474 research outputs found

### Goncharov's relations in Bloch's higher Chow group CH^3(F,5)

Using Totaro-Bloch-Kriz's linear fractional cycles Gangl and Muller-Stach
recently prove the 5-term relations for the dilogarithm in Bloch's higher Chow
group CH^2(F,3) and the Kummer-Spence relations in some group G(F) over an
arbitrary field F where G(F) is isomorphic to CH^3(F,5) up to torsions under
the Beilinson-Soule vanishing conjecture that CH^2(F,n)=0 for n>3. In this
paper we show that Goncharov's 22-term relations for the trilogarithm also hold
in G(F).Comment: 16 pages. This is a simplified versio

### Surjectivity of $p$-adic regulator on $K_2$ of Tate curves

I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of
Tate curve to the $p$-adic etale cohomology group when the base field is
contained in a cyclotomic extension of $Q_p$. This implies the finiteness of
torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence

### A geometric proof that SL_2(Z[t,t^-1]) is not finitely presented

We give a new proof of the theorem of Krstic-McCool from the title. Our proof
has potential applications to the study of finiteness properties of other
subgroups of SL_2 resulting from rings of functions on curves.Comment: This is the version published by Algebraic & Geometric Topology on 11
July 200

### Bi-relative algebraic K-theory and topological cyclic homology

It is well-known that algebraic K-theory preserves products of rings.
However, in general, algebraic K-theory does not preserve fiber-products of
rings, and bi-relative algebraic K-theory measures the deviation. It was proved
by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative
cyclic homology agree. In this paper, we show that, with finite coefficients,
bi-relative algebraic K-theory and bi-relative topological cyclic homology
agree. As an application, we show that for a, possibly singular, curve over a
perfect field of positive characteristic p, the cyclotomic trace map induces an
isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic
homology groups in non-negative degrees. As a further application, we show that
the difference between the p-adic K-groups of the integral group ring of a
finite group and the p-adic K-groups of a maximal Z-order in the rational group
algebra can be expressed entirely in terms of topological cyclic homology

### General linear and functor cohomology over finite fields

In recent years, there has been considerable success in computing Ext-groups
of modular representations associated to the general linear group by relating
this problem to one of computing Ext-groups in functor categories. In this
paper, we extend our ability to make such Ext-group calculations by
establishing several fundamental results. Throughout this paper, we work over
fields of positive characteristic p.Comment: 66 pages, published version, abstract added in migratio

### Cohomology for Frobenius kernels of $SL_2$

Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$
defined over an algebraically field of characteristic $p>2$. In this paper we
give for $r\ge 1$ a complete description of the cohomology groups for
$(SL_2)_r$. We also prove that the reduced cohomology ring
\opH^\bullet((SL_2)_r,k)_{\red} is Cohen-Macaulay. Geometrically, we show for
each $r\ge 1$ that the maximal ideal spectrum of the cohomology ring for
$(SL_2)_r$ is homeomorphic to the fiber product G\times_B\fraku^r. Finally,
we adapt our calculations to obtain analogous results for the cohomology of
higher Frobenius-Luzstig kernels of quantized enveloping algebras of type
$SL_2$.Comment: published version; a section for the case p=2 is adde

### Transfer maps and nonexistence of joint determinant

Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first
defined by Bass and Tate (1972) for simple extensions of fields via tame symbol
and Weil's reciprocity law, but their functoriality had not been settled until
Kato (1980). On the other hand, functorial transfer maps for the Goodwillie
group are easily defined. We show that these natural transfer maps actually
agree with the classical but difficult transfer maps by Bass and Tate. With
this result, we build an isomorphism from the Goodwillie groups to Milnor's
$K$-groups of fields, which in turn provides a description of joint
determinants for the commuting invertible matrices. In particular, we
explicitly determine certain joint determinants for the commuting invertible
matrices over a finite field, the field of rational numbers, real numbers and
complex numbers into the respective group of units of given field.Comment: Some minor revisions have been made after the 1st versio

### Tetrahedra of flags, volume and homology of SL(3)

In the paper we define a "volume" for simplicial complexes of flag
tetrahedra. This generalizes and unifies the classical volume of hyperbolic
manifolds and the volume of CR tetrahedra complexes. We describe when this
volume belongs to the Bloch group. In doing so, we recover and generalize
results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to
the work of Fock and Goncharov.Comment: 45 pages, 14 figures. The first version of the paper contained a
mistake which is correct here. Hopefully the relation between the works of
Neumann-Zagier on one side and Fock-Goncharov on the other side is now much
cleare

### On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

We show that if $G$ is an infinitesimal elementary supergroup scheme of
height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally
homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms
$\rho: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine
supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further
identify the cohomological support variety of a finite-dimensional
$G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. We then discuss how our
methods, when combined with recently-announced results by Benson, Iyengar,
Krause, and Pevtsova, can be applied to extend the homeomorphism
$\mathcal{N}_r(G) \cong |G|$ to arbitrary infinitesimal unipotent supergroup
schemes.Comment: Fixed some algebra misidentifications, primarily in Sections 1.3 and
3.3. Simplified the proof of Proposition 3.3.

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