All Questions
Tagged with sheaf-cohomology ag.algebraic-geometry
221 questions
1
vote
0
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91
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Continuity of motivic cohomology under direct limit
Given the motivic complexes $\mathbb{Z}(n)$ on the big Zariski site of finite type smooth $k$-schemes denoted by $FinSm_k$, we pullback it to the smooth $k$-schemes i.e. $Sm_k$. For example for a ...
- 5,901
1
vote
0
answers
199
views
Künneth formula for local cohomology with support
In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
- 473
5
votes
2
answers
527
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Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
user329228
6
votes
2
answers
524
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Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
- 8,998
7
votes
1
answer
334
views
Extending $G$-torsors on open subsets of affine space
Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). ...
- 984
8
votes
1
answer
2k
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Cohomology of Grothendieck topology
My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...
- 2,050
6
votes
2
answers
632
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Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 1,343
6
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0
answers
1k
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Intuition behind exceptional inverse image?
The story is probably well-known: given a map $f:X\to Y$ of spaces (say schemes, but there are many other contexts), we have two classical operations between sheaves on $X$ and those on $Y$: the ...
- 28.2k
4
votes
0
answers
583
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Etale cohomology of a nodal (cuspidal) curve
Let $k$ be a separably closed field, and $X/k$ be a curve (not necessarily complete) with a single singularity, a simple node $x$. Suppose $\ell$ is a prime number invertible in $k$, how do we compute ...
- 547
2
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0
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201
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Higher cohomology of projective bundles
Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...
- 439
4
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2
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484
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Removing Noetherian condition from cohomology and base change
This question is related to a question I asked a few days ago. Since there seems to be no (at least for me) satisfying reference for cohomology and base change as stated by Vakil in his script in ...
- 445
6
votes
1
answer
557
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Cohomology and base change without Noetherian assumption
In the "The Rising Sea" by Vakil one can find the base change theorem for proper morphisms over a locally Noetherian base (28.1.6). He later indicates (28.2.M) how one could exchange the ...
- 445
3
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0
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641
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fppf/ etale Cohomology calculate with Cech cohomology
Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map
let $\phi: R \to S$). Then the so called Amitsur complex
$R \to S^{\...
- 5,986
6
votes
2
answers
445
views
Representability of flat cohomology by a group scheme
In his paper "Supersingular K3 surfaces", Artin states the following theorem (Theorem 3.1) without proof:
Let $\pi:X \to S = \mathrm{Spec}(k)$ be a smooth proper surface with $k$ an ...
- 10.5k
3
votes
2
answers
371
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Extension between vector bundles inducing non-zero map on cohomology
Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i(...
- 7,377
6
votes
2
answers
789
views
Reference request: Kleiman's proof of Snapper's Lemma
On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as
a special case of Snapper's Lemma, see &...
- 1,805
3
votes
1
answer
258
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Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
3
votes
1
answer
332
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Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
- 1,805
2
votes
1
answer
279
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Help about "Varieties with small Dual Varieties" by L.Ein
I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
- 1,343
1
vote
1
answer
454
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Relationship between $H^1(X, \mathbb{T})$ and complex line bundles
Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, ...
3
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0
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446
views
Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole
Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by the two affine ...
- 1,479
1
vote
0
answers
469
views
Dimension of global holomorphic sections of a line bundle
Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
- 35
1
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0
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104
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$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality
This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...
- 1,168
4
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0
answers
211
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Does cohomology and base change hold if supported at a point?
I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and ...
- 1,104
3
votes
1
answer
556
views
Elementary way to compute Hodge numbers of Grassmanian
I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf ...
- 1,093
10
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0
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964
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intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
- 546
1
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0
answers
160
views
Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
- 1,479
4
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1
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289
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Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$
I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...
- 5,986
2
votes
0
answers
405
views
Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)
I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2
& Example 12.9.2):
Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point
$(0:...:1)...
- 5,986
4
votes
2
answers
1k
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Sheaf cohomology commutes with colimits of sheaves
Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...
- 5,986
6
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0
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637
views
Calculation in prismatic cohomology
In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...
- 91
1
vote
0
answers
161
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Surjectivity of multiplicative map (in more specific case)
(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.)
Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $...
- 111
1
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0
answers
56
views
local acyclicity when restricting to an hypersurface
Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...
- 3,472
2
votes
0
answers
136
views
A infinity structure on Yoneda Ext group
I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
- 213
1
vote
1
answer
1k
views
Injectivity of the cohomology map associated to the pullback of line bundles
Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
- 321
1
vote
1
answer
2k
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Pullback map on global sections surjective
Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...
- 5,986
1
vote
0
answers
191
views
Group cohomology of sheaves under closed immersion
Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...
- 11
6
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1
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761
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The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
- 294
2
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1
answer
385
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Very weak Riemann-Roch on curves (by J. Kollar)
I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on curves)...
- 5,986
3
votes
1
answer
460
views
Does local cohomology commute with pullback?
Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.
For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
- 473
8
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0
answers
257
views
Global functions on a product of schemes over artinian ring
For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras
$$
c:A(X)\otimes_R A(Y)\to A(X\times_SY)
$$
...
- 4,492
3
votes
0
answers
188
views
Sheaf cohomology of the complement of a schubert variety
Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
- 473
8
votes
1
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506
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Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?
Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...
- 1,198
1
vote
1
answer
415
views
Multiplicative structure for sheaf cohomology of flag varieties
Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...
8
votes
1
answer
623
views
Can one determine the trace map for a nonsingular projective variety explicitly?
I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the ...
- 615
5
votes
0
answers
268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
4
votes
1
answer
244
views
Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$
Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...
- 790
1
vote
0
answers
152
views
What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
- 435
4
votes
0
answers
240
views
Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence
Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$.
In the beginning he states that there exisits ...
- 141
6
votes
1
answer
284
views
Formal character of local cohomology groups with support in Schubert cells
Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. ...
- 473