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Before I start I'd like to mention that I am following Gathmann's Notes

Gathmann points out in Definition 4.16 how his more general definition of an affine variety (Ringed topological space that is isomorphic to the vanishing set of a set of polynomials) also agrees with theorems he has proved earlier.

So I went back and attempted to prove the previous theorems using the more general definition of an affine variety and I am stuck on Proposition 4.10

Proposition 4.10: Let $X$ and $Y$ be affine varieties, and let $\pi_X : X \times Y \rightarrow X$ and $\pi_Y : X \times Y \rightarrow Y$ be the projection morphisms from the product onto the two factors. Then for every affine variety $Z$ and two morphisms $f_X : Z \rightarrow X$ and $f_Y : Z \rightarrow Y$ there is a unique morphism $f : Z \rightarrow X \times Y$ such that $f_X = \pi_X \circ f$ and $f_Y = \pi_Y \circ f$.

In particular, suppose I am given $X, Y$ which are affine varieties with their respective sheaves of $K$-valued functions, $\mathcal O_X$ and $\mathcal O_Y$. What would the sheaf of $K$-valued functions look like for $X \times Y$.

Since I have no idea what the sheaf looks like, this is only a shot in the dark at rest of the proof. But suppose $X, Y$ are affine varieties that are isomorphic to $X', Y'$ such that both $X'$ and $Y'$ are vanishing sets of sets of polynomials. Then I'm guessing $X \times Y$ would be isomorphic to $X' \times Y'$. So if I'm given morphisms $f_X : Z \rightarrow X $ and $f_Y: Z \rightarrow Y$, I can extend them to the morphisms $f_{X'}:Z \rightarrow X'$ and $f_{Y'}: Z \rightarrow Y'$ respectively.

My earlier instance of proving the theorem for vanishing sets (since for that proof, assuming $Z$ is a general affine variety is not a problem) will tell me that there is a unique morphism $f': Z \rightarrow X' \times Y'$ such that $f_{X'} = \pi_{X'} \circ f'$ and $f_{Y'} = \pi_{Y'} \circ f'$ If $g: X' \times Y' \rightarrow X \times Y$ is an isomorphism then $f = g \circ f' : Z \rightarrow X \times Y$ is a morphism.

I'm stuck at this point and don't know how to proceed to showing that $f$ has the required properties and is unique. Any help would be appreciated. Thank you very much! Sorry if this post is too long.

Infinite_Fun
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  • (Haven't looked at the notes, so if this seems like crazy talk, apologies). If $X=spec(A)$ and $Y=spec(B)$ are affine varieties over $k$, then $X\times_k Y=spec(A\otimes_k B)$ (fiber product), and over a one point space you would expect a fiber product to be the same as the product. Weird things can happen in algebraic geometry (e.g., the fiber product of two one point spaces need not be a one point space, $L\otimes_K L$ is not a field if $L$ is a field extension of $K$), but whatever definitions you are using, this should give you a good starting candidate for $X\times Y$. – Aaron Jul 07 '17 at 06:26
  • The coordinate ring of the product of two (reduced) algebraic varieties is $k[X]\otimes_k k[Y]$ which, under certain hypothesis about the field ($k$ perfect), is isomorphic to $k[x_1,\dots,x_n,y_1,\dots,y_m]/(I_X+I_Y)$, where $I_X$ and $I_Y$ are the radical ideals associated to each variety in $\mathbb{A}k^{n+m}$. Now, I assume Gathmann shows how to construct the sheaf $O{X\times Y}$ from the coordinate ring, right? See this post for exampIe: https://math.stackexchange.com/questions/1662307/ideal-of-product-of-affine-varieties – user347489 Jul 07 '17 at 08:00
  • Thank you very much. The tensor products and so on do go over my head since I don't really know much commutative algebra beyond what's covered in those notes and things like Nullstellensatz that I've taken on faith. Is there a more elementary way to describe the topology on $X \times Y$ and its sheaf? – Infinite_Fun Jul 07 '17 at 21:14

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