Galois descent of determinants in the ramified case

In the local, unramified case the determinantal functions associated to the group-ring of a finite group satisfy Galois descent. This note examines the obstructions to Galois determinantal descent in the ramified case.Comment: 12 page

Relative K_0, annihilators, Fitting ideals and the Stickelberger phenomena

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures Coates-Sinnott and Brumer

The Arf-Kervaire Invariant of framed manifolds

This work surveys classical and recent advances around the existence of exotic differentiable structures on spheres and its connection to stable homotopy theory

On the motivic spectra representing algebraic cobordism and algebraic K-theory

We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\mathbb{C}$ and passing to spaces of $\mathbb{C}$-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic $K$-theory and periodic algebraic cobordism are $E_\infty$ motivic spectra.Comment: 28 pages; minor revisions and added application

Refined and l-adic Euler characteristics of nearly perfect complexes

We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics.Comment: 24 page

Derived Langlands VI: Monomial resolutions and 22-variable L-functions

In this brief essay a construction of the $2$-variable L-function of Langlands is sketched in terms of monomial resolutions of admissible representations of reductive locally $p$-adic Lie groups

Burns' equivariant Tamagawa invariant T Omega^{loc}(N/\Q,1) for some quaternion fields

We verify one of Burn's equivariant Tamagawa number conjectures for some families of quaternion fields

Derived Langlands V:The simplicial and Hopflike categories

This is the fifth article in the Derived Langlands series which consists of one monograph and four articles. In this article I describe the Hopf algebra and Positive Selfadjoint Hopfalgebra (PSH) aspects to classification of a number of new classes of presentations and admissibility which have appeared earlier in the series. The paper begins with a very estensive. partly hypothetical, of the synthesis of the entire series. Many of the proofs and ideas in this series are intended to be suggestive rather than the finished definitive product for extenuating circumstances explained therein

Non-factorisation of Arf-Kervaire classes through RP∞∧RP∞{\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to ${\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.Comment: 5 page

Derived Langlands IV:Notes on Mc(G){\mathcal M}_{c}(G)-induced representations

This is Part IV of a thematic series currently consisting of a monograph and four essays. This essay examines the form of induced representations of locally p-adic Lie groups G which is appropriate for the abelian category of ${\mathcal M}_{c}(G)$-admissible representations. In my non-expert manner, I prove the analogue of Jacquet's Theorem in this category. The final section consists of observations and questions related to this and other concepts introduced in the course of this series