Sheaves on Stacks

Here is a technically straightforward manner in which to introduce various categories of sheaves on algebraic stacks, and it is my intention to introduce sheaves on algebraic stacks in the stacks project along these lines. Please take a look and leave a comment if you see a problem with this approach.

Suppose that C is a site. Using conventions as in the stacks project:

  1. If p : X —> C is a stack in groupoids over C, then we declare a family of morphisms {x_i —> x}_{i ∈ I} in X to be a covering if and only if {p(x_i) —> p(x)}_{i ∈ I} is a covering of the site C. In this way X becomes a site.
  2. If f : X —> Y is a 1-morphism of stacks in groupoids over C, then f is a continuous and cocontinuous as a functor of sites. Hence f induces a morphism of topoi f : Sh(X) —> Sh(Y) with the property that the pull back of a sheaf G on Y is defined by the simple rule (f^{-1}G)(x) = G(f(x)). This construction is compatible with composition of 1-morphisms of stacks in groupoids.
  3. Finally, if a : f —> g is a 2-morphism in the 2-category of stacks in groupoids over C, then a induces a 2-morphism a : f —> g in the 2-category of topoi.

In other words, this is a perfectly reasonable way to associate a site to each and every stack over C.

Next, let C = (Sch/S)_{fppf} be the category of schemes with the fppf topology as in the stacks project. An algebraic stack X is a category fibred in groupoids over C. Hence the construction above gives us a site X_{fppf} which we will call the fppf site of X. According to the remarks above this has a suitable 2-functoriality with regards to morphisms of algebraic stacks.

Variants: If X is an algebraic stack, then p : X —> C is also a stack fibred in groupoids over C endowed with the Zariski, smooth=etale (see this post), or syntomic topology. Hence we obtain variants X_{Zar}, X_{smooth}, and X_{syntomic} satisfying functorialities as above. Note that the underlying category is X in each case.

Here are some (I think) properties of these definitions:

  1. if x is an object of X with U = p(x), then X_{fppf}/x is equivalent (as a site) to (Sch/U)_{fppf}. Hence given a sheaf F on X_{fppf} the cohomology groups H^p(x, F) are just fppf cohomology groups of some sheaf on (Sch/U)_{fppf}. This also works with the other topologies.
  2. when the topology is etale=smooth or Zariski, then H^p(x, F) can be computed on the small etale or Zariski site of U.
  3. In general X does not have a final object and does not have fibre products. If the diagonal of X is representable (by schemes) then X has all fibre products.
  4. Assume the diagonal of X is representable. Let x_0 be an object of X such that U_0 = p(x_0) is a scheme surjective, flat, locally of finite presentation over X. The representable sheaf h_{x_0} surjects onto the singleton sheaf * in Sh(X_{fppf}). Moreover, the fibre products h_{x_0} \times_{*} h_{x_0}, h_{x_0} \times_{*} h_{x_0} \times_{*} h_{x_0}, etc are representable by x_1, x_2, etc with p(x_1) = U \times_X U, p(x_2) = U \times_X U \times_X U, etc. It follows formally from this (compare with Lemma Tag 01GC and Lemma Tag 01GY) that there is a spectral sequence E_1^{p, q} = H^q(x_p, F) => H^{p + q}(X_{fppf}, F) and by the above H^q(x_p, F) corresponds to fppf cohomology of F over the scheme U_p.
  5. There is a similar spectral sequence for the smooth=etale topology if the morphism U_0 –> X is surjective and smooth and the diagonal of X is representable.
  6. If X is general there is still a spectral sequence with E_1^{p, q} = H^q(U_p, F), but then the U_p are algebraic spaces.

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