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!)
Conversation
What's a scheme?
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Yeah (I donjn principle know what a scheme is, I've just never really grokked *why*)
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so, a wiser man than me (allen knutson) once made the observation that schemes generalize, say, affine varieties in three orthogonal ways:
1. patching affine pieces
2. allowing nilpotents
3. working over a non-algebraically closed field or more general base
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these are each good and useful and people have reasons for wanting them but they are good and useful for different reasons and depending on what you want to do you may find you only need one or two. personally i have never needed 1, only use 2 and 3
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a non-algebraically closed field is most obviously useful in number theory and working over a general base is useful for lots of reasons but i'm not really the person to ask about that. nilpotents let you do things like define intersection multiplicities and tangent spaces