## Algebraic varieties , isomorphism, normal, reduced… [closed]

Hello, could someboty help me, every advice or hint will be very helpful (or if you can tell me some good book). Thank you so much!

1. X and Y are two varieties and there are points P of X et Q of Y such that the local rings O(X,P) and O(Y,Q) are isomorphic as C-algebras. I should show that there are open sets U in X ( P is in U) and V in Y ( Q is in V) and an isomorphism of U to V that sends P to Q. How can I construct that isomorphism? Is this true for projective varieties or only for affine?

• If Y is a complex algebraic variety and Z an open subvariety of Y. How can I prove that C(Z) = C(Y)
• The ring of regular functions on A² \ V(x) is isomorphic to C[x,y][1/x] ( where A² = Spec C[x,y] and C[x,y][1/x] is the localisation of C[x,y] in the multiplicative set {1,x,x^2,,,,}
• The ring of regular functions on A² \ {0} is isomorphic to C[x,y]
• How to conclude that A² \ {0} is neither affine or projective
2. If X is a connected algebraic variety, I should prove that X is reduced ( resp. integral, resp. normal) if and only if, for every closed point p of X the local ring O(X,p) is reduced ( resp. integral, normal).

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Some good books: Eisnebud-Harris, Hartshorne, Mumford, Shafarevich.... – Donu Arapura Mar 28 2011 at 21:53
You should do your homework on your own. – Martin Brandenburg Mar 28 2011 at 22:15