# Algebraic geometry: difference between variety approach and scheme approach?

This would be an elementary question and sorry if this is duplicate one - but I could not find any satisfactory answer anywhere else. :-(

I'm learning algebraic geometry not for its own but for the application to other fields, like algebraic topology. As I continue to study, I got an impression that algebraic geometry largely can be divided into, roughly speaking, something with variety and something with schemes.

I know that varieties are closely related with schemes and vice versa, and how do they relate to each other. But I got an impression that these belong to separate subfields of study. Is this really so?

May I consider these "variety approach" and "scheme approach" as separate things and safely concentrate on just one, say the scheme approach?

These two approaches seem to use somewhat different languages. Study on varieties seems to be involved with hard commutative algebra while scheme approach is not. I even could read some papers on algebraic geometry and makes sense out of it while not being familiar with commutative algebra yet. Just some exposure and experience with categorical language was enough.

Can anyone clarify these, especially the difference between "variety approach" and "scheme approach" and what these are aimed for?

(I think I know what I want from algebraic geometry and it's very scheme-oriented. But I'm not sure how much time and efforts are worth to spend on studying varieties.)

In short: "varieties are nice schemes over fields". In particular, the study of varieties is a subscase of the study of schemes. A variety (depending on who you ask) over a field $k$ is a finite type separated (usually reduced, sometimes geometrically integral!) scheme over $\text{spec}(k)$.

The reason that the two fields look so different, is purely in language. When one studies varieties on their own, it's likely that one is using a source that uses the classical (read "pre-Grothendieck") language. This language suffices to talk about such well-behaved schemes, but fails to distinguish between the more sophisticated properties of general schemes (e.g. a point $\text{spec}(k)$ and a 'fuzzy point' $\text{spec}(k[x]/(x^m))$).

That said, much of general scheme theory actually can be reduced to the study of varieties. This is because, for a sufficiently nice schemes (most that we encounter) we can think about them as being 'fibered' over varieties. By this, I mean that if $X$ is a scheme, with an equipped map of schemes $X\to S$ which satisfies some mild properties, then all of the fibers $X_s$ for $s\in S$ are varieties over $k(s)$. This allows one to think about general schemes as being 'families of varieties indexed by other schemes'.

It is a mistake to confuse 'variety land' with 'commutative algebra land'. What is more likely happening, is that when you are looking at texts on varieties, they happen to be focusing (perhaps at the beginning) on affine varieties. This is why the following correct identification can be confused with the above incorrect one: 'affine scheme land' IS 'commutative algebra land'. This holds true for schemes over arbitrary bases, and is made precise by an (anti)equivalence of categories between affine schemes (over an affine base) and algebras over that affine base.

In terms of learning it, historically varieties came before fields. They are also the most tenable examples of schemes (besides, perhaps, number rings), and so are useful to have in your back pocket. The classical language, what you would most likely learn varieties in, has the advantage of being simpler, and perhaps easier to see the geometry. Unfortunately, it is often times sloppy, and hard to analyze the fine structure results that you'd need.

In this way, I think that a cursory reading of a book on varieties, and almost more helpful a book on complex manifolds, is a helpful first step to studying schemes. That said, there will be no technical loss (only a loss of intuition) by starting directly with schemes.

• Hi Alex. Does your definition of a variety over an algebraically closed field $k$ — as a finite-type separated scheme over $\operatorname{Spec}(k)$ — cover the four usual types of varieties over $k$: affine, quasi-affine, projective and quasi-projective? (I believe that we can focus only on quasi-projective varieties because the other kinds of varieties are a special case.) – Berrick Caleb Fillmore May 8 '16 at 18:57
• Am I therefore right to say this: Let $k$ be an algebraically closed field. For every quasi-projective variety over $k$, its associated scheme over $\operatorname{Spec}(k)$ has the properties above; and every scheme over $\operatorname{Spec}(k)$ with the properties above is isomorphic, as a locally ringed space, to the scheme associated with some quasi-projective variety over $k$? Thanks! – Berrick Caleb Fillmore May 8 '16 at 19:07