In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal and Serre dimension. We prove some basic properties these dimensions, compare them open problems.

In his letter (Serre, 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions A×B/B × → Fp, where B is the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p)...

Assume that R is a semi-local regular ring containing an infinite perfect field. Let K be the field of fractions of R. Let H be a simple algebraic group of type F4 over R such that HK is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If charK 6= 2 this means precisely that the f3 invariant of HK is trivial. We prove that the kernel of the map H ét ...

— We construct explicitly some analytic families of étale (φ,Γ)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular ...

We consider integrable vertex models whose Boltzmann weights (Rmatrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras Uq(ĝ). These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed. If q...

In our previous paper [10], we established a theta correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces attached to real orthogonal groups. This correspondence is realized via theta functions associated to explicitly constructed ”special” Schwartz forms. These forms give rise to relative Lie algebra cocycles for ...

We prove the Cartan–Eilenberg stable elements theorem and construct a Lyndon–Hochschild–Serre type spectral sequence for pro-fusion systems. As an application, we determine continuous mod-[Formula: see text] cohomology ring of [Formula: any odd prime text].

Abstract. Let X be a smooth scheme of finite type over a field K, let E be a locally free OX -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor, HomGrA(−,−), on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of HomGrA(OX ,−). When X is a smo...

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b+ > 1 satisfy the duality Gr(α) = ±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles...

In this note we prove the Serre duality theorem for the cohomology of coherent sheaves on a projective scheme. First we do the case of projective space itself. Then on an arbitrary projective scheme X, we show that there is a coherent sheaf ω◦ X , which plays the role in duality theory similar to the canonical sheaf of a nonsingular variety. In particular, if X is Cohen-Macaulay, it gives a dua...