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I am trying to compute tensor product of finitely generated algebras over a field. I was able to compute few special tensor products. Is there a general technique which allows one to compute the tensor product for any two finitely generated algebras over a field. I'm looking for this because im interested in computing fibre product of schemes.

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  • $\begingroup$ It depends on what you mean by "compute." It's not hard to write down a presentation of the tensor product, but that doesn't necessarily tell you much. $\endgroup$ Nov 24, 2018 at 20:57
  • $\begingroup$ @Qiaochu If we take $A_1=\frac{K[X_1,...,X_n]}{(f_1,...,f_l)}$ and $A_2=\frac{K[Y_1,...,Y_m]}{(g_1,...,g_r)}$, then is it possible to express $A_1\otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables? $\endgroup$
    – user567863
    Nov 24, 2018 at 21:39
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    $\begingroup$ @Qiaochu: yes, it's $K[x_1, \dots x_n, y_1, \dots y_m]/(f_1, \dots f_{\ell}, g_1, \dots g_k)$. $\endgroup$ Nov 25, 2018 at 1:15
  • $\begingroup$ @Qiaochu Thanks, I will try to prove it. $\endgroup$
    – user567863
    Nov 25, 2018 at 1:24

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