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159,085 questions
18
votes
2
answers
2k
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Are Jacobians principally polarized over non-algebraically closed fields?
How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian.
So what I am asking is: if I have a curve $C$ over a non-...
- 7,108
2
votes
2
answers
2k
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Hitting times for an N-dimensional random walk on a lattice with (strictly positive) random integer steps
Please consider a random walk on a finite N-dimensional lattice with vectors $(x_1, ..., x_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away ...
- 599
2
votes
1
answer
336
views
Topologies making a class of functions continuous [closed]
Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\...
- 237k
9
votes
2
answers
670
views
Effects of "weak" vs. "strict" categories in Eckmann-Hilton arguments
A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an ...
CommunityBot
- 1
15
votes
3
answers
3k
views
Existence of fine moduli space for curves and elliptic curves
For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
- 7,108
8
votes
3
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606
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Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
- 5,827
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
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2
votes
1
answer
308
views
Image of a hyperspecial subgroup hyperspecial?
Suppose that $F$ is a nonarchimedean local field, $G_1$ and $G_2$ are connected (linear) algebraic groups over $F$, and $\phi:G_1\to G_2$ is a surjective homomorphism of algebraic groups. Suppose $H$ ...
- 41.4k
38
votes
2
answers
1k
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Size of the smallest group not satisfying an identity.
Given $F = F(x_0,\ldots,x_n)$ the free group on $n+1$ generators. Define a function $M: F\rightarrow \mathbb{N}$ such that $F(w) = l$, if the smallest group in which $w$ is not an identity is of size ...
- 8,246
4
votes
1
answer
599
views
When are two natural transformations of infinity-categories equivalent?
Suppose
C and D are two ∞-categories (quasi-categories),
$F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the ...
- 66.8k
9
votes
2
answers
1k
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Maximum Order of elements in $GL(n,Z)$
Hi,
I know that $\mathrm{GL}(n,\mathbb{Z})$ has an element of order $m$ iff $\Phi(m)\leq n$, where $\Phi(m) = \varphi(m)$ if $p_1^{\alpha_1}\neq 2$ or $m=2$, $\Phi(m) = \varphi(m)-1$ if $p_1^{\...
- 14k
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
2
votes
2
answers
718
views
Algebra / unital associative algebra: better terminology?
In Bourbaki an algebra over a commutative ring $k$ is defined to be a $k$-module $A$ together with a $k$-bilinear map $A \times A \rightarrow A$. We then have the obvious notion of morphisms of $k$-...
- 5,243
11
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2
answers
1k
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Can projective hypersurfaces contain linear spaces? How big?
I am in this, rather friendly, situation:
I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of ...
- 16k
5
votes
2
answers
302
views
In what degrees does Ext(S/(f),S) vanish?
Let $S=k[x_0,...,x_n]$ be the polynomial ring over a field $k$ and $f\in S$ non-zero and homogeneous. Is it true that $Ext^m(S/(f),S)$ is zero?
This would help me to show that $Ext^m(S/fI,S)\cong Ext^...
- 30.6k
8
votes
1
answer
1k
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Geometric Intuition for Big Monodromy
In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...
- 19.2k
2
votes
1
answer
278
views
Schemy question..
Here is where I got lost .. I have a scheme Y over k (an algebraically
closed field), in it I have an irreducible closed subscheme X of
finite type (do I need finite type?). I also know that X is
...
- 5,163
3
votes
2
answers
1k
views
What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?
One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CW-complexes I ...
- 1,355
1
vote
1
answer
392
views
Why is it important to have disjoint sets in a union for the union to make sense w.r.t the order types?
This question has been bugging me for quite some time now.
Say we have some $\beta$ smaller than some $\gamma$ and a sequence
$\beta$$\epsilon$ : $\epsilon$ smaller than cf($\beta$) cofinal in $\...
- 237k
8
votes
2
answers
988
views
cocompact discrete subgroups of SL_2
How can one construct families of cocompact discrete subgroups of the topological group $\text{SL}_2(\mathbb{C})$?
Here quaternion algebra's might help, I believe, but I have some difficulties with ...
- 5,163
2
votes
1
answer
422
views
Dirichlet L series and integrals
If $f : t \to e^{-xt}$ with $x \geqslant 1$, and $d_n$ is the number of positive integers that divide $n$, I can show that
$$ \lim_{\epsilon \to 0^+} \sum_{n\geqslant 1}\frac{\epsilon^2 d_n f(\...
- 25.8k
31
votes
2
answers
4k
views
Number theory textbook based on the absolute Galois group?
I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting ...
- 32.6k
2
votes
2
answers
424
views
characterization of a submodule
In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar ...
- 1,411
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 ...
- 9,283
1
vote
1
answer
146
views
equivalence of submodules
I have Z^3/M = Z^3/N = Z_k where M,N are submodules of Z^3 and Z_k is cyclic order k.
I would like to say some SL_3(Z) transformation takes M to N. Is this true? How to show?
- 47.3k
5
votes
5
answers
866
views
Remembering arrows' directions in basic Category Theory
Is there an easy way of remembering the direction of arrows between morphisms in Categories?
The direction of arrows so confuses me: products and co-products, (EDIT- Also, pull-backs, pushouts, ...
- 25.6k
10
votes
3
answers
2k
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Is this naive test to tell whether a complex elliptic curve has complex multiplication effective?
I have a question about a naive test to tell whether a complex elliptic curve $E$ has complex multiplication.
Recall that the endomorphism ring $End(E)$ of $E$ is isomorphic to either $\mathbb{Z}$ or ...
- 4,646
4
votes
2
answers
998
views
About higher Ext in R-Mod
So, in $R-Mod$, we have the rather short sequence
$\mathrm{Ext}^0(A,B)\cong Hom_R(A,B) $
$\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of "good" factorizations ...
- 1,987
23
votes
9
answers
4k
views
What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
CommunityBot
- 1
38
votes
2
answers
8k
views
Intuition for Primitive Cohomology
In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then ...
- 673
5
votes
1
answer
511
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Divisors and vector bundles in various categories
I'm taking a first course on complex manifolds, and am trying to square what I hear with what I know of (real) differential geometry. Please forgive me if this question is misguided!
Here are two ...
- 673
5
votes
1
answer
6k
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Algebraic equivalence VS Numerical Equivalence - An Example.
This question is arose from the question
Difference between equivalence relations on algebraic cycles
and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface.
...
CommunityBot
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16
votes
0
answers
1k
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Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
4
votes
1
answer
1k
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inverse m-matrix
The following is a result by C.R. Johnson appearing every now and then in the literature.
Let $A$ be an $n \times n$ inverse $M$-matrix. Then
All principal minors of $A$ are positive.
Each ...
- 65.4k
8
votes
1
answer
626
views
Estimate population size based on repeated observation
I take the bus to work every day. Every bus has a serial number, but unlike in the German Tank Problem, I don't know if they are numbered uniformly $1...n$.
Suppose the first $k$ buses are all ...
- 1,618
12
votes
8
answers
1k
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Why does randomness work in numerical algorithms?
There are successful numerical algorithms that involves a sequence of random numbers, like Monte Carlo methods or simulated annealing. I can follow the lines of proofs of their convergence, and ...
CommunityBot
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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 ...
- 4,277
-4
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3
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What is information-theoretic lower bound? [closed]
Hi all,
Please tell me what is information-theoretic lower bound. what does it really means
Thank you
- 21
10
votes
4
answers
967
views
What m minimizes E(|m-X|^3) for a random variable X?
Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...
- 3
8
votes
1
answer
2k
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Normal bundle to a curve in P^2
Let $C$ be a smooth curve of degree $d$ in $\mathbb{P}^2$ over $\mathbb{C}$. Say $C$ is defined by $p(x,y,z)=0$, with $p$ a homogeneous degree $d$ polynomial.
In vector calculus one learns that the ...
- 57.6k
1
vote
1
answer
300
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Is the real Jacquet module of a Harish-Chandra module still a Harish-Chandra module?
Casselman defined the real Jacquet module for a Harish-Chandra module, if we view the Jacquet module as a module corresponding to the Levi subgroup, the question is is it still a Harish-Chandra module?...
- 57.6k
28
votes
4
answers
2k
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What is the relationship between various things called holonomic?
The following things are all called holonomic or holonomy:
A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...
- 243
0
votes
1
answer
216
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Q-isogeny and Q-torsion subgroup
What is meant by a Q-isogeny and the Q-torsion subgroup? (And by Q, I mean rational 'Q')`
`
- 65.4k
2
votes
3
answers
454
views
Can there exist two non-equivalent equivariant actions of a group on vector bundle?
Can there exist two non-equivalent equivariant actions of a group $G$ on vector bundle over a $G$ space?
- 483
20
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1
answer
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Every Manifold Cobordant to a Simply Connected Manifold
I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold.
I believe that some sort of surgery will do the trick. Roughly speaking, I ...
- 27.5k
6
votes
2
answers
737
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Néron theory for motives of arbitrary weight
SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...
CommunityBot
- 1
13
votes
0
answers
943
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Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
CommunityBot
- 1
7
votes
2
answers
436
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Taking roots in simple linear algebraic groups
Suppose $G$ is a simple (linear) algebraic group over an algebraically closed field of characteristic zero, that $n$ is a positive natural number, and that $g\in G$. Can we always find an $h\in G$ ...
- 3,131
5
votes
3
answers
1k
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Functional calculus for direct integrals
Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as
$T = \int^\oplus T_x$ for ...
- 1,540
11
votes
2
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1k
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Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?
Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of quivers)...
- 3,061