I whould like to konw why the following map is an isomorphism :
$ K[X\_1,...,X\_n]/(f\_1,...,f\_r) \otimes K [Y\_1,...,Y\_n]/(g\_1,...g\_k) 
\simeq K[X\_1,...,X\_n, Y\_1,...,Y\_m] / (f\_1,...,f\_r,g\_1,...g\_k) $ which $ K $ is an algebraically closed field.
Here is my suggestion :
We put : $ \varphi \ : \ K[X\_1,...,X\_n]/(f\_1,...,f\_r) \times K [Y\_1,...,Y\_n]/(g\_1,...g\_k) \to K[X\_1 , ... , X\_n, Y\_1 , ... , Y\_m ] / (f\_1 , ..., f\_r , g\_1 , ... , g\_k) $ défined by : $ \varphi (P,Q) = PQ $ which is bilinear. So, according to the universal property of the tensor product, we obtain the the $ K $ - linear map :
$ f : K[X\_1,...,X\_n]/(f\_1,...,f\_r) \otimes K [Y\_1,...,Y\_n]/(g\_1,...g\_k) 
\simeq K[X\_1,...,X\_n, Y\_1,...,Y\_m] / (f\_1,...,f\_r,g\_1,...g\_k) $
so the question : why $ f $ is an isomorphism ? How can we show it. And sorry about my english language, i'm not a british men, sorry.
Thanks a lot.