Completion of DVR
$begingroup$
Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
Let $hat{R}$ to be the $(t)$-adic completion of $R$.
Then it seems that every element $alphainhat{R}$ can be written in
$$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
where $a_inotin(t)$ so that $a_i$'s are unit.
$t$ just works as it is an indeterminate w.r.t $R^times$.
Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
So this is my question: is it really possible to write the elements of $R^times$ as in (1)?
abstract-algebra commutative-algebra
asked Dec 17 '18 at 5:11
user190964user190964
755
$endgroup$
add a comment |
$begingroup$
Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
Let $hat{R}$ to be the $(t)$-adic completion of $R$.
Then it seems that every element $alphainhat{R}$ can be written in
$$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
where $a_inotin(t)$ so that $a_i$'s are unit.
$t$ just works as it is an indeterminate w.r.t $R^times$.
Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
So this is my question: is it really possible to write the elements of $R^times$ as in (1)?
abstract-algebra commutative-algebra
asked Dec 17 '18 at 5:11
user190964user190964
755
$endgroup$
add a comment |
$begingroup$
Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
Let $hat{R}$ to be the $(t)$-adic completion of $R$.
Then it seems that every element $alphainhat{R}$ can be written in
$$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
where $a_inotin(t)$ so that $a_i$'s are unit.
$t$ just works as it is an indeterminate w.r.t $R^times$.
Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
So this is my question: is it really possible to write the elements of $R^times$ as in (1)?
abstract-algebra commutative-algebra
asked Dec 17 '18 at 5:11
user190964user190964
755
$endgroup$
Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
Let $hat{R}$ to be the $(t)$-adic completion of $R$.
Then it seems that every element $alphainhat{R}$ can be written in
$$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
where $a_inotin(t)$ so that $a_i$'s are unit.
$t$ just works as it is an indeterminate w.r.t $R^times$.
Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
So this is my question: is it really possible to write the elements of $R^times$ as in (1)?
abstract-algebra commutative-algebra
abstract-algebra commutative-algebra
asked Dec 17 '18 at 5:11
user190964user190964
755
asked Dec 17 '18 at 5:11
user190964user190964
755
asked Dec 17 '18 at 5:11
user190964user190964
755
asked Dec 17 '18 at 5:11
user190964user190964
755
asked Dec 17 '18 at 5:11
user190964user190964
755
755
add a comment |
add a comment |
1 Answer
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$begingroup$
Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
and we may represent elements of this inverse limit concretely as compatible sequences
$$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
$$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)
Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.
Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
$endgroup$
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
add a comment |
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1 Answer
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$begingroup$
Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
and we may represent elements of this inverse limit concretely as compatible sequences
$$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
$$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)
Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.
Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
$endgroup$
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
add a comment |
$begingroup$
Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
and we may represent elements of this inverse limit concretely as compatible sequences
$$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
$$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)
Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.
Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
$endgroup$
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
add a comment |
$begingroup$
Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
and we may represent elements of this inverse limit concretely as compatible sequences
$$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
$$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)
Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.
Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
$endgroup$
Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
and we may represent elements of this inverse limit concretely as compatible sequences
$$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
$$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)
Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.
Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
edited Dec 17 '18 at 6:39
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
answered Dec 17 '18 at 5:24
StahlStahl
16.7k43355
16.7k43355
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
add a comment |
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
$begingroup$
Really thanks for indicating references.
$endgroup$
– user190964
Dec 21 '18 at 7:27
add a comment |
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