We develop the notion of stratifiability in the context of derived categories and the six operations for stacks in the work of Laszlo and Olsson. Then we reprove Behrend's Lefschetz trace formula for stacks, and give the meromorphic continuation of the L-series of stacks defined over a finite field. We give an upper bound for the weights of the cohomology groups of stacks, and as an application, prove the decomposition theorem for perverse sheaves on stacks with affine diagonal, both over finite fields and over the complex numbers. Along the way, we generalize the structure theorem of mixed sheaves and the generic base change theorem for stacks. We also give a short exposition on the lisse-analytic topoi of complex analytic stacks, and give a comparison between the lisse-etale topos of a complex algebraic stack and the lisse-analytic topos of its analytification.

Given a singular scheme X over a field k, we consider the problem of resolving the singularities of X by an algebraic stack. When X is a toroidal embedding or is etale locally the quotient of a smooth scheme by a linearly reductive group scheme, we show that such &ldquo stacky resolutions &rdquo exist. Moreover, these resolutions are canonical and easily understandable in terms of the singularities of X.

We give three applications of our stacky resolution theorems: various generalizations of the Chevalley-Shephard-Todd Theorem, a Hodge decomposition in characteristic p, and a theory of toric Artin stacks extending the work of Borisov-Chen-Smith. While these applications are seemingly different, they are all related by the common theme of using stacky resolutions to study singular schemes.

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which provides a conceptual simplification of the classical construction and which is expected to produce better results for non-smooth varieties. In this paper, we introduce a generalization of the saturated de Rham-Witt complex which allows coefficients in a unit-root $F$-crystal. We define our complex by a universal property in a category of so-called de Rham-Witt modules. We prove a number of results about it, including existence, quasicoherence, and comparisons to the de Rham-Witt complex of Bhatt-Lurie-Mathew and (in the smooth case) to crystalline cohomology and the classical de Rham-Witt complex with coefficients.

In order to shed light on Orlov’s conjecture that derived equivalent smooth, projective varieties have isomorphic Chow motives, we examine the zeta functions of derived equivalent varieties over finite fields; in this setting Orlov’s conjecture predicts equality of zeta functions. It is demonstrated that derived equivalent smooth, projective varieties over finite fields that are abelian or satisfy a certain condition on their cohomology. This condition is satisfied, for example, by a surface or Calabi–Yau 3–fold.

One of our approaches to comparing the zeta functions of derived equivalent varieties over finite fields comes from using the Lefschetz Fixed Point Theorem to turn the question into one of comparing the -adic ́etale cohomology of varieties. Cohomology groups are not in general preserved under the action of Fourier–Mukai equivalences on cohomology, but cohomological structures we call even and odd Mukai–Hodge structures, which are a realization of the Mukai motive, are preserved. Investigation into when isomorphism of these cohomological structures implies equality of zeta functions gives us our cohomological condition for equality of zeta functions.

We also develop a relative version of the map Fourier–Mukai transforms induce on cohomology and define a relative notion of even and odd Mukai–Hodge structures, and show these structures are preserved in a situation arising from the derived equivalence of smooth, projective varieties with semiample (anti-)canonical bundles. Using this result, it is demonstrated that when derived equivalent smooth, projective varieties have semiample (anti-)canonical bundles, the fibers over any fixed geometric point in their shared (anti-)canonical variety must also have isomorphic even and odd Mukai–Hodge structures. Hence, for any such varieties over finite fields, if their geometric fibers satisfy any of the conditions identified for isomorphism of Mukai–Hodge structures to imply equality of zeta functions, then the varieties themselves also have equal zeta functions.

The theme of this dissertation is the Brauer group of algebraic stacks. Antieau and Meier showed that if $k$ is an algebraically closed field of $\on{char} k \ne 2$, then $\Br(\ms{M}_{1,1,k}) = 0$, where $\ms{M}_{1,1}$ is the moduli stack of elliptic curves. We show that if $\on{char} k = 2$ then $\Br(\ms{M}_{1,1,k}) = \Z/(2)$. In another direction, we compute the cohomological Brauer group of $\G_{m}$-gerbes; this is an analogue of a result of Gabber which computes the cohomological Brauer group of Brauer-Severi schemes. We also discuss two kinds of algebraic stacks $X$ for which not all torsion classes in $\H_{\et}^{2}(X,\G_{m})$ are represented by Azumaya algebras on $X$ (i.e. $\Br \ne \Br'$).

The goal of this Thesis is to develop the theory of Picard and Albanese 1-motives attached to a variety X defined over a perfect field of positive characteristic, and to relate these 1-motives to the etale cohomology groups of X. This should be viewed as a generalization of the classical theory of Picard and Albanese varieties attached to a smooth and proper variety X. Moreover, giving such a theory allows us to relate the dimension- and codimension-one etale cohomology groups in the most natural (`motivic') way possible; in particular, independence-of-l type results in dimension- and codimension-one are automatic once one has developed such a theory.

In the case of a base field of characteristic zero, the corresponding 1-motives have been constructed and studied in previous work of Barbieri-Viale and Srinivas. When one deals with a positive-characteristic base field, new difficulties arise due to the fact that resolution of singularities in positive characteristic is still an open problem. This forces us to introduce new methods, especially a strong form of de Jong's results that allows us to resolve (in a weak sense) an arbitrary separated finite type k-scheme by a smooth Deligne-Mumford stack. A large part of this thesis is devoted to preliminary results on divisors and cycle class maps for Deligne-Mumford stacks that we need when applying the methods of Barbieri-Viale and Srinivas with stacks rather than schemes.

In the end, we manage to construct the Picard 1-motives of an arbitrary separated finite type k-scheme with no additional assumptions, and prove various functoriality and compatibility properties for these 1-motives. The situation with the Albanese 1-motives is more complicated; over an arbitrary perfect field, we only manage to show that the Albanese 1-motives of X exist after possibly base extending X via a finite field extension. We show, however, that in the case that k is a finite field or an algebraically closed field, no such field extension is necessary. The case of a finite field uses a method for descending 1-motives along an extension of finite fields when the 1-motive is only given up to isogeny. This method may be of some independent interest.