All Questions
Tagged with sheaves or sheaf-theory
979 questions
4
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How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$
For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
- 415
4
votes
1
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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 ...
- 6,038
4
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1
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409
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Why are Regular Categories assumed to be finitely complete?
Regular categories may equivalently defined as those with:
finite limits
coequalizers of kernel pairs
pulback stable regular epis
or
finite limits
pullback stable regular epi/mono factorization
...
- 352
4
votes
2
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416
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Is any constant Zariski sheaf already a Nisnevich sheaf?
Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (...
- 1,906
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1
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373
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flat descent for perverse sheaves
Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
- 3,472
4
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1
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478
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Euler Characteristic of Coverings via Sheaf Theory
Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works),
$f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic.
By the ...
- 2,554
4
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1
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250
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When do adjunctions preserve equivalence?
Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors $F_{\...
- 3,290
4
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1
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301
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What extra structure does the group of automorphisms of a torsor carry?
Let $Y$ be a space, and $G$ a group. For simplicity we can take $G$ to be finite, $Y$ to be a point, if this avoids technical issues. Then for $X/Y$ a $G$ torsor, so a sheaf of sets with $G$ action on ...
- 1,949
4
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1
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435
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Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
- 395
4
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1
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231
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Morphisms of flat families of sheaves
$X$: projective scheme over a scheme $S$.
$E, F$: $\mathscr{O}_X$-modules, flat/$S$
$\phi$: $E \rightarrow F$ : morphism s.t. $\phi_t$: $E_t \rightarrow F_t$ is zero morphism for all $t \in S$
Then,...
- 479
4
votes
1
answer
262
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Interesting (non) examples of singular support
I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
- 41
4
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1
answer
409
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Does the nearby cycle functor commute with the Verdier duality?
I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
- 21.8k
4
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1
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259
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$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$
Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the sheaf $f^\ast \mathcal I$...
- 41
4
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1
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752
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Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves
In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a ...
- 456
4
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3
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489
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Why is continuity required for sheaf-theoretic definitions of a structure on a space
For example, I take differentiability, analyticity, and algebraicity(of a function).
All(more or less) imply continuity. So when we define a differentiable function on $\mathbb R^n$ or an analytic ...
- 3,699
4
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1
answer
168
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Sheaves and gratings
A grating is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows.
A grating (carapace in french) is defined by a topological space $X$, a module (or a ...
- 18.7k
4
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1
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199
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Conformal groupoid
I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...
- 439
4
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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 $S$ ...
user333445
4
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1
answer
293
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Functorial isomorphisms
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $ \mathcal{C} $ gives a colimit preasheaf $ \mathcal{F}^{+} $ ...
- 41
4
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1
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674
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Jordan–Hölder sequence for $\mu$-semi stable sheaves
Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class.
I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
- 1,286
4
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1
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637
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On the Leray spectral sequence and sheaf cohomology
I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
- 1,227
4
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1
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291
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Exactness of $j_!$ in abelian category recollement
Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...
- 6,025
4
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1
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154
views
Is an objectwise subframe a sub-inf-lattice in a topos?
Suppose a sheaf $F$ on a site $(\mathsf C,J)$ has the property that for each $X$, $FX$ is a subframe of the subobject poset of $X$.
I think $F$ is a subsheaf of $\Omega$ in the sheaf topos $\mathcal ...
- 10.5k
4
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1
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1k
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On push-forward of the constant sheaf for fibrations
Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
- 21.8k
4
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1
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511
views
Nearby cycles and specialisation - properties
I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
- 5,562
4
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1
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504
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About the construction of the Universal Enveloping Lie Algebroid
Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T_X$ is a Lie algebroid, locally free as a $\mathcal{O}_X$ module, and its Universal Enveloping Lie Algebroid $\mathfrak{U}(...
- 63
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1
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781
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Restricting a Soft Sheaf to an Open is again Soft?
Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :)
Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are ...
- 329
4
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1
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639
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Morphisms between pure complexes of sheaves
I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
- 16.9k
4
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1
answer
191
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Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
Context and Notation
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...
- 1,611
4
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1
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216
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$G$-torsor for topological space compared to that for sheaf of groups
I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
- 365
4
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1
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263
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Relating deformations of a scheme to deformations of its singular locus
Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
- 8,998
4
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1
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604
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Extension of a first order deformation of a sheaf
Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all ...
- 1,391
4
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1
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549
views
Does this condition reduce to the correct notion of irreducibility on schemes?
Consider the category of sheaves (of sets) on the affine étale site. It's a well known fact that a morphism of schemes is a Zariski-open immersion if and only if it is an étale monomorphism, so we ...
- 19.6k
4
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1
answer
289
views
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
$\...
- 6,038
4
votes
1
answer
550
views
Dualizing sheaf and determinant of cohomology
Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent ...
- 141
4
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1
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893
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Holomorphic logarithmic De Rham complex
Let $X$ be a complex variety of dimension $n$ and $D$ a smooth hypersurface.
Let $\Omega_X(logD)^*$ be the holomorphic logarithmic De Rham complex: $\omega\in \Omega_X(logD)^k$ is a form of degree $k$...
4
votes
1
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390
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Is every soft sheaf of countable $\mathbb Q$-vector spaces a direct sum of skyscraper sheaves?
Let $X$ be a finite-dimensional compact metrizable space (these properties might partially be irrelevant; on the other hand, the case $X=[0,1]$ is already interesting to me).
Let $\mathcal F$ be a ...
- 3,184
4
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2
answers
695
views
Colimits of covers
Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show ...
- 15.5k
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0
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48
views
Resolution of constant sheaf by $L^2$ function sheaves
Let $X$ be a compact Hausdorff space equipped with a Radon measure of full support.
Then $U\mapsto L^2(U)$ is a fine sheaf, hence can be taken for a first step in an acyclic resolution of the constant ...
- 460
4
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0
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178
views
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
4
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0
answers
102
views
Topos as a totally cocomplete object in a 2-category CART
In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...
- 1,347
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0
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216
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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 ...
- 6,038
4
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0
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278
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Are manifolds "naturally" ringed or locally ringed spaces?
My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.
On the one hand, it's reasonable to ...
- 711
4
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0
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318
views
Is the restriction of an injective sheaf on a closed subscheme still injective?
Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$.
Question. Is $i^*\mathcal{I}$ still an ...
- 1,479
4
votes
0
answers
195
views
Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
- 351
4
votes
0
answers
120
views
Understanding a step in proof of sheaf version Verdier duality
Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes ...
- 1,856
4
votes
0
answers
201
views
Infinity-categorical exceptional push-forward
Classically, if $f:X\to Y$ is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor $f_!:Sh(X;k)\to Sh(Y;k)$ among $k$-valued sheaves for, say, a ...
- 4,189
4
votes
0
answers
138
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Examples of non-hypercomplete sheaves on affine schemes
Let $A$ be a commutative ring and let $\mathcal{O}$ be a sheaf of $E_{\infty}$-ring spectra on $\mathrm{Spec} A$ such that $\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$. Lurie provides a ...
- 12.1k
4
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0
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205
views
Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 1,761
4
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0
answers
347
views
How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?
$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
- 435