253 research outputs found
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
On Oliver's p-group conjecture
Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain
characteristic subgroup X(S) always contains the Thompson subgroup J(S). We
obtain a reformulation of the conjecture as a statement about modular
representations of p-groups. Using this we verify Oliver's conjecture for
groups where S/X(S) has nilpotence class at most two.Comment: 9 pages; terminology altered to conform to current usag
On a multiplicative version of Bloch's conjecture
A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of
0-cycles of a smooth complete complex variety decomposes, then the top-degree
coherent cohomology group decomposes similarly. In this note, we prove that (a
weak version of) the converse holds for varieties of dimension at most 5 that
have finite-dimensional motive and satisfy the Lefschetz standard conjecture.
The proof is based on Vial's construction of a refined Chow-Kunneth
decomposition for these varieties.Comment: To appear (in slightly different form) in Beitrage zur Algebra und
Geometrie, 8 pages, comments welcome. arXiv admin note: text overlap with
arXiv:1602.0494
Hyperdescent and \'etale K-theory
We study the \'etale sheafification of algebraic K-theory, called \'etale
K-theory. Our main results show that \'etale K-theory is very close to a
noncommutative invariant called Selmer K-theory, which is defined at the level
of categories. Consequently, we show that \'etale K-theory has surprisingly
well-behaved properties, integrally and without finiteness assumptions. A key
theoretical ingredient is the distinction, which we investigate in detail,
between sheaves and hypersheaves of spectra on \'etale sites.Comment: 89 pages, v3: various corrections and edit
On a strong form of Oliver’s p-group conjecture.
We introduce a stronger and more tractable form of Olivers p-group conjecture, and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group Sn and of the general linear group satisfy both the strong conjecture and its reformulation
Quillen stratification for the Steenrod algebra
Let A be the mod 2 Steenrod algebra, and let Q denote the category of
exterior sub-Hopf algebras of A, where the morphisms are given by inclusions.
The restriction maps Ext_A (Z/2,Z/2) --> Ext_E (Z/2,Z/2), for E in Q, can be
assembled into a map
i:Ext_A (Z/2, Z/2) --> lim_Q Ext_E (Z/2,Z/2).
There is an action of A on this inverse limit, and i factors through the
invariants under this action, giving a map
g:Ext_A (Z/2, Z/2) --> ( lim_Q Ext_E (Z/2,Z/2) )^A.
We show that g is an F-isomorphism.Comment: 29 pages, published versio
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