All Questions
22,546 questions
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What is the fastest known algorithm for evaluating a homogeneous binary polynomial?
This question was initially posted on math.stackexchange.com, but there is no appropriate answer, hence I have the right to publish it here again.
Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a ...
- 1,833
2
votes
0
answers
165
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Definition for "almost simple" linear algebraic groups
Proposition 2.18 from "Elementary abelian $p$-subgroups of algebraic groups" by R. Griess. used the term "simply connected almost simple linear algebraic group $G$" without ...
- 217
2
votes
1
answer
215
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How can one test whether a given analytic curve in the plane is algebraic or not?
Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
- 2,154
2
votes
0
answers
138
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Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
6
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135
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Reconstructing a scheme from its quotient stack
Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth).
Can we reconstruct $X$ from its quotient stack $[X/G]$?
It seems that we can expect $X$...
- 645
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0
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76
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What exactly does it mean for the moduli space of stable sheaves to have a universal family étale locally?
It is known that in general, the moduli space $M$ of (Gieseker) stable sheaves do not have a universal family. Therefore, a map $f \colon S \to M$ does not necessarily induce a family of sheaves.
On ...
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0
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85
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Taking hyperplane section remains dominant
I was reading Kollar's book "Rational curves on algebraic varieties". This is from Chapter IV, proposition $1.3$. I don't understand the proof from $1.3.3$ to $1.3.1$, Page- 182. Suppose I ...
- 59
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0
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81
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Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
1
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1
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207
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Is the vector bundle over a vector bundle, a vector bundle over the base scheme?
Suppose we have $E\to X$, a vector bundle over $X$ and $E'\to E$ a vector bundle over $E$. Composing the structure maps gives a smooth scheme $E'\to X$ over $X$. My question is, when is this a vector ...
- 187
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Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
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158
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Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 23
1
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0
answers
75
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Parametrized moduli spaces of semistable bundles by varying Kähler classes
Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
- 137
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1
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72
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Relating the order of a polynomial to the resultant in the context of formal power series
I urgently need to understand how to begin or the complete proof of the following statement:$\DeclareMathOperator{\Res}{Res}$
While reading the paper here on page one, in the introduction, the author ...
394
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115
answers
110k
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Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
4
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1
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176
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Grothendieck construction on fibred categories/stacks
This question is related to a previous question of mine, which has so far gone unanswered.
For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
- 383
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69
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Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 6,048
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1
answer
224
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Proper smooth pushforward of vector bundle is a vector bundle?
Suppose $X$ and $Y$ are algebraic varieties over a field $k$, and $f:X
\to Y$ is proper smooth. Then for a vector bundle $E$, and any $i\ge 0$, do we have $R^if_*(E)$ locally free? I know we need the ...
- 785
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2
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218
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Smooth toric variety which is a cube is a bott tower (reference request)
According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. ...
- 156k
1
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88
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Equivariant resolution of singularity making a pullback of a line bundle admit a root
I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
- 323
4
votes
1
answer
250
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Galois action on the pro-algebraic completion of the singular fundamental group
Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\text{an}}, x)$ generally does not carry an action of the absolute Galois group $\operatorname{...
- 199
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1
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131
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Poset definition of dimension
Let $\mathsf{k}$ be an algebraically closed field and $X$ an abstract variety (an integral separated scheme of finite type over $\mathsf{k}$).
Is there any way to define the usual dimension of $X$ ...
- 3,318
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246
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Fundamental group of degree 4 del Pezzo surface minus 16 (-1)-curves [Reference request]
Let $S$ be a degree $4$ del Pezzo surface (over $\mathbb{C}$).
That is, $5$ points blow-up of $\mathbb{P}^2$, or $4$ points blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$.
The classical fact is that $...
- 111
5
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1
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633
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Consistency of ZFC with inaccessible cardinals but no measurable cardinals
Let $S$ be a set and k a infinite field. The injection $S \to k\mathrm{Alg}(k^S, k)$ (sending a point to the evaluation in it) is a bijection if and only if $S$ is a non-measurable cardinal (see for ...
- 6,962
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1
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883
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
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0
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98
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A question about the sheaf supported on the zero section
Let $X$ be the total space of the cotangent sheaf on $\mathbb{P}^{2}$ and $i \colon \mathbb{P}^{2} \hookrightarrow X$ be thezero section. Suppose that $E$ is a coherent sheaf on $X$ which is set-...
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92
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Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
3
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190
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Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 328
1
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0
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99
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Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
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0
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146
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Can we find curves with many rational points using linear algebra?
Probably this is impossible, but let us try.
Working over $\mathbb{Q}[x_1,...,x_n]$.
Let $T_i$ be $n$ sets of rationals with cardinality $B$.
Assume we are given $n-2$ linear equations $f_i$ which are ...
- 25.4k
4
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1
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326
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Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 127
5
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1
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367
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Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 6,048
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8
answers
3k
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Object of proven finiteness, yet with no algorithm discovered?
I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...
- 1,053
8
votes
1
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653
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Status of a conjecture in Grothendieck's "Crystals and the de Rham Cohomology of Schemes"
Let $X/\mathbb{C}$ be a scheme over the complex numbers. In "Crystals and the de Rham cohomology of schemes," Grothendieck constructs the infinitesimal ringed site $(X_{\operatorname{inf}}, \...
- 333
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1
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128
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Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$
Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.
How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
- 233
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0
answers
127
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Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
1
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1
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291
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General algebraic definition of mirror symmetry
I'm trying to understand the following statement of Hori-Vafa from the algebraic perspective:
The mirror of the Hirzebruch surface $\mathbb{F}_{n}$ is the Landau-Ginzburg model $x+y+\frac{a}{x}+\frac{...
- 305
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219
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Quotient of K3 surfaces by non-symplectic automorphism of finite order
Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie non symplectic in sense of that that the induced action on $H(X,K_X=H^0(X, \Omega_X^2)$ is not trivial) of finite order.
...
- 6,048
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98
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Does the smooth locus of any toric variety built from a fan always contain a rational point?
Let $k$ be an arbitrary field and $X$ be a toric variety built from a fan, defined over $k$.
Does the smooth locus of $X$ always contain a $k$-rational point? Why?
- 639
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0
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
6
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1
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189
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Fully faithful embeddings of derived category of projective space into derived category of a higher dimensional projective space
Let $N>n$. Are there are any known cases of a fully faithful embedding $D^b(\mathbb{P}^n) \hookrightarrow D^b(\mathbb{P}^N)$?
- 123
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1
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257
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Reflections on affine quadric hypersurfaces
Let $f\colon\mathbb{Z}^n\otimes\mathbb{Z}^n\to\mathbb{Z}$ be a non-degenerate symmetric bilinear form and consider the affine quadric hypersurface $$
X:=\{f(x,x)+2=0\}\subseteq\mathbb{Z}^n.
$$
For ...
- 341
6
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1
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407
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Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
3
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0
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125
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Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
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0
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102
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Image of K3 surface under finite map with pure ramification rational
Let $X$ be a projective K3 surface and $f: X \to Y$ a non etale, finite map, restricting to etale on non empty open $U \subset Y$ of degree prime to char of alg closed base field of $k$. Assume ...
- 6,048
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0
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112
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Irregularity of surfaces for dominant maps
I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang:
Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an ...
- 6,048
2
votes
0
answers
133
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Dual of finite reflexive modules
Let $A$ be a commutative ring and $M$ be a finite reflexive $A$-module, i.e. the natural map $M\to (M^{\vee})^{\vee}$ is an isomorphism. Can we deduce that the dual $M^{\vee}$ is also finite?
- 151
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110
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Wobbly divisor in the moduli space of rank 2 degree 1semi-stable vector bundles over a curve of genus 2
I am looking at Nigel Hitchin's lecture "Higgs fields in low genus" on the occasion of Oscar Garcia-Prada's 60th birthday. In the rank 2 odd degree case, he mentions a map $f$ from the ...
1
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0
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102
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On descending a section of a morphism between schemes from formal completion to étale local
Here's the case, which arises from the context of doing infinitesimal deformation. Given a DVR $(S,\mathfrak{m},\kappa)$ we have the completion $\hat{S}$ with respect to $\mathfrak{m}$. Say we have a ...
- 11
2
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0
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105
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Torsion Freeness of Sheaf of Kähler Differentials
Let $X$ be an irreducible scheme over some base field $k$. Consider the sheaf of Kähler differentials $\Omega_{X/k}$. Let $w: \Omega_{X/k} \to j_* \Omega_{K(X)/k}$ be natural map induced by enbedding ...
- 6,048
0
votes
0
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170
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Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
