All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,492 questions
36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
3
votes
3
answers
384
views
Basic question about completion of local ring
Let $(A,m)$ be a Noetherian local ring and $\tilde{A}$ its completion by the maximal ideal $m$.
Are the following three statements true?
(i)
If $\tilde{A}$ is free over $A$, then $A\cong \tilde{A}$
(...
- 1,144
5
votes
1
answer
151
views
Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 16.2k
1
vote
0
answers
104
views
Is a normal domain a filtered colimit of Noetherian normal domains?
As described in the title, is any normal domain a filtered colimit of Noetherian normal domains? It will be great if one can show this, even with additional conditions, or if one can provide a ...
- 1,024
1
vote
0
answers
215
views
Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 460
2
votes
1
answer
262
views
Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
- 333
2
votes
0
answers
169
views
Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
CommunityBot
- 1
40
votes
1
answer
1k
views
Rigid non-archimedean real closed fields
Update. The question has been recently answered in the positive by David Marker and Charles Steinhorn (as in indicated in Marker's answer). Note that Remark 3 below is now expanded by reference to a ...
- 3,530
3
votes
4
answers
944
views
What conditions are needed for $-\otimes_A B$ to be faithful?
For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-...
5
votes
0
answers
211
views
On the natural map $\mathrm{Br}(R) \rightarrow \mathrm{Br}(S)$ of Brauer groups
$\DeclareMathOperator\Br{Br}$Let $R$ be a commutative ring, and let $\Br(R)$ be the Brauer group of $R$ as defined by Auslander and Goldman. Let $S$ be a commutative $R$-algebra, and consider the ...
- 63.9k
3
votes
0
answers
181
views
Conditions for an open mapping between spectra
Let $(A,\mathfrak{m})$ be a Locally Noether Ring, and $\hat{A} = \varprojlim A/ \mathfrak{m}^{n}$ .Furthermore, let $f : A \to \hat{A}$ be a canonical morphism, and consider the mapping $f^{*} : Spec(\...
- 912
13
votes
2
answers
1k
views
When does a quasicoherent sheaf vanish?
Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the ...
- 2,806
9
votes
1
answer
319
views
Nonzero module with vanishing derived fibers
What's an example of a nonzero $R$-module with vanishing derived fibers at all points of $\mathrm{Spec}(R)$?
This was asked in When does a quasicoherent sheaf vanish?
but the answer there only says ...
- 2,346
1
vote
1
answer
67
views
An example of a commutative ring $R$ which has a proper right ideal which is not a right SIP $R$-module
Recall that a module $M_R$ ($R$ is a ring with unity) is called SIP if the intersection of any two summands of $M$ is also a summand. I asked before if there exists a commutative ring which is not an ...
- 356
2
votes
1
answer
181
views
Idempotent algebras over absolutely flat ring
Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...
- 2,346
2
votes
1
answer
232
views
Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 63.9k
3
votes
0
answers
33
views
Algorithm to determine closedness of orbits?
Consider a reductive group $G$ acting on an affine variety $X$. It is known that for every $x\in X$, we have $G.x\subseteq\overline{G.x}$ is open dense. Then $\partial({G.x})\subseteq X$ is a closed ...
- 930
5
votes
0
answers
185
views
Stone–Weierstrass theorem for topological fields
It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull ...
- 12.9k
3
votes
1
answer
168
views
Does there exists a "local slice" for an action $ \widehat{\mathbb{G}}_a $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?
Every action $ \beta $ of $ \mathbb{G}_a $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_0 \ast x) = \...
- 912
1
vote
1
answer
104
views
Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...
- 16.2k
3
votes
1
answer
242
views
Points of multiplicative groups
Let $R$ be a discrete valuation ring with residue field $k$. Denote by $\mathbb G_m:= R[x,1/x]$ the multiplicative group over $R$ and $\mathbb G_{m,k}:= k[x,1/x]$. If $B$ is a flat local $R$-algebra, ...
- 532
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
0
votes
0
answers
176
views
$\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, with each pair of $\{f,g,x\}$ not generating $\mathbb{C}(x,y)$
Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$.
Write,
$f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$
and
$g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$,
for some $n,m ...
- 2,837
9
votes
0
answers
180
views
How should we picture the set of monomial orders (= positive monoid orders on $\mathbb{N}^k$)?
Motivation: So apparently there's some sort of sport competition currently going on where I live, which leads to countries being given an element of $\mathbb{N}^3$ called a “medal count”, and not ...
- 32.4k
14
votes
0
answers
601
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
CommunityBot
- 1
0
votes
0
answers
53
views
A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 333
25
votes
7
answers
3k
views
When can we prove constructively that a ring with unity has a maximal ideal?
Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
- 35.4k
1
vote
1
answer
51
views
Exceptional Lenz-Barlotti classes IVa.3 and IVb.3
On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...
- 1,300
4
votes
1
answer
185
views
Order of pole of Poincaré series
Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
- 12.9k
3
votes
0
answers
189
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
1
answer
92
views
On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 6,262
2
votes
0
answers
213
views
Open problems in differential algebra and affine algebraic geometry
I am currently doing a PhD in differential algebra and affine algebraic geometry at the University of Buenos Aires. I've been struggling to find a list of interesting and big open problems in these ...
- 867
3
votes
0
answers
110
views
When is a ring complete with respect to its nilradical?
Let $R$ be a commutative ring and let $I$ be its nilradical. When is $R$ complete with respect to $I$?
For example, if $I$ is finitely generated, there exists $N$ such that $I^N = 0$ and thus $R$ is ...
- 2,338
1
vote
1
answer
149
views
$F=\mathbb{C}(u,v)$ satisfying: For every $a,b \in \mathbb{C}[y],c,d \in \mathbb{C}[x]$: $\mathbb{C}(x,y)=F(ax+b)=F(cy+d)$
Let $u,v \in \mathbb{C}[x,y]$, where $u$ and $v$ are algebraically independent over $\mathbb{C}$ and $F=\mathbb{C}(u,v)$. Of course, $d:=[\mathbb{C}(x,y):F] < \infty$.
Denote the following ...
- 2,837
1
vote
1
answer
86
views
Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 16.2k
2
votes
1
answer
112
views
Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 16.2k
0
votes
1
answer
473
views
A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
- 2,837
1
vote
0
answers
59
views
If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$,
$F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
- 2,837
9
votes
3
answers
2k
views
Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 11
0
votes
1
answer
219
views
Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 148k
6
votes
1
answer
224
views
Integral preimages of topologically noetherian, affine schemes
Let $A\to B$ be a ring homomorphism, $d\in \mathbb{Z}_{>0}$ and let $C=\bigotimes^d_A B$ the $d$-fold tensor product of $B$ over $A$. Then $\mathfrak{S}_d$, the symmetric group of $d$ elements, ...
- 1,144
2
votes
0
answers
93
views
Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
- 480
25
votes
3
answers
2k
views
Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
- 1,726
0
votes
0
answers
42
views
When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)
Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$,
$1 \leq i \leq n$, $1 \leq j \leq m$, be such that
$f_i(x,0) \neq 0$ and $g_j(x,0)=0$.
Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
- 2,837
2
votes
1
answer
205
views
Localization of quasi-excellent rings are quasi-excellent?
If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ?
I think Matsumura's commutative ring theory book says that localization of ...
- 1,813
2
votes
1
answer
245
views
Characterization of a certain class of modules-broader than Noetherian
Let $R$ be a commutative ring with $1$.
An $R$-module $K$ has the 'S' property if $K/T \simeq K$ (i.e. isomorphic) implies that the submodule $T$ is trivial.
By Fitting's lemma any Noetherian module ...
- 63.9k
0
votes
0
answers
31
views
Formalization of the independance of products in a (commutative) semigroup
1/ It is well known that associativity implies that the result of the product of an ordered finite set of elements in a semigroup does not depend of the order of composition of the partial products.
...
- 2,655
1
vote
1
answer
135
views
Artinian Gorenstein subrings with same socle degree
I am looking for examples of Artinian Gorenstein subalgebras with the same socle degrees.
More precisely, let $A$ be an Artinian Gorenstein $k$-algebra (with $k$ algebraically closed of characteristic ...
- 50.8k
3
votes
1
answer
501
views
Description of prime ideals in $\operatorname{Spec}(\mathbb{Z}[x_1, \dots, x_n])$
Edit: This seems to be wrong, as pointed out by Will Sawin in the comments.
The prime ideals of $\mathbb{Z}$ and $\mathbb{Z}[x]$ are well-known. It is also not too hard to compute the underlying set ...
- 35.4k