17
votes
Accepted
Can you give an example of two projective morphisms of schemes whose composition is not projective?
Here is a locally Noetherian separated counterexample. I also give some motivation for this construction afterwards.
Definition. Let $Z$ be an infinite chain of affine lines: $Z = Z_1 \amalg_{p_1} ...
- 28.7k
3
votes
Is an equivariant projective morphism equivariantly-projective?
No. Let $Y$ be a point and $X$ be $\mathbb P^1$ with two points glued together. Let $G=\mathbb G_m$ act on $X$ by its usual action on $\mathbb P^1$ fixing those two points, and acting trivially (of ...
- 148k
1
vote
Accepted
Decomposition of a morphism with positive dimensional fibers
I am posting my comment as an answer. This already fails for relative dimension $1$ when the base scheme has dimension $n$ at least $3$.
Let $k$ be a field. Let $n\geq 3$ be an integer. Denote $\...
1
vote
Accepted
Extending locally free sheaves and compatibility with fibers
I am just posting my comment as an answer.
In fact, the natural homomorphisms $\mathcal{O}_{X_o}\to (j')_*\mathcal{O}_{U_o}$ and $\mathcal{O}_{X\times B} \to j_*\mathcal{O}_U$ are both isomorphisms....
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