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Hartshorne does not seem to bring this concept up so far in his AG book but I am guessing that one may define an "abstract variety", in a similar way as one defines an abstract manifold from DG.

When we study varieties we care about two things about them. First, the topology on the variety and, secondly, the regular functions defined on them (or their open subsets). Thus, perhaps, we can define an abstract variety to be a topological space $X$ together with a collection of functions $X\to k$ which we call "regular".

Why is this a useful concept, assuming we can define it? Consider the product $\mathbb{P}^n \times \mathbb{P}^m$ with the Segre embedding $\sigma$ into $\mathbb{P}^{nm+n+m}$. I find it really annoying that if I want to work with the product, $\mathbb{P}^n \times \mathbb{P}^m$, which is pretty easy to grasp, I instead have to work with $\sigma( \mathbb{P}^n \times \mathbb{P}^m) $. Which is conceptually more difficult.

What if we use the bijection of sets, $\sigma: \mathbb{P}^n \times \mathbb{P}^m \to \sigma (\mathbb{P}^n \times \mathbb{P}^m)$ and par transport de structure onto the set $\mathbb{P}^n \times \mathbb{P}^m$? We define the topology on $\mathbb{P}^n \times \mathbb{P}^m$ by pulling back the open sets. And we define that a function $f:U\to k$, where $U\subseteq \mathbb{P}^n \times \mathbb{P}^m$ is "regular" if and only if $f\circ \sigma^{-1} : \sigma(U) \to k$ is regular in the usual sense.

I am guessing that schemes are the abstraction of varieties, but I did not study them yet. Can one get away with this simple generalization of what a variety is?

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    $\begingroup$ I'm afraid you're in for quite a shock if you're hoping that schemes will be as simple as manifolds. Read ahead and you'll see. $\endgroup$ – Matt Samuel Dec 14 '14 at 0:06
  • $\begingroup$ One book that defines "abstract varieties" is Milne's. $\endgroup$ – Hoot Dec 14 '14 at 0:07
  • $\begingroup$ @Matt: Actually, I think manifolds are hard. I am not a fan of DG. I found their definitions to be very messy, and many of the arguments in DG are way too heavily reliant on pictures. I could never find myself to appreciate DG. $\endgroup$ – Nicolas Bourbaki Dec 14 '14 at 0:11
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    $\begingroup$ @ILoveMr.Paul The definition of schemes is far, far messier. It's similar except that instead of coordinate charts you have open sets that are homeomorphic to the spectrum of prime ideals of a ring, and it comes with a sheaf of rings (which corresponds to the regular functions). Maybe you'll appreciate it more though; to each their own. $\endgroup$ – Matt Samuel Dec 14 '14 at 0:17
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    $\begingroup$ @ILoveMr.Paul I think you may be confusing differential geometry with physics. I don't know what it was like in the past, but it's certainly as rigorous as anything else now. $\endgroup$ – Matt Samuel Dec 14 '14 at 0:25
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The easiest way to define an abstract variety is over an algebraically closed field $k$, as a ringed space locally isomorphic to affine varieties over $k$, with the concrete definition of an affine variety as a zero locus with its natural sheaf of regular functions.

This definition already isn't so great over an arbitrary field $k'$, where you have nullstellensatz problems. There it's still workable, however, to model your varieties locally on spectra with only closed points, ringed space built out of the maximal ideals of reduced finite-dimensional commutative algebras over $k'$.

This may not capture everything you really want to know, even while working over a field, so you can pass to the full spectrum of a reduced finite-dimensional $k'$-algebra as the local model for your varieties. This includes the prime ideals as points on an equal footing with the maximal ideals, and permits sensible talk of generic points among a great number of other benefits. At this point you're really talking about a reasonably special scheme, so that in this third context one might say that a variety is a reduced scheme of finite type over a field. But there's not universal agreement on this definition.

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