arXiv subjects
Publications and source records attributed to Shane Kelly.
We give a brief introduction to tensor triangulated geometry, a brief introduction to various motivic categories, and then make some observations about the conjectural structure of the tensor triangulated spectrum of the Morel-Voevodsky stable homotopy category over a finite field.
This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.
We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, étale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.
In the first half of this article we define a new weight homology functor on Voevodsky's category of effective motives, and investigate some of its properties. In special cases we recover Gillet-Soulé's weight homology, and Geisser's Kato-Suslin homology. In the second half, we consider the notions of "co-étale" and "reduced" motives, and use the notions to a prove a theorem comparing motivic homology to étale motivic homology. Due to the first author's Ph.D. thesis arXiv:1305.5349 we do not have to restrict to smooth schemes.
In this note we observe that we can remove the hypothesis of resolution of singularities from the isomorphism constructed by Suslin between the étale cohomology with compact support and Bloch's higher Chow groups over an algebraically closed field. We also show that this isomorphism can be obtained using Ivorra's realisation functor.
We show how a theorem of Gabber on alterations can be used to apply work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X)[1/p] = 0$ for $n < - \dim X$ where $X$ is a quasi-excellent noetherian scheme, $p$ is a prime that is nilpotent on $X$, and $K_n$ is the $K$-theory of Bass-Thomason-Trobaugh. This gives a partial answer to a question of Weibel.
This thesis presents a way to apply this theorem of Gabber to a large portion of Voevodsky's work in order to lift the assumption that resolution of singularities holds. This gives unconditional versions of many of his and others' theorems provided we work Z[1/p] linearly, where p is the exponential characteristic of the base field. One example of the many applications we give is a partial answer to a 1980 conjecture of Weibel. Another is the removal of the hypothesis of resolution of singularities from a result of Suslin that compares Bloch's higher Chow groups and etale cohomology. Voevodsky's main tool in applying resolution of singularities is the cdh topology. We enlarge it slightly in order to apply this theorem of Gabber, presenting in this thesis a topology that we name the ldh topology, where l is a prime. We compare the cdh and ldh topologies using the concept of a "presheaf with traces", providing conditions under which the cdh and ldh sheafifications of a presheaf agree, as well as its cdh and ldh cohomologies. As far as applying resolution of singularities to motives goes, Voevodsky's most important theorem can be rephrased as a cdh descent condition, and we are led to ask for conditions under which certain objects in the Morel-Voevodsky stable homotopy category satisfy ldh descent. In order to compare cdh and ldh descent, we generalise the notion of a "presheaf with traces" to the concept of an "object with traces". We build on some results of Pelaez on the functoriality of the slice filtration to show that this concept of an "object with traces" interacts well enough with the slice filtration to provide the ldh descent that we need.
Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using étale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.