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added 4 characters in body; edited title
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Harry Gindi
Harry Gindi
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The structure of the mudule relative differential forms module of Kähler differentials of R[[x]] over R

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers. 

We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of $R((x))$ over $R$----remember that we take $R$ to be the rationals which is of char $0$. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general $R$'s?

The structure of the mudule relative differential forms of R[[x]] over R

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of $R((x))$ over $R$----remember that we take $R$ to be the rationals which is of char $0$. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general $R$'s?

The structure of the module of Kähler differentials of R[[x]] over R

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers. 

We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of $R((x))$ over $R$----remember that we take $R$ to be the rationals which is of char $0$. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general $R$'s?

some latex corrections
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Martin Brandenburg
Martin Brandenburg
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It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take R to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a R((x))-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of R((x)) over R----remember that we take R to be the rationals which is of char 0. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general R's?

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of $R((x))$ over $R$----remember that we take $R$ to be the rationals which is of char $0$. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general $R$'s?

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take R to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a R((x))-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of R((x)) over R----remember that we take R to be the rationals which is of char 0. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general R's?

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of $R((x))$ over $R$----remember that we take $R$ to be the rationals which is of char $0$. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general $R$'s?

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zcqc
zcqc
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The structure of the mudule relative differential forms of R[[x]] over R

It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take R to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a R((x))-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of R((x)) over R----remember that we take R to be the rationals which is of char 0. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general R's?