35 research outputs found
On the algebraic K-theory of the complex K-theory spectrum
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and
let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary
homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy
groups. We show that up to a finite summand, these groups form a finitely
generated free module over a polynomial algebra F_p[b], where b is a class of
degree 2p+2 defined as a higher Bott element.Comment: Revised and expanded version, 42 pages
Cohomological descent theory for a morphism of stacks and for equivariant derived categories
In the paper we answer the following question: for a morphism of varieties
(or, more generally, stacks), when the derived category of the base can be
recovered from the derived category of the covering variety by means of descent
theory? As a corollary, we show that for an action of a reductive group on a
scheme, the derived category of equivariant sheaves is equivalent to the
category of objects, equipped with an action of the group, in the ordinary
derived category.Comment: 28 page
On the vanishing of negative K-groups
Let k be an infinite perfect field of positive characteristic p and assume
that strong resolution of singularities holds over k. We prove that, if X is a
d-dimensional noetherian scheme whose underlying reduced scheme is essentially
of finite type over the field k, then the negative K-group K_q(X) vanishes for
every q < -d. This partially affirms a conjecture of Weibel.Comment: Math. Ann. (to appear
On the Whitehead spectrum of the circle
The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and
Williams shows that the homotopy groups in low degrees of the space of
homeomorphisms of a closed Riemannian manifold of negative sectional curvature
can be expressed as a functor of the fundamental group of the manifold. To
determine this functor, however, it remains to determine the homotopy groups of
the topological Whitehead spectrum of the circle. The cyclotomic trace of B
okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an
expression for these homotopy groups in terms of the equivariant homotopy
groups of the homotopy fiber of the map from the topological Hochschild
T-spectrum of the sphere spectrum to that of the ring of integers induced by
the Hurewicz map. We evaluate the latter homotopy groups, and hence, the
homotopy groups of the topological Whitehead spectrum of the circle in low
degrees. The result extends earlier work by Anderson and Hsiang and by Igusa
and complements recent work by Grunewald, Klein, and Macko.Comment: 52 page
Vortices and Jacobian varieties
We investigate the geometry of the moduli space of N-vortices on line bundles
over a closed Riemann surface of genus g > 1, in the little explored situation
where 1 =< N < g. In the regime where the area of the surface is just large
enough to accommodate N vortices (which we call the dissolving limit), we
describe the relation between the geometry of the moduli space and the complex
geometry of the Jacobian variety of the surface. For N = 1, we show that the
metric on the moduli space converges to a natural Bergman metric on the Riemann
surface. When N > 1, the vortex metric typically degenerates as the dissolving
limit is approached, the degeneration occurring precisely on the critical locus
of the Abel-Jacobi map at degree N. We describe consequences of this phenomenon
from the point of view of multivortex dynamics.Comment: 36 pages, 2 figure
On the curvature of vortex moduli spaces
We use algebraic topology to investigate local curvature properties of the
moduli spaces of gauged vortices on a closed Riemann surface. After computing
the homotopy type of the universal cover of the moduli spaces (which are
symmetric powers of the surface), we prove that, for genus g>1, the holomorphic
bisectional curvature of the vortex metrics cannot always be nonnegative in the
multivortex case, and this property extends to all Kaehler metrics on certain
symmetric powers. Our result rules out an established and natural conjecture on
the geometry of the moduli spaces.Comment: 25 pages; final version, to appear in Math.
On the existence of a compact generator on the derived category of a noetherian formal scheme
In this paper, we prove that for a noetherian formal scheme X, its derived
category of sheaves of modules with quasi-coherent torsion homologies D_qct(X)
is generated by a single compact object. In an appendix we prove that the
category of compact objects in D_qct(X) is skeletally small.Comment: 13 page
Lifting and restricting recollement data
We study the problem of lifting and restricting TTF triples (equivalently,
recollement data) for a certain wide type of triangulated categories. This,
together with the parametrizations of TTF triples given in "Parametrizing
recollement data", allows us to show that many well-known recollements of right
bounded derived categories of algebras are restrictions of recollements in the
unbounded level, and leads to criteria to detect recollements of general right
bounded derived categories. In particular, we give in Theorem 1 necessary and
sufficient conditions for a 'right bounded' derived category of a differential
graded(=dg) category to be a recollement of 'right bounded' derived categories
of dg categories. In Theorem 2 we consider the particular case in which those
dg categories are just ordinary algebras.Comment: 29 page