Mixing
Dimension of affine affine algebras as a module
3
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Suppose that $Acong mathbb{R}[f_1,dots,f_d]$ is a (commutative) affine $mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $mathbb{R}[x_1,dots,x_N]$. When is $A$ a finite-dimensional $mathbb{R}$-module?
Some examples are
- Simple $mathbb{R}$-algebras,
$mathbb{R}[x]/[x^n]$ (which is an $n$-dimensional vector space)...
commutative-algebra modules dimension-theory
asked Jan 23 at 12:50
N00berN00ber
324111
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add a comment |
3
$begingroup$
Suppose that $Acong mathbb{R}[f_1,dots,f_d]$ is a (commutative) affine $mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $mathbb{R}[x_1,dots,x_N]$. When is $A$ a finite-dimensional $mathbb{R}$-module?
Some examples are
- Simple $mathbb{R}$-algebras,
$mathbb{R}[x]/[x^n]$ (which is an $n$-dimensional vector space)...
commutative-algebra modules dimension-theory
asked Jan 23 at 12:50
N00berN00ber
324111
$endgroup$
$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51
add a comment |
3
3
3
$begingroup$
Suppose that $Acong mathbb{R}[f_1,dots,f_d]$ is a (commutative) affine $mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $mathbb{R}[x_1,dots,x_N]$. When is $A$ a finite-dimensional $mathbb{R}$-module?
Some examples are
- Simple $mathbb{R}$-algebras,
$mathbb{R}[x]/[x^n]$ (which is an $n$-dimensional vector space)...
commutative-algebra modules dimension-theory
asked Jan 23 at 12:50
N00berN00ber
324111
$endgroup$
Suppose that $Acong mathbb{R}[f_1,dots,f_d]$ is a (commutative) affine $mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $mathbb{R}[x_1,dots,x_N]$. When is $A$ a finite-dimensional $mathbb{R}$-module?
Some examples are
- Simple $mathbb{R}$-algebras,
$mathbb{R}[x]/[x^n]$ (which is an $n$-dimensional vector space)...
commutative-algebra modules dimension-theory
commutative-algebra modules dimension-theory
asked Jan 23 at 12:50
N00berN00ber
324111
asked Jan 23 at 12:50
N00berN00ber
324111
edited Jan 24 at 9:50
N00ber
asked Jan 23 at 12:50
N00berN00ber
324111
asked Jan 23 at 12:50
N00berN00ber
324111
asked Jan 23 at 12:50
N00berN00ber
324111
324111
$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51
add a comment |
$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51
$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51
add a comment |
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$begingroup$
What are $f_i$'s? By the way, your second example is missing $x$.
$endgroup$
– Youngsu
Jan 23 at 14:29
$begingroup$
They're polynomials in a polynomial ring (coming from Hilbert's basis theorem)...I made the updates in the question. Thanks for pointing it out :)
$endgroup$
– N00ber
Jan 24 at 9:51