We use supernatural bundles to build $\mathbf{GL}$ -equivariant resolutions supported on the diagonal of $\mathbb{P}^{n}\times \mathbb{P}^{n}$ , in a way that extends Beilinson’s resolution of the diagonal. We thus obtain results about supernatural bundles that largely parallel known results about exceptional collections. We apply this construction to Boij–Söderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications.

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

For a site ${\mathcal S}$ (with enough points), we construct a topological space X$_($)$ and a full embedding φ$^*$ of the category of sheaves on ${\mathcal S}$ into those on X$_($) (i.e., a morphism of toposes φ:Sh (X$_($) →Sh(${\mathcal S}$)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms coh$^*$(${\mathcal S}$,A) =coh$^**$(X$_($), φ$^*$A) for any Abelian sheaf A on ${\mathcal S}$. As a particular case, this will give for any scheme Y a topological space X and a functorial isomorphism between the étale cohomology H$^**$(et{Y},A) and the ordinary sheaf cohomology H$^**$(X,φ$^**$A), for any sheaf A for the étale topology on Y.

We prove that for a non-discrete space X, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i) X is paracompact and each point is Gδ, or (ii) X is a completely paracompact Morita k-space. Consequently, if X is a non-discrete completely paracompact space in which each point is a Gδ-set or it is also a Morita k-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a space X there exists a point x ∈ X and a family ϕ of supports on X such that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result for Rn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.