All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
1
vote
1
answer
877
views
Direct image of reflexive sheaf via finite, flat map
Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free ...
- 1,813
0
votes
0
answers
98
views
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-...
21
votes
2
answers
2k
views
Naive question about constructing constructible sheaves.
In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each ...
- 28.7k
0
votes
0
answers
123
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Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 6,038
4
votes
1
answer
323
views
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 ...
- 15.9k
5
votes
1
answer
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 ...
- 3,998
2
votes
1
answer
326
views
Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
- 306
2
votes
1
answer
232
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
1
vote
1
answer
127
views
Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 39.3k
4
votes
0
answers
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 $X,Y$ ...
- 6,038
3
votes
0
answers
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
4
votes
1
answer
236
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Ampleness verifiable over faithfully flat cover
Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for ...
- 63.9k
24
votes
1
answer
837
views
Is there a useful theory of D-modules on smooth (non-analytic) manifolds?
D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
- 16.2k
2
votes
1
answer
217
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Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
- 2,727
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 148k
0
votes
0
answers
116
views
How can I calculate $\chi(\mathscr{O}(P))$
Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set
\begin{gather*}
\overline{X}=L \otimes X.
\end{gather*}
Assume $\overline{X}$ also irreducible. Now, ...
- 69
2
votes
1
answer
257
views
Understanding spaces is the same as understanding (sheaves of) functions on the space
I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written:
[...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and
understanding vector bundles (...
- 1,109
2
votes
0
answers
137
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details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
1
vote
0
answers
141
views
Homeomorphic endomorphism of schemes inducing equivalence of sheaves
Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
- 6,038
8
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0
answers
644
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Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
CommunityBot
- 1
5
votes
1
answer
350
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 28.7k
4
votes
1
answer
355
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Bott & Tu differential forms Example 10.1
In Bott & Tu's "Differential forms", Example 10.1 states:
$\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...
- 5,534
1
vote
0
answers
126
views
Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
3
votes
0
answers
205
views
Category of sheaves of vector spaces on BG
Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...
- 53
0
votes
0
answers
133
views
Higher direct images of locally constant etale sheaf under smooth proper map locally constant
Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...
- 6,038
5
votes
1
answer
286
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 6,038
0
votes
2
answers
331
views
Vakil exercise on sheaf associated to the divisor of rational section
This is exercise 15.4.G. of Vakil's notes.
Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...
- 29
39
votes
6
answers
9k
views
What is the inverse image sheaf necessary for in algebraic geometry?
Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \...
- 18.4k
2
votes
0
answers
170
views
Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
3
votes
1
answer
260
views
Etale cohomology of relative elliptic curve
Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...
- 148k
0
votes
0
answers
57
views
Lifting of quadrics containing hyperplane section for projectively normal curves
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 6,038
3
votes
1
answer
226
views
Čech cohomology refinement mapping
Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...
CommunityBot
- 1
1
vote
0
answers
272
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Vakil's Generalization of qcqs Lemma
(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma)
In the most recent notes of Vakil, this is problem 15....
- 10.4k
4
votes
0
answers
216
views
When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
- 63.9k
3
votes
1
answer
550
views
Characterization of étale locally constant sheaves over a normal scheme
I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...
- 148k
0
votes
0
answers
156
views
A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,038
1
vote
0
answers
221
views
Line bundles on curve with nodal singularity
Let $C$ be be an irreducible reduced curve over alg closed field $k$ with only one single nodal singularity $x$ and $f:N \to C$ it's normalization with $f^{-1}(x)=\{x_1,x_2\}$ (as set), and an iso ...
- 6,038
98
votes
10
answers
14k
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equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 4,492
7
votes
1
answer
787
views
Definition of the cotangent complexes of Artin stacks
I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...
CommunityBot
- 1
2
votes
1
answer
752
views
why is counit called the trace map
Let $f: X \to Y$ be a morphism of schemes, then
$f_*$ and $f^*$ form an adjoint pair inducing natural
correspondence
$\text{Hom}_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F})=
\text{Hom}_{\mathcal{O}_Y}(...
- 148k
4
votes
2
answers
340
views
Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
- 1,611
66
votes
4
answers
11k
views
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
- 1,446
2
votes
1
answer
245
views
Compatibility of Beck Chevalley condition: sheaves
Given a (not necessarily Cartesian) square of spaces
$$\require{AMScd}\begin{CD}
X @>g>> \overline{X} \\
@VVfV @VV\overline{f}V \\
Y @>\overline{g}>> \overline{Y}
\end{CD}$$
does the ...
- 409
3
votes
1
answer
466
views
Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
- 28.7k
8
votes
2
answers
2k
views
Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?
In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, ...
- 4,492
8
votes
1
answer
1k
views
Fpqc sheafification and localisation
I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
- 405
1
vote
1
answer
176
views
Field of fractions of etale stalk of Dedekind domain (Example from Milne's LEC)
Let $X=\operatorname{Spec}(A)$ be an affine Dedekind domain with field of fractions $K$. Let $\widetilde{A}$ be the integral closure of $A$ in separable closure $ K^{\text{sep}}$. A closed point $x$ ...
- 28.7k
2
votes
1
answer
290
views
Calculate stalk of etale derived pushforward sheaf (Milne's LEC)
Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale ...
- 6,038
2
votes
2
answers
429
views
If $\mathcal{F}$ globally generated, then counit map $f^*f_* \mathcal{F} \to \mathcal{F} $ surjective
Let $f: X \to S$ be a morphism, and $\mathcal{F}$
be quasi-coherent $\mathcal{O}_X$-module generated by global sections (eg if
$X$ projective, then this holds for the twisted sheaf $\mathcal{F}(n)$ ...
- 39.3k