##
**Derived \(\ell\)-adic categories for algebraic stacks.**
*(English)*
Zbl 1051.14023

Mem. Am. Math. Soc. 774, 93 p. (2003).

The main goal of this carefully written and well explained book is to prove the following Lefschetz trace formula for stacks (conjectured by the author in [Invent. Math. 112, 127–149 (1993; Zbl 0792.14005)]): Let \(\Phi_q\) denote the arithmetic Frobenius acting on the \(\ell\)-adic cohomology of a smooth algebraic stack \(\mathfrak{X}\) over the finite field \(\mathbb{F}_q\); then
\[
q^{\text{ dim}\, \mathfrak{X}} \, \text{ tr} \, \Phi_q| H^*(\mathfrak{X},\mathbb{Q}_l) = \# \, \mathfrak{X}(\mathbb{F}_q).
\]
For example, if \(\mathfrak{X} = B\mathbb{G}_m\), this formula becomes \(\frac{1}{q}\sum_{i=0}^\infty \frac{1}{q^i} = \frac{1}{q-1}\). If \(\mathfrak{X}\) is the stack \(\mathfrak{M}_1\) of curves of genus one, this formula may be interpreted as a type of Selberg Trace Formula: It gives the sum \(\sum_k \frac{1}{p^{k+1}}\text{ tr} \, T_p| \mathcal{S}_{k+2}\), where \(T_p\) is the \(p^{\text{ th}}\) Hecke operator on the space of cusp forms of weight \(k+2\), in terms of elliptic curves over the finite field \(\mathbb{F}_p\).

In order to prove the Lefschetz Trace Formula the author formulates and proves a much more general version: Firstly, he constructs the category of ‘absolutely convergent mixed constructible \(\mathcal{Q}_{\ell}\)-complexes’ which may be plugged into the Lefschetz Trace Formula as coefficient sheaves and, secondly, he passes from the absolute situation to the relative case of a morphism of finite type algebraic \(\mathbb{F}_q\)-stacks. While the main and hard part of this book is to develop this rather abstract and involved \(\ell\)-adic formalism of derived categories for algebraic stacks and to prove for instance the Smooth Base Change Theorem, the proof of the general version of the Lefschetz Trace Formula finally is quickly reduced to the ‘usual’ Lefschetz Trace Formula for the geometric Frobenius acting on the cohomology with compact support of a lisse \(\mathbb{Q}_{\ell}\)-sheaf on a smooth \(\mathbb{F}_q\)-variety and to the main theorem of Borel’s paper [Ann. Math. (2) 57, 115–207 (1953; Zbl 0052.40001)].

In order to prove the Lefschetz Trace Formula the author formulates and proves a much more general version: Firstly, he constructs the category of ‘absolutely convergent mixed constructible \(\mathcal{Q}_{\ell}\)-complexes’ which may be plugged into the Lefschetz Trace Formula as coefficient sheaves and, secondly, he passes from the absolute situation to the relative case of a morphism of finite type algebraic \(\mathbb{F}_q\)-stacks. While the main and hard part of this book is to develop this rather abstract and involved \(\ell\)-adic formalism of derived categories for algebraic stacks and to prove for instance the Smooth Base Change Theorem, the proof of the general version of the Lefschetz Trace Formula finally is quickly reduced to the ‘usual’ Lefschetz Trace Formula for the geometric Frobenius acting on the cohomology with compact support of a lisse \(\mathbb{Q}_{\ell}\)-sheaf on a smooth \(\mathbb{F}_q\)-variety and to the main theorem of Borel’s paper [Ann. Math. (2) 57, 115–207 (1953; Zbl 0052.40001)].

Reviewer: Bernhard Köck (Southampton)

### MSC:

14G15 | Finite ground fields in algebraic geometry |

14A20 | Generalizations (algebraic spaces, stacks) |

14D20 | Algebraic moduli problems, moduli of vector bundles |

18E30 | Derived categories, triangulated categories (MSC2010) |

14F99 | (Co)homology theory in algebraic geometry |