In this survey article we describe some geometric results in the theory of noncommutative
rings and, more generally, in the theory of abelian categories. Roughly speaking and by …

Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has
found numerous applications in various parts of mathematics, most prominently in …

Motivated by problems on the\'etale cohomology of Rapoport--Zink spaces and their
generalizations, as well as Fargues's geometrization conjecture for the local Langlands …

We give a sufficient condition for an Ext-finite triangulated category to be saturated.
Saturatedness means that every contravariant cohomological functor of finite type to vector …

The central theme of this book is an exposition of the geometric technique of calculating
syzygies. It is written from a point of view of commutative algebra, and without assuming any …

Let $ G $ be a finite group of automorphisms of a nonsingular three-dimensional complex
variety $ M $, whose canonical bundle $\omega _M $ is locally trivial as a $ G $-sheaf. We …

A Historical background....................................... xi A. 1 The conjectural theory described by
Beilinson............ xi A. 2 Voevodsky's motivic complexes........................ xii A. 3 Morel and …

Following the idea of [Far16], we develop the foundations of the geometric Langlands
program on the Fargues--Fontaine curve. In particular, we define a category of $\ell $-adic …

A1-homotopy theory is the homotopy theory for algebraic varieties and schemes which uses
the a ne line as a replacement for the unit interval. In the paper I present in detail the basic …

In this paper the authors investigate homological and homotopical aspects of a concept of
torsion which is general enough to cover torsion and cotorsion pairs in abelian categories …