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 genera
tors 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 topolog
y 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.
