All Questions
2,036 questions
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Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
- 501
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0
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137
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Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
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0
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106
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Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM
Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...
- 912
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2
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912
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Sheaf isomorphism
Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\...
- 331
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293
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Specialization map and fibration
Let $\pi:X \to \Delta$ be a proper, surjective, flat morphism (here $\Delta$ is the unit disc), smooth over $\Delta \backslash \{0\}$ and possibly singular central fiber. There is a fibrewise ...
- 2,323
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1
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158
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Locus of osculating planes
Let $Q\subset \mathbb P^n$ ($n\geq 3$) be a smooth quadric. Let us denote $$Z=\{ [P]\in G(3,n+1), \exists l\subset Q\mathrm{\ line\ s.t.\ } 2l\subset Q\cap P\}$$ the locus of osculating planes. Is ...
- 1,463
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324
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Simultaneous triangularizability over a commutative ring
Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $...
- 3,741
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235
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Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
- 2,837
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303
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Berthelot-Ogus comparison isomorphism
On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,
We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
- 595
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1
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234
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Zeroes of elementary polynomials without involving closed-form solutions
Consider the following two polynomials, where $n$ is an integer:
$$
p_n(x) = x^3-\frac1nx-\frac2n, \\
q_n(x) = x^2-\frac2n.
$$
For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
- 21
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0
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162
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Motivic complex on arithmetic schemes
If we believe the finite generation of motivic cohomology for regular arithmetic schemes like $X$ then we can see that (using Quillen-Lichtenbaum)for infinitely many primes $l$ we have an isomorphism ...
- 5,901
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203
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How generic are Cayley graphs of non-Abelian groups with logarithmic girth?
Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...
- 1,893
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1
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506
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Gaussian measures on non-separable spaces
Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
- 8,512
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363
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Coherent locally free sheaves on projective varieties
Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$....
user138661
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155
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
I need the reference to a detailed proof the following fact.
Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
- 494
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1
answer
537
views
Functional equations of zeta functions over global fields
The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in \...
- 7,062
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2
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788
views
Defining algebraic manifold without referring to schemes
Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions.
Is it true that there exists a smooth integral ...
- 125
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275
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Corollary 1.6 in Mumford's Geometric Invariant Theory
I do not understand fully the proof of Corollary 1.6 from Mumford's Geometric Invariant Theory (§3: Linearization of an invertible sheaf, p 35):
Corollary 1.6
$\DeclareMathOperator\Spec{Spec}\...
- 5,946
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198
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Proper direct image and minimal extension
Let us denote by $i:U\rightarrow X$ an open inclusion and $\mathcal{F}$ a coherent sheaf on $U$.
Is the map $i_!\mathcal{F} \rightarrow i_*\mathcal{F}$ always injective ?
I ask this question because ...
- 81
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0
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102
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Maximum number of bounded primitive integer points in a zero-dimensional system
Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
- 1,836
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416
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When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 61
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299
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Application of Galois descent
I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...
- 5,946
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206
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Semi-orthogonal decomposition of Verra threefold
Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It ...
- 1,982
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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.
...
- 5,946
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189
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Finite and Locally Free Map is Open
I have a question about a statement from Szamuely's "Galois Groups and Fundamental Groups" in the exerpt below (look up at page 152):
Let $\phi:X \to S$ be a finite and locally free morphism of ...
- 5,946
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161
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Special elliptic pencil of an Enriques surface (arguments in a proof)
I have a couple of questions about arguments in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $Y$ ...
- 5,946
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371
views
Hilbert scheme of relative subschemes of lenght 2
Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed ...
- 155
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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
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282
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Smooth absolutely irreducible (?) genus 1 plane pointless curve over $\mathbb{F}_{13}$
We got a family of genus 1 plane curves that may violate a bound
in a paper.
Explicitly: Let $F(x,y)$ be the degree 39 polynomial with integer coefficients:
...
- 25.4k
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2
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350
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Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
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996
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Torsion in the (co-)homology of a smooth projective variety - what is known in general?
There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...
- 3,058
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0
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241
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Good factors of L-function
Let $X/K$ be a smooth projective variety over a number field $K$. I have seen two definitions of the L-function $L^i(s)$ attached to it's cohomology groups $H^i(X)$.
Both definitions agree that at a ...
- 3,555
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0
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188
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Affine scheme as algebraic space
We working in the following with Knutson's definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):
...
- 5,946
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263
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Hopf lemma for line bundles on curves in algebraic geometry
In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...
- 73
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285
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a problem about ideals of polynomial rings
Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let $(...
- 1,990
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1
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484
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what are the subgroups of an algebraic group with codimension one [closed]
let G be an algebraic group. which subgroups of G are codimension one subgroups.
- 225
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202
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Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
- 13.6k
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3
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641
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polynomial branched cover of the sphere with specified monodromy
We know by the Riemann Existence Theorem that any Riemann surface can arise holmorphically as the branched cover of a sphere:
Which Riemann surfaces arise from the Riemann existence theorem?
Do ...
- 22.8k
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699
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When are Segre- and Veronese embeddings Gorenstein?
Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m_1}\times\cdots\times\mathbb P^{m_n}$, is there a characterization in terms of $m$ and $n$, or the $m_i$, for the ...
- 83
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194
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Locus of roots of all convex combinations of two monic polynomials, II
This post contains a revised conjecture to a conjecture I posed previously which was shown to be false.
Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]$...
- 1,719
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175
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Algebraic numbers with a polynomial property
In my research I faced with an intricate construction of an algebraic number with some properties.
Problem. For which classes of polynomials $P(X,Y)\in \mathbb{Z}[X,Y]$, we have the following property....
- 515
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202
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What is the dimension of this subvariety of the Grassmannian?
Well, actually, what are the dimensions of the following two subvarieties of the Grassmannian. Let $N$ be a positive integer.
Let $V \subseteq \mathbb{C}^N$ be a linear subspace of dimension $N-k$ ...
- 980
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2
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296
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Are orbits of an affine algebraic monoid affine?
Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
- 4,902
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0
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234
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Separability of a simple ring extension
Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
- 2,837
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654
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Sheafification map is surjective
This is not a research level problem for sure. But, similar question was asked by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there....
- 6,023
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0
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165
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Discriminant ideal in a member of Barsotti-Tate Group
Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$
is an inductive system
...
- 5,946
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0
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74
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Generators of fixed function fields under involutions
I am trying to prove something for function fields in two generators and only one involution but I don't know if it is true, if true, I would like to generalize, here it is.
Let $K=k(\eta_1,\eta_2)$ ...
1
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1
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437
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Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$
Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has ...
- 1,555
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1
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211
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Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
- 5,946
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140
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On finite dimensional commutative algebras and regular sequences
Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
- 1,227