848 research outputs found
Analytic geometry over F_1 and the Fargues-Fontaine curve
This paper develops a theory of analytic geometry over the field with one
element. The approach used is the analytic counter-part of the Toen-Vaquie
theory of schemes over F_1, i.e. the base category relative to which we work
out our theory is the category of sets endowed with norms (or families of
norms). Base change functors to analytic spaces over Banach rings are studied
and the basic spaces of analytic geometry (like polydisks) are recovered as a
base change of analytic spaces over F_1. We end by discussing some applications
of our theory to the theory of the Fargues-Fontaine curve and to the ring Witt
vectors.Comment: Small corrections have been made in the last section of the paper and
some typos have been correcte
Relative polynomial closure and monadically Krull monoids of integer-valued polynomials
Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on
D. For any f in Int(D), we explicitly construct a divisor homomorphism from
[f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of
copies of (N_0,+). This implies that [f] is a Krull monoid.
For V a discrete valuation domain, we give explicit divisor theories of
various submonoids of Int(V). In the process, we modify the concept of
polynomial closure in such a way that every subset of D has a finite
polynomially dense subset.
The results generalize to Int(S,V), the ring of integer-valued polynomials on
a subset, provided S doesn't have isolated points in v-adic topology.Comment: 12 pages; v.2 contains corrections, in that some necessary conditions
on those subsets S, for which we consider integer-valued polynomials on
subsets, are impose
Completed representation ring spectra of nilpotent groups
In this paper, we examine the `derived completion' of the representation ring
of a pro-p group G_p^ with respect to an augmentation ideal. This completion is
no longer a ring: it is a spectrum with the structure of a module spectrum over
the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In
order to explain the origin of some of these higher homotopy classes, we define
a deformation representation ring functor R[-] from groups to ring spectra, and
show that the map R[G_p^] --> R[G] becomes an equivalence after completion when
G is finitely generated nilpotent. As an application, we compute the derived
completion of the representation ring of the simplest nontrivial case, the
p-adic Heisenberg group.Comment: This is the version published by Algebraic & Geometric Topology on 26
February 200
The \'etale symmetric K\"unneth theorem
Let be an algebraically closed field, a
prime number, and a quasi-projective scheme over . We show that the
\'etale homotopy type of the th symmetric power of is -homologically equivalent to the th strict symmetric power of the
\'etale homotopy type of . We deduce that the -local \'etale
homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary
Eilenberg-Mac Lane space.Comment: revised version, comments welcome
Segal's conjecture and the Burnside rings of fusion systems
Peer reviewedPostprin
Topological modular forms with level structure
The cohomology theory known as Tmf, for "topological modular forms," is a
universal object mapping out to elliptic cohomology theories, and its
coefficient ring is closely connected to the classical ring of modular forms.
We extend this to a functorial family of objects corresponding to elliptic
curves with level structure and modular forms on them. Along the way, we
produce a natural way to restrict to the cusps, providing multiplicative maps
from Tmf with level structure to forms of K-theory. In particular, this allows
us to construct a connective spectrum tmf_0(3) consistent with properties
suggested by Mahowald and Rezk.
This is accomplished using the machinery of logarithmic structures. We
construct a sheaf of locally even-periodic elliptic cohomology theories,
equipped with highly structured multiplication, on the log-\'etale site of the
moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf
with level structure.Comment: 53 pages. Heavily revised, including the addition of a new section on
background tools from homotopy theor
- …