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Questions tagged [fibre-product]

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How to understand fibres of morphisms of schemes.

Let $f:X\to Y$ be a morphism of schemes, and let $k(y)$ to be the residue field of the point $y$. The fibre of the morphism $f$ over the point $y$ is defined to be the scheme $X_y=X\times Spec(k(y))$. ...
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0answers
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EGA I (Springer), Proposition 0.4.5.4.

I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I. When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...
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0answers
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Hartshorne II.3.22c following the hint

I know this exercise can be solved in several ways, such as Ravi Vakil's proof or this answer, but I'd like to try to follow the given hint if possible. The exercise says the following: Let $f:X→...
2
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1answer
46 views

The existence of fibre product of schemes

As we know, if $X\to S$ and $Y\to S$ are two morphisms of schemes, then the fibre product always exists. But considering, for a non-surjective morphism $f:X\to S$ and a closed point $s\in S\setminus f(...
2
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1answer
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A question on the fibered product of schemes

In Hartshorne's "Algebraic Geometry", on p.99, at the end of the proof of Lemma 4.5, the author mentions that in general, if $\xi \in X \times_Y X$ and $p_1(\xi) = p_2(\xi)$, where $p_1$ and $p_2$ are ...
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0answers
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Fibred product of a (vector bundle) set with itself

Maybe the question is stupid but I can't find a rasonably solution. Let $\pi:E\rightarrow M$ be a bundle and let $f:N\rightarrow M$ be a map bewteen manifolds. The fibred product bundle is the bundle ...
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2answers
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Fiber Product is an embedded submanifold

Let $\pi:E\to B$ a smooth submersion and $\phi:F\to B$ a smooth map. Defining the fiber product of $E$ and $F$ with respect to $B$ as the set: $$E\times_B F:=\{(e,f)\in E\times F\mid \pi(e)=\phi(...
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0answers
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fibered product or quotient group?

This might be more of a mathoverflow question, but here it goes... For a number field $K$, Marcolli in http://www.its.caltech.edu/~matilde/QSManabelian.pdf page 10 calls $G^\text{ab}_K \times_{\hat{\...
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0answers
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Abstract deformations of affine schemes are affine?

I have a question about the deformation theory as presented in Hartshorne's book. It is about the deformation problem $D$, maybe better known as abstract deformations, concerning deformations without ...
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0answers
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Generalizing the Chinese remainder theorem with fibred products [duplicate]

For a commutative ring $R$ and ideals $I,J\subset R$ with $I+J$ we have an isomorphism $$R/(I\cap J)\cong R/I\times R/J.$$ But I'm wondering how much can be said if $I+J\neq R$. We still have an ...
5
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1answer
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Short exact sequence of topological groups which is split, but not topologically split

Consider an exact sequence of locally compact groups $$1 \to A \overset{\iota}{\to} B \overset{\pi}{\to} C \to 1.$$ Naturally, I assume the homomorphisms are continuous. I should probably also assume ...
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2answers
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Notation for fibred product of multiple morphisms?

The fibred product of two morphisms $f:\ X\ \longrightarrow\ S$ and $g:\ Y\ \longrightarrow\ S$ is often denoted $X\times_SY$, or sometimes notation like $X{\ }_f\hspace{-2pt}\times_gY$ is used if ...
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1answer
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If $h_u \times_G F$ is representable for every $u \in \text{Ob}(C)$ and $G$ is representable then so is $F$.

This is from the Stacks Project. I have so far the following pullback squares: $$ \require{AMScd} \begin{CD} D @>{Q}>> C^{op}\\ @V{P}VV @V{H}VV \\ C^{op} @>{h_u \times_G F}>> \text{...
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1answer
33 views

$(h_u \times_{\xi, G, a} F)(X) = \{(f, \xi) : x \xrightarrow{f} u, \xi' \in F(X), G(f)(\xi) = a_x(\xi')\}$ using definition of fibre product.

I understand the fibre product definition now and have proved that for functors $F, G, H : C^{op} \to \text{Sets}, \ a : F \to G, \ b : H \to G$ that the fibre product can be defined by: $$(F \...
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1answer
68 views

If $f: a \to b$ is representable and $g: b \to c$ is representable, then so is the composition $g\circ f$.

I'm studying from the Stacks Project. They say the proof is omitted. And they don't indicate the difficulty. I've tried piecing together commuting squares etc, but have found nothing useable. Here'...