Newest Questions
162,007 questions
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Commuting totalizations with filtered colimits in the plus-construction for $\infty$-presheaves on a topological space
Let $\mathrm{An}$ be the $\infty$-category of anima (i.e., $\infty$-groupoids).
Let $X \colon \mathcal{O}(U)^{\mathrm{op}} \to \mathrm{An}$ be a presheaf on a topological space $U$.
Now, consider a ...
- 7,876
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0
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33
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Hasse principle for rational number times a quadratic form
Motivation:
Hasse principle for rational times square
local-global principle for units
Does the Hasse norm theorem easily imply the global squares theorem?
Let $K/F$ be a quadratic extension of global ...
- 41
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15
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Applicability of Besicovitch Covering Theorem in different norms in $\mathbb{R}^n$
This question is probably quite simple, but I would still appreciate a straightforward answer, whether positive or negative. Thank you in advance.
I alredy know that the Besicovitch Covering Theorem ...
4
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30
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Kashiwara-Schapira Stack of microsheaves
I will assume familiarity with the notions of singular support as defined by Kashiwara and Schapira in Sheaves on Manifolds. Let $M$ be a manifold, one can define a functor :
$\mu Sh^{pre} : Op_{T^*M, ...
- 111
2
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2
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80
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Hyperplane sections of general type surfaces
Let $X$ be a smooth projective complex surface with $K_X$ very ample. Let $D\in |K_X|$ be a general element.
Does there exist another smooth projective variety $D'$ with a finite flat morphism $D\to D'...
- 493
1
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90
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Dirichlet series of $\zeta'(s)\zeta'(1-s)$
I want to find the Dirichlet series coefficients of $\zeta'(s)\zeta'(1-s)$.
I know that $$\zeta'(s)=-\sum_{n=1}^{\infty}\frac{\log n}{n^s},$$provided $\Re(s)>1$. I would have thought it would be ...
- 11
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1
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48
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Expected number of steps for a random walk to return with a single barrier
Here is the problem and a possibly noobie question I am trying to figure out.
You have a biased random walk on $\mathbb{Z}$ with jump to the left probability $p=0.7$ and jump to the right $q=1-p=0.3$. ...
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1
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43
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What is known about the algebras $\mathrm{End}_{U_q(\mathfrak{sl}_2)}(L(d)^{\otimes n})$?
Let $A_{d,n}=\mathrm{End}_{U_q(\mathfrak{sl}_2)}(L(d)^{\otimes n})$. For $d=1$, this is the Temperley-Lieb algebra $A_{1,n}=TL_n(q+q^{-1})$. For $d=2$, Scrimshaw calls this the tangle algebra, first ...
- 502
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29
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Nonnegative submartingales: convergence to infinity in probability
Consider a nonnegative submartingale $X_n$, with uniformly bounded jumps, and jump variance uniformly bounded from below. Can we conclude that it converges to infinity in probability? Also, is it ...
- 1,907
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1
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112
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Categorical structure guaranteed to exist, but not necessarily preserved
Background
I'm currently studying arithmetic universes (AUs), which are defined to be list-arithmetic pretoposes (see "Joyal's arithmetic universe as list-arithmetic pretopos" by Maietti). ...
- 335
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51
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For endorsement [closed]
I am a research scholar working in the area of General Topology, and I would like to submit it to arXiv under the math.GN category. As this is my first submission in this category, I need an ...
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50
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Algorithm that allows to get any value of A000123 if finite number of values of A002577 are known
Let
$T(n,k)$ be an integer coefficients such that $$ T(n,k) = \begin{cases} 1 & \text{if } k = 0 \vee n = k \\ \displaystyle{ \sum\limits_{i=k-1}^{n-1} T(n-1,i) T(i,k-1) } & \text{otherwise} \...
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1
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55
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Combining two Dirichlet polynomials into one polynomial
I want to write $$\sum_{m_1,m_2\le M}\frac{a(m_1)}{m_1^s}\frac{a(m_2)}{m_2^{1-s}}$$ as one single Dirichlet polynomial that is, I want to find the Dirichlet polynomial $\sum_{n\le N}f(n)n^{-s}$ such ...
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1
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85
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Factorization of polynomials vanishing on quadrics: divisibility by the defining equation
Let $P(x,y,z)$ be a real polynomial that vanishes on the ellipsoid defined by $$ g(x,y,z) := \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1 = 0.$$ How can one rigorously justify the ...
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55
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Equivalent definitions of volume of representations (or characteristic classes of flat bundles)
Statement of the problem:
Let $M$ be a closed connected oriented manifold with fundamental group $\Gamma$ (considered as covering transformations acting on its universal cover). Let $G$ be a (...
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6
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1
answer
142
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Circle action on free loop space of a classifying space
It is a folklore result that if $G$ is a discrete group and $BG$ its classifying space, then the free loop space $L(BG)$ is homotopy equivalent to $EG\times_{Ad} G$ where $EG$ is the universal $G$-...
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1
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131
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A structured recursive formula for the complete homogeneous symmetric polynomial
I recently discovered a recursive, closed-form summation formula that appears to compute the complete homogeneous symmetric polynomial $h_n(x_0, x_1, \dots, x_{m-1})$, but in a more structured and ...
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36
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From global to local: strong failure of regularity (quasirandomness) in random bipartite graphs
Let $n$ be a positive integer and $\varepsilon > 0$ with $\varepsilon \ll 1$ and $n \gg 1/\varepsilon$. Let $G = (A, B, E)$ be a bipartite random graph with $|A| = |B| = n$, where each edge between ...
- 379
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56
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Fejér-Riesz inequality for $H^p$ on the unit disk for more general curves
Let us consider the Hardy space $H_p$ on the unit disk, and a function $f \in H_p$. There's an inequality by Fejér and Riesz stating that:
\begin{equation}
\int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2}...
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0
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58
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Projective module over Robba ring
Is a projective module finitely generated over Robba ring free?
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49
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Can we have a descending powerset class in Stratified ZF?
Working in $\sf Stratified \ ZF + Class \ Comprehension$, where class comprehension is that of Morse-Kelley. Is there anything to forbid having a class $C$ that meet these two conditions:
$\forall x \...
- 12.4k
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22
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Sigma Algebra- All of Statistics [migrated]
I am reading the book All of Statistics by Larry Wasserman. In the first chapter he defines probability axioms and says that it is not always possible to assign probabilities to all the subsets of a ...
6
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96
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Is there a model category structure for C-homotopy?
After reading Harry Altman's question about homotopy with regards to arbitrary connected spaces I started wondering about the extent to which the notion defined there is compatible with more recent ...
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43
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What do invariant subspaces of graph-related matrices encode?
The question that I have posted here is as follows:
The eigenvalues of the adjacency matrix or laplacian matrix for a given graph $G$ has been relatively well studied. These provide a rather ...
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1
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60
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Ordinary Quivers for Infinite-Dimensional Algebras
I am wondering if there is an analogue to the ordinary quiver for a finite-dimensional, basic, connected $\Bbb K$-algebra. In particular, I am interested in an ordinary quiver for the polynomial ring $...
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50
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Maximal density of overlapping circle packing
I'm looking for a way to prove that on a plane if we place points with a minimal distance of $d$ and each point is the center of a circle of radius $d$ then the density of the plane is no more than $\...
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43
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When is −∥x−y∥+∥A(x−y)∥ a CPD kernel ? - Difference of CPD radial kernels remains CPD (Bernstein functions)
I’m studying a family of translation-invariant kernels built from two norms and would like to understand exactly when their difference remains conditionally positive-definite. In particular, we focus ...
- 1
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0
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23
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Uniform entropy bounds for unions of VC subgraph classes
I'm working with VC subgraph classes of functions, say $\mathcal{F}$ and $\mathcal{G}$, which are both uniformly bounded and admit envelopes $F$ and $G$, respectively. I came across a useful lemma (...
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178
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Periodic cyclic homology and Hochschild homology
Suppose we have a quasi-compact quasi-separated scheme $X$ and consider $D_{QCoh}(X)$ the dg enhancement of derived category of quasi coherent sheaves on $X$. Then the Hochshild Homology $HH_*(X)$ is ...
- 481
2
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1
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196
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Endomorphisms of groups schemes involving $\mathbb{G}_a$ and flat base change
Let $R$ be a ring of positive characteristic. Let $G$ be a commutative affine and smooth group scheme over $R$. I consider two abelian groups:
$$M(G):=\mathrm{Hom}(G,\mathbb{G}_{a,R})$$
and
$$N(G):=\...
- 1,509
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88
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Convergence of mollifiers of a Lipschitz function on a codimension 1 subspace
Let $f:\mathbb{R}^2\to \mathbb{R}$ be $L$-Lipschitz. Let
$f_\varepsilon:=f*\eta_\varepsilon$ be its smooth $\varepsilon$-mollification, where
$\eta_\varepsilon(x)=\frac{1}{C\varepsilon^2}\eta(|x|/\...
- 1,322
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47
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Index of singular point for 2nd order ODE
There exists classification of equilibrium points for 2nd order ODEs like
$$
\ddot x = f(x, \dot x), \quad \text{where} \quad x \in \mathbb{R}.
$$
The classification includes stable/unstable nodes, ...
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78
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On Schuricht and Schönherr approach for proving the divergence theorem on general Borel sets
This question stems from a ZBmath search I did yesterday evening, and it is somewhat related to the following MathOverflow question: "On which regions can Green's theorem not be applied? ".
...
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2
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1
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314
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Identity for A000123
Let
$a(n)$ be A000123 (i.e., number of binary partitions: number of partitions of $2n$ into powers of $2$), whose ordinary generating function is $$ \frac{1}{1-x} \prod\limits_{j=0}^{\infty} \frac{1}{...
- 5,890
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78
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Existence of normal element whose spectrum coincides with a realizable spectrum
Let $A$ be a $C^*$ algebra. Assume that $x\in A$ is a given element.
Does there exist a normal element $n\in A$ whose spectrum coincides with the spectrum of $x$ namely $\sigma(n)=\sigma(x)$? If the ...
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4
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1
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332
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An elementary problem on univariate polynomials
Suppose that $p(x) = \sum_{i=0}^d a_i x^i$ is a univariate polynomial of degree $d$ over the field of real numbers. Suppose that $d > 2$, and that $a_d, a_{d-1} \not = 0$.
Assume that for some $\...
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45
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Subharmonic functions of general bilinear forms
It is well known that $|\nabla u|^2$ is subharmonic whenever $\Delta u = 0$. This is the usual Bochner identity.
I am looking for generalizations of this in the following sense: Let $u$ be a smooth ...
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594
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Where can I find a reference table for ordinal arithmetic?
I am working on a calculator for ordinal arithmetic.
Ordinal arithmetic is extremely tricky. So much so that there have been academic papers on the very subject that got some results wrong. Not to ...
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67
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When is the Kobayashi metric Riemannian?
Let $X$ be a complex manifold. On the tangent space $T_X$ we have the Kobayashi infinitesimal pseudonorm $F_X$, which is in general upper semi continuous as a function on the total space of $T_X$, and ...
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109
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Blockwise measure-positive subsets of $\mathbb{N}$
$\def\N{\mathbb{N}}$There are different notions of when a set $A\subseteq \N$ has "positive measure", to which I want to compare a third notion. We say that $A\subseteq \N$ is
blockwise ...
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2
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63
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Are there Chern number inequalities for symplectic manifolds?
There are some inequalities of Chern numbers of complex manifolds or algebraic varieties. For example, for surfaces of general type we have the Bogomolov-Miyaoka-Yau inequality and the Noether ...
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3
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107
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Locally compact topology on subsets of Cantor set
Cross-posting this question from MSE.
Let $C\subseteq \mathbb R$ denote the Cantor set and let $A\subseteq C$ be dense with $|A| = |\mathbb R|$, and assume $A$ doesn't contain its infimum. Is there a ...
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81
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Image of the evaluation map of family rational curves
Let $X$ be a smooth projective Fano variety of dimension $N$. Let $Y$ be a component of $\text{Mor}^d(\mathbb{P}^1, X)$. Let $V$ be a irreducible component of the closed subset of $Y$ consisting non-...
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1
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86
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Hermite-type convex interpolation
Let $f : [0, 1] \to \mathbb{R}$ be a smooth strictly convex function.
Let $0 \leq x_1 < \dots < x_n \leq 1.$
I am interested in whether there exists a polynomial $p$ such that it is convex on $[...
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0
answers
61
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Name of a certain type of "generalisation"
This question is sort of metamathematical, so I hope it's appropriate for the site.
Let's say that Proposition A says that all objects satisfying property X have property Y. Now say that Proposition B ...
- 1,171
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1
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62
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$\text{div}(\mathbf {A}\nabla u)=0$ with bounded $\mathbf A$: least Holder continuity of $u$?
Let $u\in H^1(B_1)$ be a weak solution to $$\text{div}(\mathbf {A}\nabla u)=0 \qquad\text{in }B_1$$ where $\lambda \mathbf I\le\mathbf A\le\Lambda\mathbf{I}$. What is the least Holder continuity of $u$...
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0
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53
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Sharpness of $L^1 - L^2$ inequality on Gauss space
Talagrand's $L^1-L^2$ inequality states that if $f \colon \mathbb{R}^n \to \mathbb{R}$ is a smooth enough function, we have for the canonical basis $\{e_i\}$ that
$$
\mathrm{Var}(f) \leq C_1 \,
\sum_{...
- 396
2
votes
0
answers
105
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Homotopy class of certain loops in $\mathit{SO}(n)$
I'm wondering if there's any known formula or algorithm for computing the class of a loop in $\pi_1(\mathit{SO}(n))$ (say, for a sufficiently large $n$). In particular, I have a family of such loops ...
- 21
2
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0
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85
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Finite and faithfully flat morphisms of Huber Rings
I am currently studying the notes of Kiran Kedlaya for Arizona Winter School 2017 on Perfectoid Spaces.
Link : https://swc-math.github.io/aws/2017/2017KedlayaNotes.pdf
I am stuck at Exercise $1.1.6$, ...
3
votes
0
answers
57
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Does a monoidal "reflective" adjunction induce adjunctions on module categories?
Let $F\colon (\mathcal{D},\otimes_{\mathcal{D}},1_{\mathcal{D}})\to (\mathcal{C},\otimes_{\mathcal{C}},1_{\mathcal{C}})$ be a (strong) monoidal functor and let $G$ be right adjoint to $F$. Then $G$ ...
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