All Questions
22,548 questions
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Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
3
votes
1
answer
272
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degeneration of a Veronese surface
Let $V$ be the Veronese surface, obtained as the image of $\mathbb{P}^2$ in $\mathbb{P}^5$ by the complete linear system of conics. I understand that $V$ can degenerate to the union of a cubic scroll $...
- 3,779
4
votes
0
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178
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Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,946
5
votes
1
answer
327
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Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf{...
- 1,349
0
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1
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178
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Cohomology of Blow-ups and Minimal Models in Higher Dimensions
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero. Consider a sequence of blow-ups:
$$X_n \xrightarrow{\pi_n} X_{n-1} \xrightarrow{\pi_{n-1}} \cdots \...
- 17
10
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2
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286
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Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $...
- 195
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114
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
1
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178
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Prop 1.3 in "Birational geometry of algebraic varieties": specialization of rational curves
I have a couple of questions about some arguments in proof of Proposition 1.3 from Birational geometry of algebraic varieties by Kollár and Mori (p 8):
Proposition 1.3. [Abh56, Prop. 4] Let $X$ be ...
- 5,946
1
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0
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88
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Identification of different components of Hilbert modular surface?
I'm wondering whether the different components of the Hilbert modular surface can be (naturally?) identified with each other, or if they're at least abstractly isomorphic. (I'd also be interested in ...
- 2,054
2
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1
answer
68
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When are solutions to a set of multi-variate quadratic equations isolated points?
Suppose I have set of $n$ multi-variate polynomial equations with $n$ unknowns $x_1, \dots, x_n$. The $n$ equations have real coefficients and are quadratic (so largest degree is $2$). How do I ...
2
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124
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Surjectivity of a restricted Hitchin map
Let $C$ be a smooth projective algebraic curve over the complex numbers and let $V$ be a stable rank 2 vector bundle on $C$. Then consider the vector bundle $\mathcal{E}nd_0(V) $ of tracefree ...
6
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0
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138
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For a proper scheme $X,$ the neutral component of automorphism group scheme $\mathrm{Aut}^0_X$ is an algebraic group
I am currently reading through M. Brion's "Notes on Automorphism Groups of Projective Varieties" found here. On page 5 he writes
We assume from now on that $X$ is proper, and we denote by $\...
- 201
4
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1
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291
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Counter example to every closed subscheme $\operatorname{Proj} A$ is of the form $\operatorname{Proj}A/I$
I was under the impression that for a positively graded ring $A$ (not necessarily generated in degree $1$) that every closed subscheme of $\operatorname{Proj}A$ was of the $\operatorname{Proj}A/I$. ...
- 391
8
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1
answer
525
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How to go about finding polynomial from specified monodromy?
I want to find the polynomial $p(x,y)=0$ that corresponds to a four-sheeted Riemann surface with monodromy $(123),(132),(124),(142)$ at four branch points. Such a surface is genus 1, but I'm ...
- 81
1
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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
1
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169
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Generalized Langlands correspondence for non-simply connected groups
Consider a connected real reductive group $G$ and its complexification $G_{\mathbb{C}}$. Define the notions of a generalized Langlands dual group $G^\vee$ for non-simply connected groups, denoting the ...
- 27
1
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0
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107
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Universal motivic measures for singular varieties
Consider a field $K$ of characteristic zero and the category $\text{Var}_K$ of varieties over $K$. We're interested in the Grothendieck ring of varieties $K_0(\text{Var}_K)$ and the concept of motivic ...
8
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259
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What is an example of Beilinson's theorem on $D^b\mathrm{Perv}$ failing for non-field coefficients?
In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $...
- 495
2
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1
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265
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On intersection theory on toric varieties
Let $\Delta$ be a polytope and consider the projective toric variety $P_{\Delta}.$
Given a curve $C \subset \mathbb{P}_{\Delta},$ which is not toric, is it true that in the Chow group we have
$$ C = \...
- 55
4
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0
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100
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Structure of points of elliptic curves in field with restricted ramification
Let $k$ be a finite field of characteristic $p$ and let $C$ be a curve over $k$. Let $E$ be a non-constant elliptic curve over $k(C)$. Taking the Néron model of $E$ and removing the singular fibers ...
- 177
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1
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94
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Embedding of finite group quotient of the projective space
Let us consider the $\mathbb{Z}_2$ action on the complex projective space $\mathbb{P}^3$ defined by the involution $[Z_0, Z_1, Z_2, Z_3] \to [Z_1, Z_0, Z_3, Z_2]$. Let $Y$ be the quotient space.
...
- 549
1
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0
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108
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Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 326
2
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0
answers
101
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Conjecture on the moduli space of stable sheaves on Calabi-Yau threefolds
Let $X$ be a smooth projective Calabi-Yau threefold over $\mathbb{C}$. Let $M_{X}(r,c_1,c_2)$ denote the moduli space of Gieseker-stable sheaves on $X$ with Mukai vector $(r,c_1,c_2)$.
Is the ...
2
votes
0
answers
279
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Why is the weight monodromy hard in mixed characteristics?
I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...
- 2,907
3
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1
answer
187
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Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 541
2
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1
answer
211
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Splitting of composition of trace and counit in derived setting
Let $X,Y$ be varieties (separated of finite type schemes) over base field $k$, $\mathcal{F}$ be constructible sheaf on $Y_{\mathrm{et}}$ and assume that we have a finite morphism $f: X \to Y$, which ...
- 5,946
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159
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Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
- 1,763
2
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0
answers
149
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Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
3
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199
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When a fully faithful functor from an abelian category to itself will be an equivalence
Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence.
If $A$ is a "nice" category, I think $i$ ...
- 253
2
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1
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262
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Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
- 343
2
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0
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62
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Base change for finding fibers of the pushforward of a line bundle along a proper non-flat morphism
Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
- 2,974
2
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1
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204
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What does the Serre functor of equivariant category of fractional CY category look like?
I am considering the following set up. Let $\mathcal{A}$ be a fractional Calabi-Yau category and denote by $S$ the Serre functor and $S^m=[n]$. Now I consider a finite group action $G$ on $\mathcal{A}$...
- 1,982
3
votes
1
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366
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Descent for étale covers of proper regular models of elliptic curves
Let $K$ be a complete (but think Henselian suffice for purposes of this question) local field of characteristic $0$ with residue field $k$ of characteristic $p>0$ and ring of integers $R=\mathcal{O}...
- 5,946
4
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0
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161
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Gauss-Manin connection, base change and projection formula
Let $X$, $Y$ be complex analytic manifolds, and $f : X \to Y$ a proper, smooth morphism (i.e. smooth fibers). In every course talking about the Gauss-Manin connection, you find the following formula :
...
- 123
1
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138
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Quotients of open subsets of the semi-stable locus
This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point.
Let $U$ be the set of irreducible non-cuspidal ...
- 23.5k
7
votes
2
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617
views
Genus 0 curves on surfaces and the abc conjecture
One of the most obvious methods to prove that a given Diophantine equation $P(x_1, \dots, x_n)=0$ has infinitely many integer solutions is to find polynomials $P_1, \dots, P_n$ in one variable $u$, ...
- 7,182
1
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0
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166
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Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
4
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1
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178
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Computing the divisor class group of toric varieties over an arbitrary field
Let $k$ be an arbitrary field and let $X$ be a toric variety over $k$, coming from a fan $\Sigma$. If $k$ is algebraically closed, then theorem 4.1.3 of Cox ,Little and Schenck’s Toric Varieties book ...
- 639
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85
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Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
4
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2
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296
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Boundedness of the preimage of sphere via homogeneous polynomials
I am stuck with the following question. Any help or reference would be greatly appreciated.
Assume $F:\mathbb R^n\to \mathbb R^m$ to be a homogeneous polynomial of degree $d$, and assume $F$ to be ...
- 311
4
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0
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249
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Bounds for torsion in Betti cohomology
Let $X\subset \mathbb{P}^{N}_{\mathbb{C}}$ be a smooth, projective variety of dimension $n$ and degree $D$. Is there an upper bound on the torsion in the Betti cohomology groups $H^{i}(X, \mathbb{Z})$ ...
- 41
3
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1
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265
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Base change in Chriss-Ginzburg
Below is a fragment of the book by Chriss and Ginzburg. Proposition 5.3.15(b) is stated in $K$-theory. My question is, does the same conclusion (and proof?) of proposition 5.3.15(b) (i.e. base change) ...
- 2,974
6
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0
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173
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How derived category behaves for pushout of schemes
If $X,Y,Z$ are schemes with two closed embeddings $Z \to X$, $Z \to Y$, then we have a pushout $X \sqcup_Z Y$ in the category of schemes. Consider the induced diagram of bounded derived category of ...
- 165
0
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0
answers
65
views
Image of a hyperplane under finite map
If we consider the following map $f: \mathbb{P}^n \to \mathbb{P}^n$ which takes $(x_0, x_1,..., x_n) \to (x_0^m, x_1^m, ..., x_n^m)$, then the image of the hyperplanes $H_i:= \{x_i= 0\}$ are also a ...
- 549
1
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0
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75
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Simplicity of Jacobian of curves of genus 2
Let $p\neq 2$ and $a, b, \in \overline {\mathbb F_{p}}\setminus \{0,1\}$ two elements distinct from each other, and let $s$ be an element which is transcendental over $\overline {\mathbb F_{p}}$.
We ...
- 187
1
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0
answers
128
views
Induced map of a GIT quotient map
Let $X$ be a smooth variety, $G$ a connected reductive group, and there is a $G$-linearlisatioin of a line bundle $L$.
Let us consider $p:X^{ss}(L)\rightarrow X/\!\!/_LG$, which is the GIT quotient ...
- 11
2
votes
2
answers
87
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Computation of ideal of functions, given by explicit quadratic equations, vanishing on $G/P$ for the exceptional Lie group $G_2.$
In Section 10.6.6 of Procesi's "Lie Groups" he writes that a theorem due to Kostant tells us that for an algebraic group $G$ and a parabolic subgroup group $P,$ the ideal of functions ...
- 201
3
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0
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181
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Conditions for an open mapping between spectra
Let $(A,\mathfrak{m})$ be a Locally Noether Ring, and $\hat{A} = \varprojlim A/ \mathfrak{m}^{n}$ .Furthermore, let $f : A \to \hat{A}$ be a canonical morphism, and consider the mapping $f^{*} : Spec(\...
- 41
9
votes
1
answer
330
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Nonzero module with vanishing derived fibers
What's an example of a nonzero $R$-module with vanishing derived fibers at all points of $\mathrm{Spec}(R)$?
This was asked in When does a quasicoherent sheaf vanish?
but the answer there only says ...
- 2,356
4
votes
0
answers
192
views
Vanishing of all higher direct images for a non-flat morphism
Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$) ...
- 2,974