Motivation for concepts in Algebraic Geometry

Question

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?

Answer 1

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)

Answer 2

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.

Answer 3

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 .

Answer 4

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.