We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a nontrivial adaptation of Deligne’s method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about “Brown Representabi...

Duality for complete discrete valuation fields with perfect residue field coefficients in (possibly p-torsion) finite flat group schemes was obtained by Bégueri, Bester, and Kato. In this paper, we give another formulation proof of result. We use the category a Grothendieck topology on it. This simplifies reduces duality to classical results Galois cohomology. A key point is that resulting site...

We study several classical duality results in the theory of tensor products, due mostly to Grothendieck, providing new proofs as well as new results. In particular, we show that the canonical mapping Y ∗ ⊗π X → (L(X, Y ), τ)∗ is not always injective, answering a problem of Defant and Floret. We use the machinery of vector measures to give new proofs of the dualitites (X⊗ε Y )∗ = N (X, Y ∗), whe...

This paper is motivated by questions such as P vs. NP and other questions in Boolean complexity theory. We describe an approach to attacking such questions with cohomology, and we show that using Grothendieck topologies and other ideas from the Grothendieck school gives new hope for such an attack. We focus on circuit depth complexity, and consider only finite topological spaces or Grothendieck...