We prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s formula for the Chow groups of $0$ -cycles on such schemes. We identify the Chow group of $0$ -cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasiprojective schemes over algebraically closed fields. This completes the missing p-part in the torsion theorem of Spieß and Szamuely.

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles $\text{CH}_{0}(X)$ and the Chow group of 0-cycles with modulus $\text{CH}_{0}(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of $\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that $\text{CH}_{0}(X|D)$ is torsion-free and there is an injective cycle class map $\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_{0}(X,D)$ is an extension of $\text{CH}_{1}(X|D)$ by $\text{CH}_{0}(X|D)$.

We show that for any commutative Noetherian regular ring $R$ containing $\mathbb{Q}$, the map $K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.

We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.

We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

We show localization, excision and descent theorems for K-theory of Deligne-Mumford stacks. Our approach employs the Nisnevich site which is a complete, regular and bounded cd-structure on the category of such stacks and restricts to the usual Nisnevich site on schemes. By combining excision with a refinement of localization sequences due to Krishna and Töen, we show that K-theory of perfect complexes on tame Deligne-Mumford stacks satisfies Nisnevich descent.

For a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.

We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.

We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

Let X be a reduced and projective singular surface over ℂ and let → X be a resolution of singularities of X. We show that CH2(X) ≅ CH2() if and only if for i = 0, 1. This verifies a conjecture of Srinivas.

We also verify Bloch's conjecture for singular surfaces assuming it holds for smooth surfaces. As a byproduct, we give an application to projective modules on certain singular affine surfaces.