We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral sequences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the RiemannRoch theorem for curves and Bott’s theorem in representation theory (following [8]) using the algebraic-geomet...

Let F̄p be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, F̄p) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an a...

Immediate predecessors of this work were a paper on two-dimensional deadbeat observers by Bisiacco and Valcher [Multidimens. Systems Signal Process., 19 (2008), pp. 287–306] and one on one-dimensional functional observers by Blumthaler [Linear Algebra Appl., 432 (2010), pp. 1560–1577] (compare also Fuhrmann’s comprehensive paper [Linear Algebra Appl., 428 (2008), pp. 44– 136]). The present pape...

Notations and conventions 296 Introduction 296 I. Serre duality and almost split sequences 300 I.1. Preliminaries on Serre duality 300 I.2. Connection between Serre duality and Auslander–Reiten triangles 304 I.3. Serre functors on hereditary abelian categories 307 II. Hereditary noetherian abelian categories with non-zero projective objects 309 II.1. Hereditary abelian categories constructed fr...

Abstract. We calculate Ext• SL2(k) (∆(λ),∆(μ)), Ext• SL2(k) (L(λ),∆(μ)), Ext• SL2(k) (∆(λ), L(μ)), and Ext•SL2(k) (L(λ), L(μ)), where ∆(λ) is the Weyl module of highest weight λ, L(λ) is the simple SL2(k)-module of highest weight λ and our field k is algebraically closed of positive characteristic. We also get analogous results for the Dipper-Donkin quantisation. To do thus we construct the Lyn...

These are lecture notes, in progress, on monomial ideals. The point of view is that monomial ideals are best understood by drawing them and looking at their corners, and that a combinatorial duality satisfied by these corners, Alexander duality, is key to understanding the more algebraic duality theories at play in algebraic geometry and commutative algebra. Sections written so far cover Alexan...

Here the solid arrows represent maps that are given, and the problem is to find a map h commuting in the diagram. Usually the map i is an inclusion, and p is some kind of “bundle map” such as a local product. Note the special cases: (i) if i is an inclusion, E = A, and f = 1A, then the extension problem asks whether A is a retract of X; and (ii) if p is surjective, X=B, and g = 1B, the lifting ...

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension L/K of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes ...