Computation of tensor product
Let $k$ be a field.
Consider the map $k[X,X^{-1},Y, Y^{-1}] to k[T, T^{-1}] $ where $X to T$ and
$Y to T$.
I'm interested in computing the tensor product
$ k[T,T^{-1}] otimes_{k[X,X^{-1},Y, Y^{-1}]} k[X^{1/n},Y^{1/n},X^{-1/n},Y^{-1/n}] $.
I'm not sure how to proceed with this computation. Any help would be appreciated
abstract-algebra tensor-products
asked 16 hours ago
grontim
308110
add a comment |
Let $k$ be a field.
Consider the map $k[X,X^{-1},Y, Y^{-1}] to k[T, T^{-1}] $ where $X to T$ and
$Y to T$.
I'm interested in computing the tensor product
$ k[T,T^{-1}] otimes_{k[X,X^{-1},Y, Y^{-1}]} k[X^{1/n},Y^{1/n},X^{-1/n},Y^{-1/n}] $.
I'm not sure how to proceed with this computation. Any help would be appreciated
abstract-algebra tensor-products
asked 16 hours ago
grontim
308110
add a comment |
Let $k$ be a field.
Consider the map $k[X,X^{-1},Y, Y^{-1}] to k[T, T^{-1}] $ where $X to T$ and
$Y to T$.
I'm interested in computing the tensor product
$ k[T,T^{-1}] otimes_{k[X,X^{-1},Y, Y^{-1}]} k[X^{1/n},Y^{1/n},X^{-1/n},Y^{-1/n}] $.
I'm not sure how to proceed with this computation. Any help would be appreciated
abstract-algebra tensor-products
asked 16 hours ago
grontim
308110
Let $k$ be a field.
Consider the map $k[X,X^{-1},Y, Y^{-1}] to k[T, T^{-1}] $ where $X to T$ and
$Y to T$.
I'm interested in computing the tensor product
$ k[T,T^{-1}] otimes_{k[X,X^{-1},Y, Y^{-1}]} k[X^{1/n},Y^{1/n},X^{-1/n},Y^{-1/n}] $.
I'm not sure how to proceed with this computation. Any help would be appreciated
abstract-algebra tensor-products
abstract-algebra tensor-products
asked 16 hours ago
grontim
308110
asked 16 hours ago
grontim
308110
asked 16 hours ago
grontim
308110
asked 16 hours ago
grontim
308110
asked 16 hours ago
grontim
308110
308110
add a comment |
add a comment |
1 Answer
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Use the lemma $R/I otimes_R S cong S/IS$, and then simplify.
Capital variables are very easy to mistype, so I'll use lower case.
Let $R=k[x,y,x^{-1},y^{-1}]cong k[x,y,z,w]/(xz-1,yw-1)$, then $k[t,t^{-1}]cong R/(x-y)$, since then $z-w = (xzw-w)-(yzw-z)$ is contained in the ideal automatically. Also $S:=k[x^{1/n},y^{1/n},x^{-1/n},y^{-1/n}]cong R[a,b]/(a^n-x,b^n-y)$.
Thus $$k[t,t^{-1}]otimes_R Scong R/(x-y)otimes_R R[a,b]/(a^n-x,b^n-y)$$
$$cong R[a,b]/(a^n-x,b^n-y,x-y)$$
$$cong k[x,y,z,w,a,b]/(xz-1,yw-1,a^n-x,b^n-y,x-y)$$
$$cong k[x,z,a,b]/(xz-1,a^n-x,b^n-x)$$
$$cong k[a,b,u]/(a^n-b^n,au-1)$$
answered 9 hours ago
jgon
13.1k21941
add a comment |
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1 Answer
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1 Answer
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active
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active
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Use the lemma $R/I otimes_R S cong S/IS$, and then simplify.
Capital variables are very easy to mistype, so I'll use lower case.
Let $R=k[x,y,x^{-1},y^{-1}]cong k[x,y,z,w]/(xz-1,yw-1)$, then $k[t,t^{-1}]cong R/(x-y)$, since then $z-w = (xzw-w)-(yzw-z)$ is contained in the ideal automatically. Also $S:=k[x^{1/n},y^{1/n},x^{-1/n},y^{-1/n}]cong R[a,b]/(a^n-x,b^n-y)$.
Thus $$k[t,t^{-1}]otimes_R Scong R/(x-y)otimes_R R[a,b]/(a^n-x,b^n-y)$$
$$cong R[a,b]/(a^n-x,b^n-y,x-y)$$
$$cong k[x,y,z,w,a,b]/(xz-1,yw-1,a^n-x,b^n-y,x-y)$$
$$cong k[x,z,a,b]/(xz-1,a^n-x,b^n-x)$$
$$cong k[a,b,u]/(a^n-b^n,au-1)$$
answered 9 hours ago
jgon
13.1k21941
add a comment |
Use the lemma $R/I otimes_R S cong S/IS$, and then simplify.
Capital variables are very easy to mistype, so I'll use lower case.
Let $R=k[x,y,x^{-1},y^{-1}]cong k[x,y,z,w]/(xz-1,yw-1)$, then $k[t,t^{-1}]cong R/(x-y)$, since then $z-w = (xzw-w)-(yzw-z)$ is contained in the ideal automatically. Also $S:=k[x^{1/n},y^{1/n},x^{-1/n},y^{-1/n}]cong R[a,b]/(a^n-x,b^n-y)$.
Thus $$k[t,t^{-1}]otimes_R Scong R/(x-y)otimes_R R[a,b]/(a^n-x,b^n-y)$$
$$cong R[a,b]/(a^n-x,b^n-y,x-y)$$
$$cong k[x,y,z,w,a,b]/(xz-1,yw-1,a^n-x,b^n-y,x-y)$$
$$cong k[x,z,a,b]/(xz-1,a^n-x,b^n-x)$$
$$cong k[a,b,u]/(a^n-b^n,au-1)$$
answered 9 hours ago
jgon
13.1k21941
add a comment |
Use the lemma $R/I otimes_R S cong S/IS$, and then simplify.
Capital variables are very easy to mistype, so I'll use lower case.
Let $R=k[x,y,x^{-1},y^{-1}]cong k[x,y,z,w]/(xz-1,yw-1)$, then $k[t,t^{-1}]cong R/(x-y)$, since then $z-w = (xzw-w)-(yzw-z)$ is contained in the ideal automatically. Also $S:=k[x^{1/n},y^{1/n},x^{-1/n},y^{-1/n}]cong R[a,b]/(a^n-x,b^n-y)$.
Thus $$k[t,t^{-1}]otimes_R Scong R/(x-y)otimes_R R[a,b]/(a^n-x,b^n-y)$$
$$cong R[a,b]/(a^n-x,b^n-y,x-y)$$
$$cong k[x,y,z,w,a,b]/(xz-1,yw-1,a^n-x,b^n-y,x-y)$$
$$cong k[x,z,a,b]/(xz-1,a^n-x,b^n-x)$$
$$cong k[a,b,u]/(a^n-b^n,au-1)$$
answered 9 hours ago
jgon
13.1k21941
Use the lemma $R/I otimes_R S cong S/IS$, and then simplify.
Capital variables are very easy to mistype, so I'll use lower case.
Let $R=k[x,y,x^{-1},y^{-1}]cong k[x,y,z,w]/(xz-1,yw-1)$, then $k[t,t^{-1}]cong R/(x-y)$, since then $z-w = (xzw-w)-(yzw-z)$ is contained in the ideal automatically. Also $S:=k[x^{1/n},y^{1/n},x^{-1/n},y^{-1/n}]cong R[a,b]/(a^n-x,b^n-y)$.
Thus $$k[t,t^{-1}]otimes_R Scong R/(x-y)otimes_R R[a,b]/(a^n-x,b^n-y)$$
$$cong R[a,b]/(a^n-x,b^n-y,x-y)$$
$$cong k[x,y,z,w,a,b]/(xz-1,yw-1,a^n-x,b^n-y,x-y)$$
$$cong k[x,z,a,b]/(xz-1,a^n-x,b^n-x)$$
$$cong k[a,b,u]/(a^n-b^n,au-1)$$
answered 9 hours ago
jgon
13.1k21941
answered 9 hours ago
jgon
13.1k21941
answered 9 hours ago
jgon
13.1k21941
answered 9 hours ago
jgon
13.1k21941
13.1k21941
add a comment |
add a comment |
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