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I want to show that the product of smooth quasi-projective varieties is smooth.

Call my varieties $A$ and $B$. Since both are smooth, at every point the dimension of the tangent space equals the dimension of the variety. Will the dimension of $A \times B$ be the sum of the dimensions of $A$ and $B$? If so, how can I show that the same thing happens to the dimension of the tangent space at any point?

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    Whatever definition of smoothness you are using, it will be local. Can you reduce the question to showing that the tensor product of two smooth local rings over a field is smooth?2011-02-24

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