Separation conditions

Let me discuss a bit the possible separation conditions to impose on algebraic stacks.

Before we talk about stacks, let’s review the conditions we have for algebraic spaces X. Here is a list:

  1. Decent. This means that every point of X can be represented by a quasi-compact monomorphism from the spectrum of a field into X.
  2. Reasonable: This means that for an affine scheme U any etale morphism U —> X has universally bounded fibres.
  3. Very reasonable: This means that there exist schemes U_i and an etale surjective morphism \coprod U_i —> X such that each U_i —> X is quasi-compact onto its image.
  4. Quasi-separated: This means that the diagonal morphism X —> XxX is quasi-compact.
  5. Locally separated: This means that the diagonal morphism X —> XxX is an immersion.
  6. Separated: This means that the diagonal morphism X —> XxX is a closed immersion.

Most algebraic geometers will work with either quasi-separated or locally separated spaces (note that in the stacks project a locally separated algebraic space is not required to be quasi-separated, e.g., any scheme is a locally separated algebraic space). On the other end of the spectrum requiring a space to be “decent” is a very mild condition that implies the points on a space behave like points on a scheme. All of the other conditions imply that X is decent (the hardest one to prove is 5 => 1 which is due to David Rydh and not yet in the stacks project). It seems that the class of all decent spaces, singled out by David Rydh, is a very nice class of algebraic spaces to work with.

Now for algebraic stacks there are going to be many, many different flavors of separation conditions. The reason is that if X is an algebraic space over S, then we can impose conditions on the diagonal Δ : X —> X x_S X but we may also impose conditions on the diagonal of the diagonal

Δ_2 : X —> X x_{Δ , X x_S X, Δ} X

Note that this is just the identity section of the inertia stack of X. So for example requiring this second diagonal to be quasi-compact is equivalent to the condition that Aut(x) —> T is quasi-separated for any object x of X over affine schemes T. Then by a standard trick (Lemmas Tag 02YI and Tag 0455) this implies that Isom(x, y) —> T is quasi-separated for any pair of objects x, y of X over T.

What David Rydh suggested to me in an email (if I understood correctly) is that we number diagonals as follows:

  1. The structure morphism X —> S is the zeroth diagonal Δ_0 of X.
  2. The usual diagonal Δ : X —> X x_S X is the first diagonal Δ_1 of X.
  3. The second diagonal is Δ_2 : X —> X x_{Δ , X x_S X, Δ} X as above.

Presumably higher diagonals will not be needed since we work with stacks, and not higher stacks. Using this terminology we can define “X is of finite presentation over S” as “X is locally of finite presentation and Δ_0, Δ_1, and Δ_2 are quasi-compact”.

Moreover, in an ancient email (Mar 6, 2006) of Martin Olsson about the definitions of stacks in the stacks project he suggested that it might be a good idea to look at those stacks which are “locally separated on the diagonal”. In the language above I think translates into saying that Δ_2 is an immersion. This means that given a, b : x —> y morphisms of objects of X over an affine scheme T the locus “a = b” is represented by an open sub scheme of T. I think Martin’s point was that this is a natural condition which is often satisfied in moduli problems.