Gersten Conjecture For Equivariant K-theory And Applications

For a reductive group scheme over a regular semi-local ring, we prove an equivarinat version of the Gersten conjecture. We draw some interesting consequences for the representation rings of such reductive group schemes. We also prove the rigidity for the equivariant K-theory of reductive group schemes over a henselian local ring. This is then used to compute the equivariant K-theory of algebraically closed fields

Equivariant cobordism of schemes

We study the equivariant cobordism theory of schemes for action of linear algebraic groups. We compare the equivariant cobordism theory for the action of a linear algebraic groups with similar groups for the action of tori and deduce some consequences for the cycle class map of the classifying space of an algebraic groups.Comment: This revised version supercedes arxiv:1006:317

Cobordism of flag bundles

Let $G$ be a connected linear algebraic group over a field $k$ of characteristic zero. For a principal $G$-bundle $\pi: E \to X$ over a scheme $X$ of finite type over $k$ and a parabolic subgroup $P$ of $G$, we describe the rational algebraic cobordism and higher Chow groups of the flag bundle $E/P \to X$ in terms of the cobordism of $X$ and that of the classifying space of a maximal torus of $G$ contained in $P$. As a consequence, we also obtain the formula for the cobordism and higher Chow groups of the principal bundles over the scheme $X$. If $X$ is smooth, this describes the cobordism ring of these bundles in terms of the cobordism ring of $X$

A cdh approach to zero-cycles on singular varieties

We study the Chow group of zero-cycles on singular varieties using the cdh topology. We define the cdh versions of the zero-cycles and albanese maps. We prove results comparing these groups for a singular variety with the similar groups on the resolution of singularities. We use these to prove some results about the known Chow group of zero-cycles on surfaces and threefolds and some cases of arbitrary dimension

On The Negative K-Theory of Schemes in Finite Characteristic

We study the negative $K$-theory of singular varieties over a field of positive characteristic and in particular, prove the vanishing of $K_i(X)$ for $i < -d-2$ for a $k$-variety of dimension $d$