Let $X$ and $Y$ be two possibly non reduced analytic spaces over a third analytic space $S$. I have not been able to find a reference where the fibre product $X\times_S Y$ is constructed !!!! Any help would be appreciated.

My guess is that the topological space is just the fibre product of topological spaces; and, at least when $S$ is a point, the structure sheaf is the sheaf associated to the pre-sheaf which on $U\times V$ is $\mathcal{O}_X(U)\otimes_{\mathbb{C}}\mathcal{O}_Y(V)$.

I really would like to have a reference!

  • 3
    $\begingroup$ The fiber product is defined for instance in the book "Several Complex Variables IV: Algebraic Aspects of Complex Analysis", page 116. $\endgroup$
    – HYL
    Apr 21, 2017 at 9:23
  • $\begingroup$ Thanks! This is what I was looking for. You are welcome to write it as an answer $\endgroup$
    – Giulio
    Apr 21, 2017 at 9:28

1 Answer 1


Apart from the reference given in the comment, you can also find a proof of the existence of the fiber product in Fischer's "Complex Analytic Geometry", Corollary 0.32.

For direct products, a more straightforward construction can be found in Grauert and Remmert's "Coherent Analytic sheaves", Chapter 1.3. As they pointed out, their method avoids analytic tensor products of the structure sheaves.


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