We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to four when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and for closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.

Inspired by work of Szymik and Wahl on the homology of Higman–Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.

Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui’s AH-conjecture for all minimal, ample groupoids with comparison.

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance, the negative K-groups and all twisted Nil-groups vanish for an additive category, if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.

Tachikawa's second conjecture for symmetric algebras is shown to be equivalent to indecomposable symmetric algebras not having any nontrivial stratifying ideals. The conjecture is also shown to be equivalent to the supremum of stratified ratios being less than $1$, when taken over all indecomposable symmetric algebras. An explicit construction provides a series of counterexamples to Tachikawa's second conjecture from each (potentially existing) gendo-symmetric algebra that is a counterexample to Nakayama's conjecture. The results are based on establishing recollements of derived categories and on constructing new series of algebras.

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$- and $\mathrm {T}(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$.

Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$-theory spectra.

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$-theory.

We prove that real topological Hochschild homology $\mathrm {THR}$ for schemes with involution satisfies base change and descent for the ${\mathbb {Z}/2}$-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher-dimensional projective spaces.

The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.

Loday’sassembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalisation that places both ingredients on the same footing. Building on Elmendorf–Mandell’s multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher-categorical language. It also allows us to extend the idea to new contexts and set up a nonabelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

We study fundamental properties of analytic K-theory of Tate rings such as homotopy invariance, Bass fundamental theorem, Milnor excision, and descent for admissible coverings.

In this paper we compute the topological Hochschild homology of quotients of discrete valuation rings (DVRs). Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of $\operatorname {THH}$ (respectively, logarithmic $\operatorname {THH}$) of complete DVRs originally due to Lindenstrauss and Madsen (respectively, Hesselholt and Madsen).

For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.

We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

We develop methods for constructing explicit generators, modulo torsion, of the $K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

We prove several results showing that the algebraic $K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.

Vorst's conjecture relates the regularity of a ring with the $\mathbb {A}^{1}$-homotopy invariance of its $K$-theory. We show a variant of this conjecture in positive characteristic.

We discuss some general properties of $\mathrm {TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm {TR}$ of $H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm {TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and étale cohomology for $K(1)$-local $\mathrm {TR}$.