Tensor product of finitely generated algebras
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.
algebraic-geometry commutative-algebra tensor-products
asked Nov 24 at 20:55
user567863
1913
add a comment |
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.
algebraic-geometry commutative-algebra tensor-products
asked Nov 24 at 20:55
user567863
1913
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.
– Qiaochu Yuan
Nov 24 at 20:57
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
1
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24
add a comment |
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.
algebraic-geometry commutative-algebra tensor-products
asked Nov 24 at 20:55
user567863
1913
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.
algebraic-geometry commutative-algebra tensor-products
algebraic-geometry commutative-algebra tensor-products
asked Nov 24 at 20:55
user567863
1913
asked Nov 24 at 20:55
user567863
1913
asked Nov 24 at 20:55
user567863
1913
asked Nov 24 at 20:55
user567863
1913
asked Nov 24 at 20:55
user567863
1913
1913
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.
– Qiaochu Yuan
Nov 24 at 20:57
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
1
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24
add a comment |
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.
– Qiaochu Yuan
Nov 24 at 20:57
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
1
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24
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.
– Qiaochu Yuan
Nov 24 at 20:57
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.
– Qiaochu Yuan
Nov 24 at 20:57
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
1
1
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24
add a comment |
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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.
– Qiaochu Yuan
Nov 24 at 20:57
@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 $A_1otimes _{K}A_2$ as a quotient of a polynomial ring in $m+n$ variables?
– user567863
Nov 24 at 21:39
1
@Qiaochu: yes, it's $K[x_1, dots x_n, y_1, dots y_m]/(f_1, dots f_{ell}, g_1, dots g_k)$.
– Qiaochu Yuan
Nov 25 at 1:15
@Qiaochu Thanks, I will try to prove it.
– user567863
Nov 25 at 1:24