All Questions
77 questions
2
votes
0
answers
100
views
On a certain radical of the formal power series ring $K[[X_1,X_2,\ldots,X_{\infty}]]$
Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$ (the symbol $X_{\infty}$ is to ...
- 563
2
votes
0
answers
281
views
Galois cohomology of cyclotomic extension
Let $K$ be a complete discrete valuation ring with algebraically closed residue field $F$ of characteristic $p > 0$. Suppose ${\Bbb Q}_p \subset K$ and the absolute ramification index v$_{\pi_K}(p) ...
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1
vote
1
answer
584
views
Etale cohomology of singular varieties?
In all the literature I have read, etale cohomology is defined for smooth varieties. Suppose $X/\mathbb{Q}$ is a singular variety over $\mathbb{Q}$, does there exist etale cohomology $H_{\text{et}}^*(...
- 2,971
1
vote
2
answers
505
views
Base change of a finite morphism
Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...
- 563
1
vote
1
answer
315
views
About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
user497064
1
vote
2
answers
471
views
Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
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1
vote
1
answer
329
views
Kummer extension of Galois modules
Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...
- 11
1
vote
0
answers
140
views
Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
- 2,907
1
vote
0
answers
119
views
On the exponent of a certain matrix $A$ in characteristic $p > 0$
Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ ...
- 563
1
vote
0
answers
551
views
An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$
We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$
${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
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1
vote
0
answers
199
views
Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
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1
vote
0
answers
179
views
Spivakovski-Popescu-Neron desingularisation
For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...
- 781
1
vote
0
answers
485
views
On the coherence of formal power series ring
Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum c_{e_1,.....
- 781
1
vote
0
answers
201
views
Number of minimal primes for UFD
Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...
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0
votes
2
answers
2k
views
Tensor products of two domains
Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite.
That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
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0
votes
1
answer
278
views
Proof of the coherence of ${\Bbb F}_q[[X_1,\dots,X_{\infty}]]$
We shall define the infinitely-many-variable formal power series ring $A = {\Bbb F}_q[[X_1,\ldots,X_{\infty}]]$ over a finite field ${\Bbb F}_q$ as the following$\colon$
$A \colon= \underset{n \geq ...
- 563
0
votes
1
answer
353
views
Strange subscheme in ${\mathrm{Spec}} R \times {\Bbb A}^1_{\Bbb C}$
Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as
$R \...
- 563
0
votes
1
answer
147
views
Closure of non-closed subset in Ring theory
We say that the ring $R$ is topological, when we are given neighbourhoods $\{O_{\lambda};\lambda \in \Lambda\}$ of $0_R$.
We say ${\cal I}$ is a closed ideal of $R$ if and only if for any sub-index ...
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0
votes
1
answer
343
views
Relative Bertini Theorem
Let
$A \colon= {\Bbb C}[S_1,\ldots,S_n]$ with $1 \leq n < \infty$
$B \colon= A[X_1,\ldots,X_d]$ with $2 \leq d < \infty$.
$O \colon= (0,\ldots,0)$ be the origin of ${\mathrm{Spec}}\,B$.
...
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0
votes
1
answer
131
views
Radical of modules [closed]
Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define
$I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$,
where proper means ...
- 781
0
votes
1
answer
149
views
Power series rings and the formal generic fibre
Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
- 563
0
votes
0
answers
82
views
How geometry changes up to Hermitian inner product on Line bundle (Kodaira embedding)
Riemann metric $g \colon= \Sigma g_{ij} dx_i \otimes dx_j$ on a Kähler manifold $M$ will define the length of a line on $M$, i.e. intrinsic geometry. The line bundle $L$ on $M$ is equipped with a ...
- 563
0
votes
0
answers
287
views
On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
0
votes
0
answers
116
views
Gauss lemma for a complete Noetherian domain
Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
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0
votes
0
answers
100
views
Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
0
votes
0
answers
243
views
Uniqueness of decomposition of completely reducible representations
Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-...
- 1,040
-3
votes
1
answer
963
views
On the maximal ideal m of the formal power series ring [closed]
Let $A \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be the formal power series ring with infinitely many variables over a field $K$. We can represent it also by the following manner$\colon$
\begin{equation*...
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