The two projection maps are different?

Question

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..."

Essentially my problem is the following claim: Let $V \xrightarrow{i} U$ a morphism of schemes and consider the two projections $V \times_{U} V \xrightarrow{p_{i}} V$ for $i=1,2$. If $i$ is not injective, "the two projections will be different".

This seems crazy to me given the symmetry in the construction, can someone please comment?

Answer 1

Let $k$ be a field. Take $U=\text {Spec}(k),V=\mathbb A^1_k$ and for $ i:\mathbb A^1_k\to \text {Spec}(k)$ take the structural map.
Then the two projections $p_i: \mathbb A^1_k\times_{\text {Spec}(k)} \mathbb A^1_k =\mathbb A^2_k\to \mathbb A^1_k$ are certainly different.

Answer 2

I guess the following is meant (and let me generalize this directly to category theory):

Let $i : V \to U$ be a morphism in a category with fiber products. Then the following are equivalent:

• $i$ is a monomorphism
• The projections $p_1,p_2 : V \times_U V \to V$ are equal.
• The diagonal morphism $\Delta : V \to V \times_U V$ is an isomorphism.

The proof is straight forward.

Of course, in concrete categories, "monomorphism" and "morphism with underlying injective map" are completely different concepts. Some authors say "injective sheaf homomorphism" when they actually mean a monomorphism of sheaves. Perhaps this also applies to schemes.