In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass ...

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The ...

Our goal for this lecture is to prove that morphisms of projective varieties are closed maps. In fact we will prove something stronger, that projective varieties are complete, a property that plays a role comparable to compactness in topology. For varieties, compactness as a topological space does not mean much because the Zariski topology is so coarse. Indeed, every subset of An (and hence of ...

For any abelian category C satsifying (AB5) over a separated, quasicompact scheme S , we construct a stack of 2-groups GL(C) over the flat site of S. We will give a concrete description of GL(C) when C is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space gl(C) of GL(C) at the origin has a structure as a Lie 2-algebra.

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme finite dimension has strong generator in sense Bondal-Van den Bergh. This extends recent result Neeman and is new even affine case. The main ingredient includes Gabber's weak local uniformization theorem notions boundedness descendability morphism schemes.

Let η be the generic point of X; we note that the constant sheaf F on X is equal to η∗F (i.e., the natural map F → η∗ηF is an isomorphism). This is because any E which is quasicompact and étale over X must also be normal and Noetherian, so the connected components of E are its irreducible components, with their generic points exactly the points of the fiber Eη. Hence we reduce to showing that t...