1. Show that a map between affine varieties can be continuous for the

Zariski topology without being regular.

2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)

to the hyperbola xy=1, but neither is isomorphic to A^{1}.

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- Jul 2nd 2008, 05:24 AMtemanratmAffine varieties
1. Show that a map between affine varieties can be continuous for the

Zariski topology without being regular.

2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)

to the hyperbola xy=1, but neither is isomorphic to A^{1}. - Jul 2nd 2008, 07:32 PMNonCommAlg
i'll assume that the base field is

define by and

Quote:

2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)

to the hyperbola xy=1.

by: now if then for some

but then: i.e. so is well-defined.clearly is a surjective

ring homomorphism. now suppose then i.e. for some therefore:

so hence so is injective and we're done!

Quote:

but neither is isomorphic to A^{1}.

2) and are not isomoprphic because, if there was an isomorphism then assuming that

we'll get thus and therefore

are constant. but then would not be surjective. Q.E.D. - Jul 3rd 2008, 08:49 AMtemanratm
Thank you a lot

I've one more problem

Let q be a power of a prime p and let F_{q} be the field with q elements.

S is a subset of F_{q}[X_{1},...,X_{n}] and let V be ots zero set in k^{n}.

where k is alg-c closure of F_{q}.

Show that the map (a_{1}, .... a_{n})->((a_{1})^q,...,(a_{n})^q) is a regular map

f:V->V (i.e., f(v)subset of V)

Verify that the set of fixed points of f(x) is the set of zeros of the elements

of S with coordinates in F_{q}.

I would be very grateful

thx in advance - Jul 4th 2008, 02:03 AMNonCommAlg
let and since

we'll have: therefore:

i.e.

Quote:

Verify that the set of fixed points of f(x) is the set of zeros of the elements

of S with coordinates in F_{q}.

so each is a root of the polynomial but every element of is also a root

of and obviously has exactly roots in thus