All Questions
6,055 questions
1
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963
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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 ...
- 2,782
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\...
- 2,782
13
votes
1
answer
7k
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Chinese Remainder Theorem for rings: why not for modules?
This is a followup to Analog to the Chinese Remainder Theorem in groups other than Z_n. . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese ...
- 33.8k
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
16
votes
4
answers
4k
views
Example of the completion of a noetherian domain at a prime that is not a domain
Let $R$ be a Noetherian domain, and let $\mathfrak{p}$ be a prime ideal; consider the completion $\hat R_{\mathfrak{p}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of the system of quotients $R/\...
- 7,217
7
votes
5
answers
2k
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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 $X$...
- 485
9
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3
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753
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If L is a field extension of K, how big is L*/K*?
Let $K$ be a field and $L$ an extension of $K$. I wonder how much larger the multiplicative group $L^\times$ of $L$ is than the multiplicative group $K^\times$ of $K$.
I know that if $L=K(t)$ and $t$ ...
- 4,280
-2
votes
1
answer
780
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commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 1
5
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1
answer
3k
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Does the category Monoid of monoids have finite coproducts?
Does the category Monoid of monoids have finite coproducts?
- 67
2
votes
3
answers
1k
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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 ...
- 21
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\...
- 422
1
vote
1
answer
1k
views
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!
...
- 422
9
votes
5
answers
3k
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Alternative proof of unique factorization for ideals in a Dedekind ring
I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
- 14.7k
10
votes
2
answers
1k
views
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) ...
- 10.7k
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 $\...
- 50.6k
11
votes
6
answers
33k
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Are submodules of free modules free? [closed]
Are all submodules of free modules free? I would like a reference to a proof or counterexample please.
- 637
4
votes
3
answers
1k
views
An example where GCD depends on the domain
First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$.
I want to find an example of an GCD-domain $R$, a subdomain $S \...
9
votes
1
answer
831
views
Different notions of associated prime (in the non Noetherian case)
Most books I have treat primary decomposition only in the Noetherian case. Atyiah-MacDonald goes a step further and prover the uniqueness theorems of primary decomposition without the Noetherian ...
- 14.7k
15
votes
1
answer
636
views
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) ...
- 12.6k
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 ...
31
votes
8
answers
21k
views
Reference book for commutative algebra
I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:
More comprehensive than Atiyah–Macdonald
More readable than Matsumura (maybe better ...
3
votes
1
answer
536
views
Question on $Ext$
Let $S$ be the polynomial ring $k[x_0,\ldots,x_n]$, $x$ one of the variables $x_i$, $I\subseteq S$ a homogeneous ideal which has a generating set $f_1,\ldots,f_r$ where $\deg_x f_i=0$ for all $i$.
...
- 83
21
votes
2
answers
1k
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What properties define open loci in excellent schemes?
Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open ...
- 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
8
votes
4
answers
2k
views
Formally étale at all primes does not imply formally étale?
All rings are assumed to be commutative and unital, with all homomorphisms unital as well.
On last week's homework, there was a mistake in one of the questions:
(2.5) Let $R\to S$ be a ...
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.
...
- 1,391
24
votes
5
answers
6k
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To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field?
In his answer to a question about simple proofs of the
Nullstellensatz
(Elementary / Interesting proofs of the Nullstellensatz),
Qiaochu Yuan referred to a really simple proof for the case of an
...
- 1,411
24
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1
answer
4k
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Minimal number of generators of a homogeneous ideal (exercise in Hartshorne)
In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
Show that a strict complete intersection is a set theoretic complete intersection.
Here are ...
- 14.7k
11
votes
4
answers
4k
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Variants of Eisenstein irreducibility
In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
- 32.6k
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
2
votes
2
answers
2k
views
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?
- 1,503
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 ...
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:...
- 3,103
19
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2
answers
5k
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Characterizations of UFD and Euclidean domain by ideal-theoretic conditions
This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ...
- 7,217
0
votes
2
answers
563
views
Primary decomposition of zero-dimensional modules
(I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) ...
- 5,243
72
votes
14
answers
22k
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Elementary / Interesting proofs of the Nullstellensatz
Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...
17
votes
3
answers
3k
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Ghost components of a Witt vector - Motivation
I'd like to know if anyone has a good explanation for where the ghost components that are used to define Witt vectors come from. A lot of sources I've read take the ghost components for their ...
- 1,098
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[...
- 6,792
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 ...
- 5,163
1
vote
1
answer
1k
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Unique factorization in polynomial rings
Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first.
Well, ...
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 ...
- 1,609
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 ...
- 705
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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 ...
10
votes
1
answer
785
views
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 ...
- 30.5k
4
votes
1
answer
1k
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On using field extensions to prove the impossiblity of a straightedge and compass construction
Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / \mathbb{...
19
votes
5
answers
7k
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When a formal power series is a rational function in disguise
Given a formal power series $f \in k[[X]]$, where $k$ is a commutative field, is there any good way to tell whether or not $f\in k(X)$?
Edit: To clarify, "good way to tell" means "computable ...
1
vote
1
answer
190
views
Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 87
13
votes
4
answers
1k
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Why is a monoid with closed symmetric monoidal module category commutative?
Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural ...
- 12.3k
37
votes
2
answers
3k
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How can I define the product of two ideals categorically?
Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms ...
- 118k
10
votes
3
answers
1k
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Can injective modules over R give non-injective sheaves over Spec R?
In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. ...
- 27.9k