All Questions
Tagged with limits-colimits algebraic-geometry
46
questions
2
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0
answers
119
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Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?
I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around.
(1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
- 1,632
2
votes
0
answers
93
views
Spec of an infinite intersection of ideals, Spec of a colimit
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). ...
- 447
2
votes
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138
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Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)
This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
- 1,785
3
votes
1
answer
142
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Writing $\operatorname{Spec}\mathbb{Z}$ as an inverse limit of finite $T_0$-spaces
I want to show that $\operatorname{Spec}\mathbb{Z}$ can be written as an inverse limit of finite $T_0$-spaces. First off, $\operatorname{Spec}\mathbb{Z} = \{(0), (2),(3),(5),...\}$, so the closed sets ...
1
vote
0
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59
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Sheaves valued in a $k$-category
Let $\mathcal{C}$ be a $k$-category which we regard as an $\infty$-category whose objects all happens to be $k$-truncated.
A sheaf $F$ valued in $\mathcal{C}$, in the $\infty$-categorical sense, is a ...
- 1,613
0
votes
1
answer
70
views
Does invertibility of a section $f$ (of a sheaf of rings) in every open set containing $x$ imply invertibility of $f$ in the stalk at $x$?
Let $\mathcal{O}$ be a sheaf of commutative rings on a topological space $X$.
Let a point $x \in X$ and a global section $f \in \mathcal{O}(X)$ be given.
Suppose that for every open $U \subset X$ with ...
- 2,461
0
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0
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128
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Meaning and examples of Grothendieck condition AB4*
What are some examples of AB4* categories? In particular, for a thesis I am writing I need to know if cochain-complexes form an AB4* category
- 566
2
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0
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99
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Spec commutes with cofiltered limits in concrete example
Currently I am learning some alg. geometry and I would like to show the following claim:
Let $\mathfrak{p}$ be some prime ideal of $A$. Then
$$ \varprojlim_{f\notin \mathfrak{p}} \operatorname{Spec}...
- 944
4
votes
2
answers
348
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Is the limit of a family of sheaves a sheaf?
So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if ...
- 697
2
votes
1
answer
211
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Is the structure presheaf of $\text{Spec}(A)$ a limit of presheaves?
In Shafarevich's Basic Algebraic Geometry, he defines the structure presheaf on $\text{Spec}(A)$ first by $\mathcal{O}(D(f))\cong A_f$ and then $\mathcal{O}(U)=\lim_{D(f)\subset U}\mathcal{O}(D(f))$, ...
- 2,660
1
vote
1
answer
58
views
Do ample sheaves descend along limits / noetherian approximation?
Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a ...
- 9,497
1
vote
1
answer
104
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Lemma 10.94 from Goertz' and Wedhorn's Algebraic Geometry I
I have a question about the proof of Lemma 10.94 from Goertz' & Wedhorn's Algebraic Geometry I:
Lemma 10.94. Let $S$ = $\operatorname{Spec}A$ be an integral noetherian scheme with
generic point $\...
- 7,561
0
votes
2
answers
456
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The stalk as a colimit
From Vakil's notes:
The notion of colimit is defined for diagrams $D:I\to\textbf{Set}$. The colimit of $D:I\to\textbf{Set}$ is the limit of $D^{op}:I^{op}\to \mathbf{Set}^{op}$. What exactly is the ...
- 12k
3
votes
0
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624
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Direct limit, inverse limit and Spec
The set-up
$k$ is a field
$T_i = \operatorname{Spec} A_i$ is an inverse system of affine $k$-schemes, where $i<j$ if $\operatorname{Spec} A_j \subset \operatorname{Spec} A_i$ (inclusion).
$X$ is a ...
- 1,207
2
votes
1
answer
133
views
Filtered colimits in $D(X_{\text{proét}})$
In §5.2 of their pro-étale paper, Bhatt and Scholze define a functor $L$ in terms of a sequential colimit, and, separately, a full subcategory $D’$ of $D(X_{\text{proét}}):=D(X_{\text{proét}},\mathbf ...
- 2,161