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 pp-adic regulator on K2K_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 SL2SL_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.