Search

We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived ∞-category of proétale sheaves, while constructible sheaves are those that are lisse on a stratification. We show that constructible sheaves satisfy proétale descent. We also establish a t-structure on constructible sheaves in a wide range of cases. We finally provide a toolset to manipulate categories of constructible sheaves with respect to the choices of coefficient rings, and use this to prove that our notions reproduce and extend the various approaches to, say, constructible ℓ -adic sheaves in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and -schemes (after Kato, Deitmar and Connes–Consani). Beside this unification, the category of blueprints contains new interesting objects as “improved” cyclotomic field extensions of and “archimedean valuation rings”. It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Titsʼ idea of Chevalley groups over , congruence schemes, sheaf cohomology, K-theory and a unified view on analytic geometry over , adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry. [Copyright &y& Elsevier]

Abstract: We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, , is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme , its derived category of sheaves of modules with quasi-coherent torsion homologies is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category (which is equivalent to ) in the case of a usual scheme. [Copyright &y& Elsevier]