Schemes 19: Products

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne..
We define fibered products of schemes, sketch their construction,
and give a few examples to illustrate their slightly odd behavior.

[newwaveinfantry]

20:40 - You have also said before that Spec(k^n) has n points, but isn't that wrong?
Take R^2. It has 3 prime ideals. The 0 ideal as well as the ones generated by (1, 0) and (0, 1), respectively. Those ideals are maximal as R^2/((1, 0)) is isomorphic to R. In general, k^n should have Krull dimension n-1 and 2^n ideals, 2^n-1 of which prime, since the whole ring is not a prime ideal.

[hausdorffm]

20:47, The isomorphism L[x]/(p1, ..., pn) = L[x]/(p1) x L[x]/(p2) .... is based on Chinese remainder theorem which requires p1, ..., p1 are coprime, which is ensured by the assumption that K/k is separable extension.

[jiaxizhao]

The example should be SpecQ[\sqrt{2}] \otimes_Q SpecQ[\sqrt{3}] right?

[patrickdasilva]

Q(sqrt 2) x_Q Q(sqrt 2) = Q(sqrt 2, sqrt 3)?? I'm confused. Q(sqrt 2) x_Q Q(sqrt 2) is isomorphic to Q(sqrt 2)[x]/(x^2-2), which is Q(sqrt 2)[x]/(x-sqrt 2) x. Q(sqrt 2)[x]/(x+sqrt2) which is itself isomorphic to Q(sqrt 2)^2. This is not Q(sqrt 2, sqrt 3)! I guess there was a mix up in the choice of examples.