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"
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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
Nice, dude! I found it helpful to make the calculations with ideals explicit:
x^2 + y^2 = 1 <==> all ideals in k[x, y] containing x^2 + y^2 - 1
intersection with y = a(x - 1) <==> all ideals in k[x] containing x^2 + a^2 (x-1)^2 - 1
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Generically these ideals are just those generated by multiples of this quadratic
Easiest to think about when k algebraically closed. Then the quadratic is (x - r)(x - s)
Becomes (x - 1)^2 when a = infty, giving the nilpotent elements
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