How can schemes see two points associated with $y = x^2$

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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




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    $k[x,y]/ (y - x^2, y) = k[x,y] / (y,x^2) = k[x]/(x^2)$. The final ring is a vector space of dimension 2, which is the multiplicity of the intersection.
    – 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.
    – Jesko Hüttenhain
    Jul 29 at 22:28










  • What does this have to do with schemes? This is the oldest idea in AG.
    – Rene Schipperus
    Jul 30 at 0:19














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

















  • 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.
    – 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.
    – Jesko Hüttenhain
    Jul 29 at 22:28










  • What does this have to do with schemes? This is the oldest idea in AG.
    – Rene Schipperus
    Jul 30 at 0:19












up vote
3
down vote

favorite









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













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












share|cite|improve this question




share|cite|improve this question








edited Jul 30 at 2:08









Eric Wofsey

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asked Jul 29 at 21:09









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  • 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.
    – 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.
    – Jesko Hüttenhain
    Jul 29 at 22:28










  • What does this have to do with schemes? This is the oldest idea in AG.
    – Rene Schipperus
    Jul 30 at 0:19












  • 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.
    – 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.
    – Jesko Hüttenhain
    Jul 29 at 22:28










  • What does this have to do with schemes? This is the oldest idea in AG.
    – Rene Schipperus
    Jul 30 at 0:19







2




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.
– Lorenzo
Jul 29 at 21:18




$k[x,y]/ (y - x^2, y) = k[x,y] / (y,x^2) = k[x]/(x^2)$. The final ring is a vector space of dimension 2, which is the multiplicity of the intersection.
– Lorenzo
Jul 29 at 21:18




1




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.
– Jesko Hüttenhain
Jul 29 at 22:28




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.
– Jesko Hüttenhain
Jul 29 at 22:28












What does this have to do with schemes? This is the oldest idea in AG.
– Rene Schipperus
Jul 30 at 0:19




What does this have to do with schemes? This is the oldest idea in AG.
– Rene Schipperus
Jul 30 at 0:19















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