All Questions
6,053 questions
7
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Krull dimension of a completion
How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull ...
- 85.6k
11
votes
2
answers
869
views
Why is the prime spectrum not useful in non-archimedean analytic geometry?
This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the notes of Conrad.
Reading Conrad's notes (and e.g. those of ...
- 57.6k
5
votes
3
answers
3k
views
Generalized Chinese Remainder Theorem
Let $U,V$ be submodules of a $R$-module $M$. Then the diagonal induces an isomorphism
$M/(U \cap V) \to M/U \times_{M/(U+V)} M/V.$
This is a (useful!) generalization of the Chinese Remainder Theorem ...
- 12.3k
3
votes
1
answer
1k
views
Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
- 161
3
votes
0
answers
325
views
Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
- 30.5k
4
votes
1
answer
1k
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Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.
Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...
- 373
10
votes
2
answers
610
views
When is tensoring with a module representable by a scheme?
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...
- 27k
5
votes
3
answers
752
views
Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
- 39.9k
14
votes
1
answer
1k
views
Two questions about Cohen-Macaulay rings
The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.
Let $A$ be a local Cohen-Macaulay ring of dimension $d$.
Let $I$ be an ...
CommunityBot
- 1
3
votes
1
answer
1k
views
Lifting results from smooth maps to essentially smooth maps.
Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is essentially finitely presented provided that $...
- 85.6k
9
votes
0
answers
281
views
Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 22.1k
15
votes
6
answers
1k
views
An example of a series that is not differentially algebraic?
Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
CommunityBot
- 1
13
votes
2
answers
967
views
Does torsion-freeness of class group localize?
Let $R$ be a local normal domain, and let $P \in Spec (R)$. It is well known that $Cl(R) \to Cl(R_P)$ is surjective. However, I do not know any example where $Cl(R)$ is torsion-free, but $Cl(R_P)$ is ...
- 10.5k
2
votes
2
answers
1k
views
Maximal ideal of codimension >1
To assuage my conscience over an unsourced statement in a paper I'm writing:
I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or ...
- 65.4k
4
votes
1
answer
3k
views
When are intersections of finitely generated field extensions finitely generated?
Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be ...
- 63.1k
7
votes
1
answer
726
views
Do all Dedekind domains have the "Riemann-Roch property"?
Let $R$ be a Dedekind domain with fraction field $K$.
Say that a Dedekind domain $R$ has the Riemann-Roch property if: for every nonzero prime ideal $\mathfrak{p}$ of $R$, there exists an element $f ...
- 10.5k
4
votes
1
answer
358
views
Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
- 44.4k
14
votes
1
answer
7k
views
When is the set of zero divisors equal to the union of the minimal primes in a reduced ring?
It is straightforward to show, for example, that the set of zero divisors of a (commutative unital) reduced Noetherian ring is precisely the union of its minimal primes. When else can we say that the ...
- 19.6k
3
votes
4
answers
1k
views
Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
CommunityBot
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4
votes
1
answer
1k
views
What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?
$\mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $\mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $\...
- 10.7k
6
votes
2
answers
738
views
A reference: the splitting principle for exterior powers of coherent sheaves?
It's well known that if E is a vector bundle with Chern roots $a_1,\ldots, a_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a_i$'s. I would like to say ...
- 30.5k
5
votes
0
answers
388
views
is there a notion of weakly noetherian?
A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
- 403
0
votes
1
answer
370
views
Proving that two local PIDs, one inside the other, with the same field of fractions are equal.
Let $R\subset S$ be two local PIDs that have the same fields of fractions. How to prove that they are equal?
CommunityBot
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3
votes
1
answer
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How to prove that the subrings of the rational numbers are noetherian?
I have managed to prove that the aforementioned subrings are in bijection with the subsets of the primes, however, I haven't been able to prove that they are all noetherian. I need help.
- 1,417
3
votes
1
answer
602
views
a question about flatness
In the book "étale cohomology" by Milne, proposition 2.5 at p.9, it said :
Let $B$ be a flat $A-$algebra where $A$ and $B$ are noetherian rings, and consider $b \in B$. If the image of $b$ in $B/mB$ ...
- 156k
1
vote
1
answer
963
views
Question on an exercise in Hartshorne: Equivalence of categories
This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.
Let $K$ be a field and $S=K[X_0,\ldots,X_n]$ a graded ring. Set $X=Proj(S)$ and ...
- 16k
6
votes
2
answers
976
views
Question on a theorem of Eisenbud's and Harris' "The geometry of schemes"
My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\...
- 57.6k
1
vote
1
answer
474
views
Expressing fiber product of affines via an ideal
Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$).
Let $Z$ be the affine scheme defined by the ideal $L$...
- 23.3k
7
votes
5
answers
2k
views
Does a locally free sheaf over a product pushforward to a locally free sheaf?
Suppose $X$ and $Y$ are two (smooth, affine) algebraic varieties. Let $\mathcal{F}$ be a locally free coherent sheaf over $X \times Y$, and let $\mathcal{G}$ be the pushforward of $\mathcal{F}$ to $X$...
- 21
5
votes
1
answer
3k
views
Does the category Monoid of monoids have finite coproducts?
Does the category Monoid of monoids have finite coproducts?
- 156k
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 65.4k
2
votes
3
answers
1k
views
Commutative Noetherian Domains of Krull Dimension One
k is an alegraically closed field and A is a commutative k-algebra. We also know that A is a Noetherian domain and its Krull dimension is one. Are there any necessary and sufficient conditions on A ...
- 57.6k
2
votes
1
answer
1k
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Example of restriction of a finite morphism which is not finite
Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\...
CommunityBot
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1
vote
1
answer
1k
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Example of inclusion which is not a finite morphism [closed]
Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
...
CommunityBot
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28
votes
3
answers
3k
views
Why is "h" the notation for class numbers?
A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\...
- 4,279
10
votes
2
answers
1k
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Complete intersections and flat families
If I have a flat family $f \colon X \to T$ such that some fiber is (locally) a complete intersection, does that imply that there is an open set $U$ in $T$ such that the fibers above $U$ are (locally) ...
- 156k
15
votes
1
answer
633
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Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
- 85.6k
3
votes
1
answer
1k
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Inverse for a permutation over GF(2)
Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous ...
- 30.5k
7
votes
2
answers
1k
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Elementary proof that projective space is a quotient
Fix an algebraically closed † ground field $k$ of any characteristic. I want to use the classical definition of projective $n$-space $\mathbb{P}^n$ as set quotient of $\mathbb{A}^{n+1}\...
- 11.3k
2
votes
1
answer
693
views
When is the restriction map on global sections an embedding
Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$.
Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume
$f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.
...
- 11.1k
13
votes
2
answers
713
views
How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 52.6k
8
votes
3
answers
1k
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Derivations of C(X)? or Why Must Supermanifolds be Smooth?
What are the derivations of the algebra of continuous functions on a topological manifold?
A supermanifold is a locally ringed space (X,O) whose underlying space is a smooth manifold X, and whose ...
- 27.5k
25
votes
3
answers
2k
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product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 65.4k
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
CommunityBot
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2
votes
2
answers
2k
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What is the transcendence degree of Q_p and C over Q?
Is the tr.deg of Q_p over Q 1? and what about C over Q?
- 65.4k
21
votes
1
answer
2k
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Does formally etale imply flat for noetherian schemes?
This is a followup to an earlier question I asked: Does formally etale imply flat? After some remarks I received on MO I noticed that this was answered to the negative by an answer to an earlier ...
CommunityBot
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6
votes
2
answers
3k
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Converse to Hilbert basis theorem?
Specifically, is it possible for a non-Noetherian ring $R$ to have $R[x]$ Noetherian? Every reference I've seen for the Hilbert basis theorem only states the direction "$R$ Noetherian $\Rightarrow$ $R[...
CommunityBot
- 1
2
votes
0
answers
450
views
Rosenlicht differentials for possibly non-reduced curves
Let $X$ be proper Cohen-Macaulay scheme of pure dimension 1 over an algebraically closed field $k$. When $X$ is moreover reduced, Rosenlicht's theory of regular differential forms gives a beautiful ...
CommunityBot
- 1
5
votes
2
answers
2k
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Weakened conditions for étale + X implies faithfully flat.
Let $F:R \to S$ be an étale morphism of rings. It follows with some work that $f$ is flat.
However, faithful flatness is another story. It's not hard to show that faithful + flat is weaker than ...
- 9,273
10
votes
1
answer
785
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How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)
Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).
Let $R$ be a ...
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