Fix a symbol $underline{a}$ in the mod-$ell$ Milnor $K$-theory of a field $k$, and a norm variety $X$ for $underline{a}$. We show that the ideal generated by $underline{a}$ is the kernel of the $K$-theory map induced by $ksubset k(X)$ and give generators for the annihilator of the ideal. When $ell=2$, this was done by Orlov, Vishik and Voevodsky.
In this paper we study a spectrum $K(mathcal{V}_k)$ such that $pi_0 K(mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_0[mathcal{V}_k]$ and show that classes in the kernel of multiplication by $[mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are varieties such that $[X] eq [Y]$, $Xtimes mathbb{A}^1$ and $Ytimes mathbb{A}^1$ are not piecewise isomorphic, but $[Xtimes mathbb{A}^1] =[Ytimes mathbb{A}^1]$ in $K_0[mathcal{V}_k]$. Along the way we present new proofs of the result of Larsen--Lunts on the structure on $K_0[mathcal{V}_k]/([mathbb{A}^1])$.
This paper contains a construction of generators and partial relations for $K_1$ of a simplicial Waldhausen category where cofiber sequences split up to weak equivalence. The main application of these generators and relations is to produce generators for $K_1$ of a (simplicial) assembler.
Let $mathcal{C}$ be a finitely bicomplete category and $mathcal{W}$ a subcategory. We prove that the existence of a model structure on $mathcal{C}$ with $mathcal{W}$ as subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where $mathcal{C}$ is a partial order and show that these are determined by the homotopy categories.
This paper works out in detail the closed multicategory structure of the category of Waldhausen categories.
