# Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.

In HigherAlgebra, the derived category $\mathcal D(X)$ is defined to be the differential graded nerve of $\mathrm{Mod}(\mathcal O_X)^\circ$, the full subcategory spanned by the fibrant objects (Note that $\mathrm{Mod}(\mathcal O_X)$ is Grothendieck abelian).

Are there good references for the theory of derived functors in this setting?

Is it known when projection formula and (flat) base change work?

Edit: Let me explain why I think that the second question is a sensible one to ask. (I must admit that I found these sources after I posted that question)

There are some rather general results for the unbounded derived category in Spaltenstein. Propositions (6.18) and (6.20) prove base change and projection formula in the case of ringed spaces (given some finiteness-condition on the ring).

Closer and at the same time further from my setting (topological stacks, but only sheaves of abelian groups and the push-forward has to have finite cohomological dimension) is Lemma (6.5.7) in this article.

One of the applications I have in mind is proving that Fourier-Mukai transformations are closed under composition - the only proof I know relies on base change and the projection formula.

• What do you mean by "smooth stack"? Differentiable stack, i.e. the stack of principal bundles for a Lie groupoid, or something else? – David Carchedi Aug 12 '14 at 20:32
• I mean a differentiable stack, i.e. a stack that has an atlas that is a submersion. (as in Heinloth's notes on differentiable stacks) – Martin Ruderer Aug 12 '14 at 20:38
• Ok, good (that's the same as a stack of principal bundles btw). – David Carchedi Aug 12 '14 at 22:06
• May I ask what your motivation is? Without knowing more info, I would guess that this is only a reasonable thing to study if the stack $X$ has an etale atlas, i.e. is an etale differentiable stack (basically an orbifold, but without separation conditions). – David Carchedi Aug 13 '14 at 21:00
• I edited the question to give a little more background and a concrete application. – Martin Ruderer Aug 14 '14 at 9:23