Tensor product of short exact sequences
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Let $k$ be a field and $R=k[x_j:jin J]/I$ be a smooth $k$-Algebra. Then $R otimes R$ is also a smooth $k$-Algebra and by the universal property of the tensor product, we obtain a (surjective) map $k[x_i:iin J] otimes k[x_j :jin J] rightarrow Rotimes R$.
Can we say anything about the kernel? If yes, how Is it related to $I otimes k[x_j :jin J] +k[x_j:iin J] otimes I$?
commutative-algebra tensor-products
asked Aug 27 at 12:06
slinshady
355110
add a comment |Â
up vote
1
down vote
favorite
Let $k$ be a field and $R=k[x_j:jin J]/I$ be a smooth $k$-Algebra. Then $R otimes R$ is also a smooth $k$-Algebra and by the universal property of the tensor product, we obtain a (surjective) map $k[x_i:iin J] otimes k[x_j :jin J] rightarrow Rotimes R$.
Can we say anything about the kernel? If yes, how Is it related to $I otimes k[x_j :jin J] +k[x_j:iin J] otimes I$?
commutative-algebra tensor-products
asked Aug 27 at 12:06
slinshady
355110
1
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $k$ be a field and $R=k[x_j:jin J]/I$ be a smooth $k$-Algebra. Then $R otimes R$ is also a smooth $k$-Algebra and by the universal property of the tensor product, we obtain a (surjective) map $k[x_i:iin J] otimes k[x_j :jin J] rightarrow Rotimes R$.
Can we say anything about the kernel? If yes, how Is it related to $I otimes k[x_j :jin J] +k[x_j:iin J] otimes I$?
commutative-algebra tensor-products
asked Aug 27 at 12:06
slinshady
355110
Let $k$ be a field and $R=k[x_j:jin J]/I$ be a smooth $k$-Algebra. Then $R otimes R$ is also a smooth $k$-Algebra and by the universal property of the tensor product, we obtain a (surjective) map $k[x_i:iin J] otimes k[x_j :jin J] rightarrow Rotimes R$.
Can we say anything about the kernel? If yes, how Is it related to $I otimes k[x_j :jin J] +k[x_j:iin J] otimes I$?
commutative-algebra tensor-products
asked Aug 27 at 12:06
slinshady
355110
edited Aug 27 at 13:58
asked Aug 27 at 12:06
slinshady
355110
asked Aug 27 at 12:06
slinshady
355110
asked Aug 27 at 12:06
slinshady
355110
355110
1
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43
add a comment |Â
1
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43
1
1
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43
add a comment |Â
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1
The kernel should look more like $Iotimes 1+1otimes I$.
– Pedro Tamaroff♦
Aug 27 at 12:30
Indeed you are right. This is also what I wanted to express, that was a bit careless off me.
– slinshady
Aug 27 at 12:43