How can schemes see two points associated with $y = x^2$
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Clash Royale CLAN TAG #URR8PPP up vote 3 down vote favorite I have read somewhere that one motivation of scheme theory is that using methods developed in scheme theory we can detect that the point associated to the intersection of $y = 0 cap y = x^2$ has multiplicity two. Can someone explain this in details to me. That would be very helpful. algebraic-geometry schemes share | cite | improve this question edited Jul 30 at 2:08 Eric Wofsey 162k 12 188 298 asked Jul 29 at 21:09 Newbie 384 1 11 2 $k[x,y]/ (y - x^2, y) = k[x,y] / (y,x^2) = k[x]/(x^2)$. The final ring is a vector space of dimension 2, which is the multiplicity of the intersection. â Lorenzo Jul 29 at 21:18 1 If you really want all the details, you might have to start studying scheme theory. I recommend "The geometry of schemes" , by Eisenbud and Harris. â Jesko Hüttenhain Jul 29 at 22:28 What does this have to do with