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Why study schemes instead of only affine/projective varieties, given by zeros of polynomials in the affine/projective space? I mean, what is gained by introducing the concept of schemes?

Thank you!

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If you read the introductory paragraph of en.wikipedia.org/wiki/Alexander_Grothendieck and the History and motivation section of en.wikipedia.org/wiki/Scheme_(mathematics), you'll get an idea why it is important. Probably it is mostly important to understand way some of the Weil Conjectures (en.wikipedia.org/wiki/Weil_conjectures) were proven. –  bgins Jan 16 '12 at 17:12
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The move from varieties to schemes involves at least two orthogonal generalizations: allowing nilpotents and working over something other than an algebraically closed field. These issues should probably be discussed separately. –  Qiaochu Yuan Jan 16 '12 at 17:18
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A third important generalisation is that the definition of scheme inherently allows gluing, whereas it is not at all obvious whether the result of gluing two quasi-projective varieties together is another quasi-projective variety! This is important for the piecewise construction of various moduli spaces... –  Zhen Lin Jan 16 '12 at 17:29
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A whole (little) book, The Geometry of Schemes by Eisenbud and Harris, has been written to answer your question. IMO you should try to get hold of it and read through at least its first few chapters. (I personally think this is not a good question for this site, for it more or less requires a chapter-length essay to be answered sensibly...) –  Mariano Suárez-Alvarez Jan 16 '12 at 17:44
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Besides, schemes are not necessarily defined over a field, but can be defined over arbitrary commutative rings, such that $\mathbb{Z}$ or another number ring. For example, taking the "fibers" of a curve over $\mathbb{Z}$ you can get curves over $\mathbb{F}_p$ and over $\mathbb{Q}$. This tool is extremely important for the study of elliptic curves and for arithmetic geometry. –  Andrea Jan 17 '12 at 14:35

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Edit: Edited to better organize the trichotomy noted in the comments of Qiaochu Yuan and Zhen Lin.

There are three distinct aspects of schemes that each have their own purpose:

(1) Affine schemes generalizing affine varieties by allowing nilpotent elements in the coordinate ring. When you look at families of affine varieties, sometimes the limiting space is only a scheme and not a variety. For example, consider the affine variety given by the equation $z^2 = txy$. This is a well-defined affine variety that is presented as a subvariety of $\mathbb{A}^4$. We could also view it as a family of affine subvarieties of $\mathbb{A}^3$ parameterized by $t$. However, this is only valid for $t \neq 0$. When $t = 0$, the equation becomes $z^2 = 0$. You could do two things to resolve this: (a) Eliminate the nilpotents and say that variety is $z = 0$ when $t = 0$. With this approach, you now have to check for nilpotents every time you degenerate a variety, and then remove them. And now there are very few theorems you can prove about two different varieties that occur in a family. (b) Accept that nilpotents are inevitable when trying to understand families of varieties, and allow them in your geometric theory. At the cost of additional abstraction and more technical baggage necessary, you are able to prove much more powerful theorems relating varieties in a family (for example, the constancy of the Hilbert polynomial in a flat family of subvarieties of projective space -- if you don't know what that means yet, that's okay, but it's a very important result).

(2) The second reason, anticipated somewhat by Kronecker, is that replacing the coordinate ring of an affine algebraic variety with an arbitrary commutative ring yields a striking connection between algebaric goemetry and algebraic number theory. In particular, many properties $\text{Spec } \mathcal{O}_K$ relate to classical number theory invariants of $\mathcal{O}_K$, the ring of integers of an algebraic number field. For example, ramification in number fields can be interpreted as ramification of the associated geoemtric map between their spectra. You can also use this approach to study algebraic varieties over fields that are not algebraically closed.

(3) Once you accept affine schemes, a general scheme is obtained as a certain ringed space that is locally isomorphic to an affine scheme. This idea goes back to Weil, and it was introduced in algebraic geometry for the same reason that general manifolds were introduced to replace the study of embedded manifolds in Euclidean space. This allows gluing constructions, without having to check whether the resulting space is embedded in an affine or projective space. Of course, the main interest is still in studying affine and projective varieties, but the generality of schemes allows one to make many more constructions to study them.

There are a couple of really good sources to turn to learn more about this:

(1) Dieudonne's classical book, History of Algebraic Geometry

(2) Ravi Vakil's book in progress, Foundations of Algebraic Geometry

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Direct link to the book in progress Foundations of Algebraic Geometry by Ravi Vakil. –  Pierre-Yves Gaillard Jan 16 '12 at 18:28

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