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I hope these questions are not too trivial.

Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c \rangle $$ is thought to be restricting to the fiber over $t=c$.

On the other hand, considering $\mathbb{R}$, $$ \mathbb{R}\otimes_{\,\mathbb{R}}\mathbb{C} $$ is thought to be a base extension.

Question 1: So tensoring is not only thought of as a restriction, but it is also thought of as an extension? Why do we need or when do we use base extensions?

Question 2: Geometrically, what are

  1. $\operatorname{Spec}(\mathbb{C}[s]\otimes_{\,\mathbb{Z}}\mathbb{C}[u,v])$?

  2. $\operatorname{Spec}(\mathbb{C}[s]\otimes_{\,\mathbb{R}}\mathbb{C}[u,v])$?

  3. $\operatorname{Spec}(\mathbb{C}[s]\oplus\mathbb{C}[u,v])$?

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In the former case, you have a quotient of the base ring $\mathbb{C}[t]$; in the latter, an extension of the base ring $\mathbb{R}$; tensoring gives you a way of changing the "ring of scalars"; change it to a quotient, you are "restricting"; change it to an extension, you are "extending". –  Arturo Magidin Jul 7 '12 at 19:30
    
Thank you Arturo. What you said seems to clarify a lot of the misunderstandings I've had for awhile... –  math-visitor Jul 7 '12 at 19:33
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