0
votes
1answer
48 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
1
vote
1answer
81 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
2
votes
2answers
188 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
0
votes
0answers
23 views

counterexample of formal power series over a commutative ring with identity [duplicate]

Let $A$ be a commutative ring with identity. Let $A[[x]]=\{\sum_{i=0}^{\infty}a_ix^i\mid a_i\in A\}$. Then it can be shown that if $f(x)\in A[[x]]$ is nilpotent, then $a_i$ are nilpotent for all $i$. ...
2
votes
0answers
95 views

Irreducible polynomials as formal power series

I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. ...
5
votes
1answer
362 views

Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form $$f(x,y) = f_{r}(x,y) + ...
-1
votes
1answer
94 views

The completness of ring and its power series ring

Let $R$ be a ring and $I$ an ideal of $R$. If $R$ is $I$-adic complete, why then $R[[x]]$ is $J:=IR[[x]]+(x)R[[x]]$-adic complete? (Commutative Ring Theory, H. Matsumura) Take a Cauchy sequence ...
1
vote
1answer
57 views

A problem on power series

Let $P(T) = \frac { 1 + T + T^2 + \cdots + T^m}{ 1 - T^2( 1 + T + T^2 + \cdots + T^n)} = \sum _{i = 0}^\infty \beta_n T^n$ be a formal power series expansion. This kind of series arose while I was ...
7
votes
1answer
164 views

Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication

Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...
5
votes
1answer
148 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
6
votes
2answers
293 views

Ring of formal power series finitely generated as algebra?

I'm asked if the ring of formal power series is finitely generated as a $K$-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
3
votes
1answer
176 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
6
votes
1answer
83 views

When is k[x,y]/I complete for the (x,y)-adic topology?

Let $k$ be a field. If necessary, add assumptions on $k$ or just take $k=\mathbb{C}$. It is easy to classify the ideals $I \subseteq k[x]$ such that $k[x]/I \to k[[x]]/(I)$ is an isomorphism, namely ...
8
votes
4answers
663 views

what is the fraction field of $R[[x]]$, the power series over some ring?

I have a question similar to 74335. Let $R$ be an integral domain. Is there a nice description of the fraction field of the power series $R[[x]]$? I know that this field can be a proper subfield of ...
1
vote
0answers
84 views

subrings A of the ring of power series k[[t]] with the condition (A : k[[t]]) $\neq${0} and k $\subset$ A

I would like to understand the structure of the subrings A of the ring of formal power series k[[t]] (where k is a field) which satisfy the condition (A : k[[t]]) $\neq$ {0} and k $\subset$ A. Are ...