# Questions tagged [fibre-product]

The tag has no usage guidance.

15 questions
2answers
67 views

### 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 $X_y=X\times Spec(k(y))$. ...
0answers
171 views

### 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 $X$ ...
0answers
62 views

1answer
61 views

### 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 ...
0answers
107 views

### 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 ...
2answers
356 views

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 \...
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'...