Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces

Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $$\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$$ has to be representable. The latter means that if $$X,Y$$ are schemes, $$h_X,h_Y$$ their Yoneda representations and we have natural maps $$h_X \to \mathcal{X}, h_Y \to \mathcal{X}$$ then the fiber product $$h_X \times_{\mathcal{X}} h_Y$$ has to be representable in "usual" sense by a scheme. That's a clear formulated technical condition that one can start to verify.

Question: What is the intuition/philosophy one should have in mind associating with this condition on representability of the diagonal $$\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$$? What makes this condition so "interesting" in order to study stacks? What is the wittness of this condition?

Recall then we talking about schemes a scheme $$S$$ is called separated if the diagonal map $$S→S×S$$ is a closed immersion. It allows to "transfer" in certain way a "category theoretical Hausdorff axiom" the algebraic geometry, since the classical one fails for Zariski topology.

For schemes the property "be separated" is is often used in following sence: Let $$f: X\to Y, g:Y \to Z$$ morphisms, composition $$g \circ f$$ has certain property P well behaving under pullbacks and $$g$$ separated. Then $$f$$ has also P.

Is the the motivation for representability of $$\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$$ of similar manner?

That is my question is which intuitive property is "transfered" by imposing the representability for the diagonal map to the world of stack?

1 Answer

The basic issue is the following

Theorem. If the diagonal is representable (by schemes, algebraic spaces, etc.), then any morphism $$S\rightarrow \mathcal{X}$$ with $$S$$ a scheme, algebraic space, etc. is representable (by schemes, algebraic spaces, etc.).

For a proof, see

Note, for instance that if the diagonal is affine, then on a scheme the intersection of two open affines is open (which is a kind of weak separability that is often useful in cohomological situations.)

A representability condition on the diagonal gives some information on the wildness of the (possible) non-separability of the stack/space considered. Being representable by a closed embedding is as good as it gets (i.e. the Hausdorff condition) but as long as you are willing to consider patchings with bad properties, you need some degree of control of its behavior. So some condition on the diagonal tames this. You might weaken from closed embeddings to affine schemes or to just schemes (this would make the patching scheme-like) or even to algebraic spaces (then the patching would look schematic form the etale toposic point of view). Beyond that, the rough idea is that very complicated behaviors might arise and one should avoid it. Thus the condition.

• yes probably I see the issue. That allows to transfer terminology from schematic world (like smooth, etale, flat and so on) to morphisms $S\rightarrow \mathcal{X}$ with $S$ scheme. For example lot of interesting stacks like DM-stack come by defintion with a cover $p:U \to \mathcal{X}_{DM}$ and one want to impose an extra schematic condition $p$ (eg in case of DM-stack $p$ has to be etale). And the most naive way might be to demand that the scheme that represents this covering map is etale and so on. I think that was the point. Thank you a lot! – katalaveino Jan 8 at 0:37
• Yes, I could not have said it better. Glad my comment helped you. – Leo Alonso Jan 8 at 13:30