Recently Active Questions
159,066 questions
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What roles do "base change" play in algebraic geometry?
It might be a not very specific problem. I just wanna know how much do we rely on the property of "base change closed". In the definition of Grothendieck pretopology, we require a collection of ...
- 19.6k
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Blow up along codimension one closed subscheme
Suppose X is dimension two locally Noetherian scheme. Y is a closed subscheme of X, with codimension 1. Denote X' to be the blow up of X along Y. Prove that the structure morphism f:X'-->X is a finite ...
- 6,543
23
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1
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967
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Do DG-algebras have any sensible notion of integral closure?
Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
- 4,909
4
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3
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854
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Variational characterization of curvature?
Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives ...
- 27.5k
14
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Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
CommunityBot
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Request for reference: Banach-type spaces as algebraic theories.
Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
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29
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2
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Examples of algebraic closures of finite index
So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
...
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Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?
There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (...
CommunityBot
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1
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What are the "special" strata of Sym^n(C^2)?
The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal ...
CommunityBot
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4
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An everywhere locally trivial line bundle
Is there a variety $X$ over $\mathbb{Q}$ and a line bundle $L$ over $X$ (other than the trivial line bundle $\mathcal{O}_X$ ) such that $L_v$ is the trivial line bundle over $X_v=X\times_{\mathbb{Q}}\...
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797
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Tensor product is to flat as Hom is to ?
Sorry if I'm missing something here, but what do we call $M$ if the functor $H_M:N\mapsto Hom(M,N)$ is exact? Is this in fact equivalent to being flat through some adjointness properties?
- 57.6k
6
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1
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310
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Sequence of Diophantine Equations
Is there some (huge) positive integer $M$ with the following property:
for any $z>M$, there exist positive integers $x, y_{1}, y_{2},..., y_{z}$
such that $x^x$ $=$ $y_{1}^{y_{1}}$+ $y_{2}^{y_{2}}$+...
CommunityBot
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5
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2k
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non-abelian groups of prescribed order
Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this ...
CommunityBot
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3
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Degree of an embedded algebraic variety
Let $X$ be an algebraic variety and $A$ is a ample divisor on $X$. Let $m$ be a sufficiently large natural number such that $X \overset{\varphi_{mA}}{\to} \mathbf{P}H^0(X, \mathcal{O}_X(mA))$ defined ...
- 57.6k
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classification of irreducible admissible representations of GL(n)
Does anyone know the classification of irreducible admissible representations of GL(n) (over real,complex and p-adic fields), or some references?
Sorry if this question is not appropriate here.
- 57.6k
4
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A hands-on description of a "completion" of the free commutative monoid on countably many generators
This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question here, but Ilya N asked me to break that post up into several questions.
Consider the free commutative ...
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23
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3
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Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 15.6k
21
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3
answers
1k
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What's the analogue of the Hilbert class field in the following analogy?
There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and ...
- 65.4k
2
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0
answers
1k
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Automorphisms of category of groups [duplicate]
Possible Duplicate:
What are the auto-equivalences of the category of groups?
Does the category of groups have any nontrivial automorphisms? (an automorphism of a category being a functor from ...
CommunityBot
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17
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2
answers
2k
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How probable is it that a rational prime will split into principal factors in a Galois number field?
Let $L$ be a Galois number field over $\mathbb{Q}$. Based on classical algebraic number theory (specifically, the Chebotarev density theorem), I can answer lots of numerical questions about how ...
- 30.5k
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How do you see the genus of a curve, just looking at its function field?
Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly ...
- 7,442
2
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1
answer
946
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categorical description of elements in a direct limit
is it possible to characterize the elements of a (special) direct limit only using the universal property? in detail:
let's first concentrate on the category of sets. by an element, I mean a morphism ...
CommunityBot
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5
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3
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Counting lattice points on an n-simplex
Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where:
$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are ...
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1
answer
363
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Efficiently sampling points from an integer lattice.
Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
- 1,618
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3
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987
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Octonionic Unitary Group?
Hi all.
I was wondering if anyone has any references on work related to the Octonionic Unitary group. I would imagine that such a group would be generated by Octonionic skew-Hermitian matrices (at ...
- 33.3k
8
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1
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563
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Is tensor product exact in abelian tensor categories with duals?
Suppose we are in an abelian tensor category with duals, where all objects have finite length. Let $0 \to A \to B \to C \to 0$ be a short exact sequence and $Z$ an object of the category. Is
$$0 \to ...
- 25.2k
5
votes
1
answer
2k
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Self-similar matrices? [closed]
Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
- 7,800
5
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1
answer
513
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Field of Definition of a Meromorphic Function
Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field,...
- 1,522
7
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2
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726
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Zeta function for curves in a manifold
Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...
- 15.1k
22
votes
7
answers
21k
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Quantitatively speaking, which subject area in mathematics is currently the most research active? [closed]
I was wondering if there is a list of the most active branches of mathematics?
If MathOverflow is a representative sample, then algebraic geometry is by far the most popular. Is this the case?
- 33.3k
7
votes
1
answer
571
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Categorical duals in Banach spaces
Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)".
Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
- 25.8k
4
votes
1
answer
479
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Distributions as presheaves?
The yoneda lemma gives us a characterization of $Psh(\mathcal{C})$ that seems very similar to the theory of distributions. That is, we have a notion of representable presheaves, similar to ...
- 19.9k
12
votes
3
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1k
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distance regular metric spaces
A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
26
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3
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Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
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1
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sheafifying a projective limit of presheaves
Let $F=(F\_n)\_n$ be an $\ell$-adic sheaf on $X\_{et}$, for a variety $X$ over an algebraically closed field $k$ of characteristic not equal to $\ell$. Does the presheaf sending $U$ to $H^i(U,F):=\lim\...
- 57.6k
3
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1
answer
306
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What are the products in the category of normed vector spaces with linear contractions?
In the category of normed vector spaces in which the morphisms are linear contractions, what do products look like?
- 7,329
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0
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442
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Generalizations of generators / hyperplanes descriptions for cones to partially-ordered fields?
Background: given a finite-dimensional real vector space V of dimension d, I can define a pointed cone in two ways: either as a set of the form $\{r_1v_1 + \cdots + r_nv_n \mid r_1, \dots, r_n \in R_{\...
- 47.3k
1
vote
1
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The topology of periodic permutohedral boundary conditions
$\mathbb{R}^n$ admits a tessellation by permutohedra. The corresponding identification of facets of a permutohedron therefore gives a well-defined space: call it $X_n$. For example, $X_2$, the hexagon ...
- 47.3k
5
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BPP being equal to #P under Oracle
Luca Trevisan here gives a randomized polynomial-time approximation algorithm for #3-coloring given an NP oracle.
In a similar vein, I was wondering if there were any results on $BPP^{NP}\stackrel{?}{...
- 2,416
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Are automorphism groups of hypersurfaces reduced ?
In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of ...
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Help me understand boundary terms in actions over nontrivial manifolds
So I have this manifold $M$, along with a metric $g_{\mu\nu}(x)$ and metric-compatible covariant derivative $\nabla_\mu$ (which is not necessarily the one corresponding to the Levi--Civita connection)....
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3
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Standard name of "atomic morphisms"?
Googling for "atomic morphism" gives me only 70 results. Is this concept so fruitless or does it have another standard name?
What I mean is a morphism $f: A \rightarrow B$ such that
$$(\forall g,h)\ ...
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Relationship between algebraic and holomorphic differential forms
I'm a little confused and in need of some clarification about the
relationship between algebraic and holomorphic differential forms:
(1) What is the exact definition of the module of differential
...
CommunityBot
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2
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What are important examples of filtered/graded rings in physics?
Hi,
what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics?
Of ...
- 33.3k
4
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1
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Can the inner structure of an object be systematically deduced from its position in the category? [closed]
Background
Even for the novice it seems comprehensible that the "inner structure" of an object is determined (up to isomorphism) by its "position" in a category, defined by the ...
CommunityBot
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6
votes
4
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Killing fields on homogeneous spaces
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...
- 27.5k
3
votes
3
answers
2k
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prime ideal factorization in an extension field
Let $p$ be a rational prime and $K$ a number field.
Dedekind's discriminant theorem tells us that
$p$ ramifies in $K$ $\iff$ $p$ divides the discriminant of $K$.
Hence if $p$ does not divide ...
CommunityBot
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6
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3
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661
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completion of category is idempotent
The question belongs to elementary category theory, so please forgive me if this is trivial. I think I even read a proof for this some weeks ago, but I can't find it.
In topology, you have the ...
- 65.4k
3
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3
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4k
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What are the prime ideals in rings of cyclotomic integers?
Is a good characterization of Spec $\mathbb{Z}[\zeta_n]$ known? Same question for its unit group.
- 118k
6
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3
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791
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'Focusing' the mass of the Probability Density Function for a Random Walk
Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability ...
- 3,069