alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
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What's a scheme?
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oof. so, there are a lot of different ways to think about this, and my preferred way of thinking about it is not the most common one (functor of points). it also depends a lot on what your existing background is. eg, do you already have a sense of what a variety is?
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I have a pet answer if you know what a manifold is already 🙃
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What is it?
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So before Grothendieck, algebraic geometers basically studied tons of EXAMPLES of "algebraic geometric objects"
These examples were all "varieties" (i.e. subsets of n-dimensional space over a field defined by polynomial equations)
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It would kinda be like if differential geometers and algebraic topologists went around studying Klein bottles, toruses, n-dimensional spheres, etc. without defining the thing that unified all of them
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Grothendieck's definition of "scheme" fixes this. A scheme is, simply, the "general algebraic geometric object" of which varieties are an important special case
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So the part that I find confusing is what non-variety objects are schemes really generalizing? It seems like they do "varieties but also (algebraically) rings". But there's no good examples of non-variety *geometric* objects they generalize
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here's the simplest example i can think of: the (scheme-theoretic) intersection of two varieties isn't always a variety. if you consider, say, a circle and a line tangent to it in the plane, their scheme-theoretic intersection has nilpotents
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the intersection reflects the tangency, that the two are intersecting "to first order" and not just at a point. this is actually *good* because we want a linear thing and a quadratic thing to intersect in "two points" and that's only true if we count with "multiplicity"
let's see how this works out in coordinates: we can take the unit circle x^2 + y^2 = 1 and the equation of a line y = a(x - 1) passing through (1, 0). substituting gives x^2 + a^2 (x^2 - 2x + 1) = 1. generically this is a quadratic equation with 2 distinct roots
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but if we take a = infinity (this just corresponds to making the line vertical) we get (x - 1)^2 = 0; now there's one root with multiplicity 2. the "principle of continuity" is that we want this to still count as "two points" and the scheme-theoretic intersection captures that
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