# Motivation for concepts in Algebraic Geometry

I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.

** Question **

Are there any well-motivated introductions to scheme theory?

My idea of what "well-motivated" means are specific enough that I think it warrants a detailed example.

** Example of what I mean by well motivated **

The only algebraic geometry books I have seen which cover schemes seem to leave out essential motivation for definitions. As a test case, look at Hartshorne's definition of a separated morphism:

Let $f:X \rightarrow Y$ be a morphism of schemes. The diagonal morphism is the unique morphism $\Delta: X \rightarrow X \times_Y X$ whose composition with both projection maps $\rho_1,\rho_2: X \times_Y X \rightarrow X$ is the identity map of $X$. We say that the morphism $f$ is separated if the diagonal morphism is a closed immersion.

Hartshorne refers vaguely to the fact that this corresponds to some sort of "Hausdorff" condition for schemes, and then gives one example where this seems to meet up with our intuition. There is (at least for me) little motivation for why anyone would have made this definition in the first place.

In this case, and I would suspect many other cases in algebraic geometry, I think the definition actually came about from taking a topological or geometric idea, translating the statement into one which only depends on morphisms (a more category theoretic statement), and then using this new definition for schemes.

For example translating the definition of a separated morphism into one for topological spaces, it is easy to see why someone would have made the original definition. Use the same definition, but say topological spaces instead of schemes, and say "image is closed" instead of closed immersion, i.e.

Let $f:X \rightarrow Y$ be a morphism of topological spaces. The diagonal morphism is the unique morphism $\Delta: X \rightarrow X \times_Y X$ whose composition with both projection maps $\rho_1,\rho_2: X \times_Y X \rightarrow X$ is the identity map of $X$. We say that the morphism $f$ is separated if the image of the diagonal morphism is closed.

After unpacking this definition a little bit, we see that a morphism $f$ of topological spaces is separated iff any two distinct points which are identified by $f$ can be separated by disjoint open sets in $X$. A space $X$ is Hausdorff iff the unique morphism $X \rightarrow 1$ is separated.

So here, the topological definition of separated morphism seems like the most natural way to give a morphism a "Hausdorff" kind of property, and translating it with only very minor tweaking gives us the "right notion" for schemes.

Is there any book which does this kind of thing for the rest of scheme theory?

Are people just expected to make these kinds of analogies on their own, or glean them from their professors?

I am not entirely sure what kind of posts should be community wiki - is this one of them?

-
I just rolled back an edit which merely made a cosmetic change to formatting, and thus bumped a 5-year old question which has been long-answered, for no go reason –  Yemon Choi Mar 15 at 13:03

I would say that the book you're looking for is probably "The Geometry of Schemes" by Eisenbud and Harris. It is very concrete and geometric, and motivates things well (though I don't think it does so in quite the detail of proving that a topological space is Hausdorff iff $X\to 1$ is separated, but I believe it does discuss separatedness and why it is good and why it captures the intuition of Hausdorff space)

-
Thanks, I had not heard of that one. I will check it out and see if it meets my criteria. –  Steven Gubkin Jan 20 '10 at 18:26
As a non-algebraic geometer, I would start with the Red Book, which gives intuition about varieties as well as schemes, and then read The Geometry of Schemes. It's a great book, but its focus is more on really understanding schemes geometrically and explicitly, and for me this was hard without a better understanding of the most basic notions of algebraic geometry, and how they correspond to the geometric notions I'm familiar with. I came back after reading the Red Book and found it much easier. Both are amazing texts. –  Tom Church Jan 20 '10 at 23:01
I checked this book out. Your right: It is much better at motivating things geometrically than I have seen. Also I like the prevalence of the functor of points approach, which I think is actually closer to the spirit of the geometry than the "normal" approach to schemes. –  Steven Gubkin Jan 21 '10 at 3:14
In many ways it is. If nothing else, you get some constructions to work better, like products are the products of the functors of points, but not of the underlying spaces (because the Zariski topology contains almost none of the geometric information we want) –  Charles Siegel Jan 21 '10 at 12:50

I would recommend Ravi Vakil's notes, which give good geometric intuition for just about everything they cover, and are forthright when the material is "just algebra" and should be regarded as such. They do, like Hartshorne, start off with a dose of abstract exercises about sheaves, but there's really no way to get around the necessity of doing so. As an example, whereas Hartshorne (in II.8) pulls the conormal and relative cotangent exact sequences for modules of differentials out of thin air (= Matsumura), Ravi's notes introduce these by emphasizing their intuitive geometric content in the smooth case, which as far as I can tell is the sort of thing you're interested in.

-

Dear Steven, I think Mumford's notes of the mid 60's, the first ever explaining schemes to ordinary mortals, are still the closest to what you want. They have become a book in 1988: The Red Book of Varieties and Schemes, published by Springer (LNM 1358).

After a first chapter on classical algebraic varieties, Mumford introduces schemes by quoting Felix Klein [in the 1880's!] and amazingly commenting "It is interesting to read Felix Klein describing what to all intents is nothing but the theory of schemes".

And then Mumford brilliantly motivates the necessity of schemes and their nilpotents for a more refined study of varieties. He illustrates his text with wonderful little drawings, among which his great picture of $Spec \mathbb Z [X]$, still admired today. For example Lieven le Bruyn has a series of very interesting articles in his blog "Never ending books" based on that drawing (and as a bonus you can see both the picture of $Spec \mathbb Z [X]$ and that of Mumford...):

http://www.neverendingbooks.org/index.php/mumfords-treasure-map.html

PS Although it is the exact opposite of what you are asking for (!), let me mention that conversely the notion of proper map in Algebraic Geometry seems to have influenced Bourbaki's point of view on proper maps in General(= point-set) Topology. He defines them as universally closed maps and, almost as an afterthought, mentions that in the case of locally compact spaces they are characterized by the property that compact subsets have compact inverse images .

-
The red book is what I would suggest as well. –  Rob Harron Jan 20 '10 at 21:11
Georges, you seem to have duplicated yourself! –  Qiaochu Yuan Jan 20 '10 at 21:11
@Qiaochu: there were some problems with the rollout of beta 6 this morning and Georges isn't able to log in. Hopefully this issue will be resolved soon. See tea.mathoverflow.net/discussion/172/… –  Anton Geraschenko Jan 20 '10 at 21:53
I've merged the two users, but the login problem is still there. –  Anton Geraschenko Jan 20 '10 at 21:57
RE your PS: I think that is great as long as you can develop an intuition for that definition. I think mathematics is filled with examples of definitions which were "too abstract", but were later shown to be great conceptual and pedagogical improvements. For instance, I think Van Kampen's Theorem is much more clearly in the "abstract" form given by Peter May in his book as a statement about colimits of groupoids: It gives a much clearer picture (for me) than the "amalgamated product" formulation of old. Even more basic, examples are given by negative numbers, abstract groups, ... etc –  Steven Gubkin Jan 21 '10 at 3:21

I think the books of Shafarevich meet your criteria. He gives analytic intuitions when he starts explaining about schemes. I had found it to be very helpful.

-