Since the last updateon October 12 we have added the following material

  1. Gabber’s argument that categories of quasi-coherent modules form a Grothendieck abelian category (for schemes, spaces, and algebraic stacks),
  2. an example of an fpqc space which is not an algebraic space,
  3. an example of a quasi-compact non-quasi-separated morphism of schemes such that pushforward does not preserve quasi-coherency,
  4. some material related to my course on commutative algebra: exercise, lemmas, shorten proof of ZMT, etc
  5. introduced lisse-etale (and flat-fppf) sites,
  6. functoriality of lisse-etale topos for smooth morphisms (and flat-fppf for flat morphisms),
  7. material on Grothendieck abelian categories, incuding existence of injectives and existence of enough K-injective complexes (following Spaltenstein and Serp\’e),
  8. cohomology of unbounded complexes and adjointness of Lf^* and Rf_*,
  9. a lot of material on D_{QCoh}(X) for an algebraic stack X, including Rf_* (on bounded below for quasi-compact and quasi-separated morphisms) and Lf^* (unbounded for general f).

In particular my suggestion in this post worked out exactly as advertised. The existence of Rf_* is straightforward. It turns out that once you prove that the category D_{QCoh}(X) as defined in the blog post is equivalent to the version of D_{QCoh}(X) in L-MB or Martin Olsson’s paper (i.e. defined using the lisse-etale site), then you immediately obtain the existence of Lf^*. Namely, the existence of the lisse-etale site is used to prove that the Verdier quotient used to define D_{QCoh}(X) is a Bousfield colocalization (technically it is easier to use the flat-fppf site to do this, because we use the fppf topology as our default topology, but one can use either).

A bit of care is needed when working with the lisse-etale site and the lisse-etale topos. As discussed elsewhere, one reason is that the lisse-etale topos isn’t functorial for morphisms of algebraic stacks. Here is a another. There is a comparison morphism of topoi

g : Sh(X_{lisse,etale}) —-> Sh(X_{etale})

The functor g^{-1} has a left adjoint denoted g_! (on sheaves of sets) and we have g^{-1}g_! = g^{-1}g_* = id. This means that Sh(X_{lisse,etale}) is an essential subtopos of Sh(X_{etale}), see SGA 4, IV, 7.6 and 9.1.1. Let K be a sheaf of sets on X_{lisse,etale}. Let I be an injective abelian sheaf on X_{etale}. Question: H^p(K, g^{-1}I) = 0? In other words, is the pullback by g of an injective abelian sheaf limp? If true this would be a convenient way to compare cohomology of sheaves on X_{etale} with cohomology of sheaves on the lisse-etale site. Unfortunately, we think this isn’t true (Bhargav made what is likely a counter example — but we haven’t fully written out all the details).

Rf_* not preserving quasi-coherence

Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated.

Let k be a field of characteristic p > 0. Let G = Z/pZ be the cyclic group of order p. Set S = Spec(k[x]) and let X = [S/G] be the stacky quotient where G acts trivially on S. Consider the morphism f : X —> S. Then Rf_*O_X is a complex with cohomology sheaves isomorphic to O_S for all p >= 0. In fact Rf_*O_X is quasi-isomorphic to ⊕ O_S[-n] where n runs over nonnegative integers.

Now consider the complex K = ⊕ O_X[m] where m runs over the nonnegative integers. This is an object of D_{QCoh}(X) but it isn’t bounded below. So we have to pay attention if we want to compute Rf_*K. Namely, in D(O_X) the complex K is also K = ∏ O_X[m]. Since cohomology commutes with products, we see that

Rf_*K = ∏ Rf_*O_X[m] = ∏ (⊕ O_S[m – n]).

In degree 0 we get an infinite product of copies of O_S which isn’t quasi-coherent.

Conclusion: Rf_* does not map D_{QCoh}(X) into D_{QCoh}(S).

Of course if f is a quasi-compact and quasi-separated morphism between algebraic spaces, then this kind of thing doesn’t happen.