sections that need tender loving care; I mention them on the front. There are many other things on my to-do list as well, including many comments you’ve made.

The notes now have a tentative title (The Rising Sea: Foundations of Algebraic Geometry). The phrase is due to Grothendieck, translated by Colin McLarty; it is the title of **Daniel Murfet’s wonderful blog**.

The index is in the process of being made. (No need to give me specific comments until it has converged!)

Finally, if you would like notes on commutative algebra that are very much from the same point of view as these notes, you may enjoy Andy McLennan’s notes, available **here**. It also contains an (explicated) English translation of Serre’s epochal FAC, and all the algebra needed as background.

January 9, 2015 at 2:31 pm

Ravi,

Thanks for posting a new version! I have only checked out whether you addressed the two (exceedingly small) errors I pointed out on the last post. It seems that in your correction of one of these errors

[note from Ravi: this error is pointed out, you are a little ambiguous. On 502here][note from Ravi: just after the statement of Theorem 19.1.1], you say:“Remark: “injective on closed points and tangent vectors at closed points” means that π is unramified (under these hypotheses)”

This is even more exceedingly small than before, but ‘means’ is ambiguous here. My first inclination would be to think that ‘means’=’is defined by’, but this is incorrect–consider the squaring map G_m->G_m, this is unramified, but not injective at closed points. Perhaps ‘is a special case of’ would be better than ‘means’.

Thanks again, and keep up the great work!

Best,

Alex

January 14, 2015 at 11:00 am

I agree, and you are right! That is one of the “to-do’s” embedded in the file, to fix this. What I meant to say was:

Remark: In this case of finite type schemes over an algebraically closed field, “injective on tangent vectors at closed points” is equivalent to “unramified”. (We will define unramified in \S 21.6.)

I just wanted to check that this was true before having it in the public version. It might also be good to have an Exercise in 21.6 saying: Show that a morphism of finite type schemes over an algebraically closed field is unramified if and only if is injective on tangent vectors. (This was mentioned in the remark immediately after Theorem 19.1.1.)

(Does this sound reasonable to you?)

January 15, 2015 at 4:10 pm

Yeah that seems fair. What is your intended solution? I assume you just want:

“Since unramifiedness is an open condition (being the complement of the support of the coherent module ) it suffices to check unramified at closed points. Since the residue field extension is trivial, there is no need to check that. Thus, we only need to check that . But, by assumption, we have a surjective map , which by Nakayama gives the desired equality.”

January 17, 2015 at 11:20 am

Yes indeed! (And the fact that for you this is the obvious solution means that you’ve completely digested how to think about this.)

January 14, 2015 at 11:03 am

I should also acknowledge that I owe responses to earlier comments by many people, and fully intend to catch up!

January 17, 2015 at 11:40 am

I intend to post the next revision at the end of January. One chance I am currently intending to make is this. For many categories, I use \textit (e.g., \textit{Sets}, or \textit{QCoh}_X), but for general facts about categories, I tend to use calligraphic font (or more precisely, \mathscr, the suggested replacement of \mathcal Sandor Kovacs). I am now thinking of using \textit for all categories for consistency. Any complaints/preferences/comments before I do this?