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Tom Bachmann

Publications and source records attributed to Tom Bachmann.

At least 19 recordsLinked to original sources
arXiv2026

Motivic Eilenberg--Mac Lane spaces at the characteristic

We provide a novel, homotopy theoretic model for the motivic Eilenberg--Mac Lane spaces $K(Z_{(p)}(n), 2n) \in Spc(k)$ over fields of characteristic p. Using this, we determine their motives and, consequently, determine the motivic Steenrod algebra at the characteristic. Applications include the Hopkins--Morel isomorphism over arbitrary base schemes as well as the computation of the p-completed motivic stable stems and algebraic cobordism groups of fields of characteristic p. Along the way we expand the theory of normed algebraic structures in motivic homotopy theory, compare them to derived algebra at the characteristic, and produce a functor computing the motivic homology of the $P^1$-loop spaces of (certain) motivic spaces in terms of the motives.

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arXiv2026

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C.

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arXiv2026

Algebraic vs. analytic: a survey

Given a pair of complex algebraic varieties, we can ask: does every analytic map admit an algebraic representative? Such questions are most interesting when the source and target are non-compact varieties, in which case the problems are closely linked with Hodge-type conjectures. Concrete avatars of this kind of problem, which we discuss from several points of view, include: which holomorphic vector bundles on an affine variety admit an algebraic structure, and which homotopy classes of maps admit algebraic representatives?

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arXiv2026

Motivic Hochschild homology of mod 2 motivic cohomology over algebraically closed fields

We compute the tensor of the multiplicative group scheme with the mod-$2$ motivic cohomology spectrum in normed motivic spectra over the complex numbers, and find that the resulting algebra is free on a generator in bidegree (2,1). This gives a motivic analog of Bökstedt periodicity. The proof proceeds by comparing the tau-inverted and tau-reduced forms of the tensor. After inverting tau, the calculation reduces to classical B{ö}kstedt periodicity via Betti realization. The reduction modulo tau is governed by a comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau. This comparison produces divided power operations and leads to mixed Cartan and Adem relations intertwining normed and topological power operations. A key input is a detailed analysis of motivic extended powers of spheres and their tau-torsion structure. In contrast with the corresponding simplicial-circle calculation due to Dundas-Hill-Ormsby-Østvær, the large families of tau-torsion classes disappear for the Gm-tensor, leaving a considerably more rigid algebraic structure.

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arXiv2026

On P^1-stabilization in unstable motivic homotopy theory

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for ${\mathbb P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

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arXiv2025

Monadic resolutions for generalized spaces

We extend the work of Bousfield and Kan on monadic resolutions of spaces to $\infty$-topoi, with applications to genuine $G$-equivariant spaces ($G$ a finite group) and motivic spaces over a perfect field. In particular, we give a proof of the principal fibration lemma in this context. We apply the principal fibration lemma to prove convergence of several kinds of monadic resolutions in unstable equivariant and motivic homotopy theory. For example, we show that, over an algebraically closed field, the unstable Adams--Novikov spectral sequence (i.e., the monadic resolution corresponding to the algebraic cobordism spectrum $\mathrm{MGL}$) converges for all nilpotent, connected, $2$-effective motivic spaces.

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arXiv2025

$\mathbb{A}^1$-invariant motivic cohomology of schemes

Voevodsky outlined a conjectural programme that his slice filtration in motivic homotopy theory should give rise to a good theory of $\mathbb{A}^1$-invariant motivic cohomology. This paper achieves his vision in the generality of arbitrary quasicompact, quasiseparated schemes, by introducing a theory of $\mathbb{A}^1$-invariant motivic cohomology which is related to Weibel's homotopy $K$-theory via an Atiyah--Hirzebruch spectral sequence, and which we compare to étale and syntomic cohomology in the style of the original conjectures of Beilinson and Lichtenbaum. In addition, it is represented by an absolute motivic spectrum and therefore satisfies cdh descent, and modules over it offer a candidate for the derived category of $\mathbb{A}^1$-invariant motives. We establish some of Voevodsky's open conjectures on slices, in particular relating the zeroth slice of the motivic sphere to homotopy $K$-theory. In the final section we prove analogous results for the Hermitian $K$-theory of qcqs schemes on which $2$ is invertible. As an auxiliary tool we introduce cdh-motivic cohomology, defined as the cdh sheafification of the left Kan extension of the motivic cohomology of smooth $\mathbb{Z}$-schemes. We offer a new approach to control the latter, independent of previous work on $\mathbb{A}^1$-invariant motivic cohomology of smooth schemes over mixed characteristic Dedekind domains: our approach is based on recent developments in $p$-adic cohomology, in particular syntomic and prismatic cohomology. The cdh-motivic cohomology is also a necessary ingredient in the last two authors' and Bouis' construction of non-$\mathbb{A}^1$-invariant motivic cohomology of qcqs schemes.

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arXiv2025

Unstable motivic and real-étale homotopy theory

We prove that for any base scheme $S$, real étale motivic (unstable) homotopy theory over $S$ coincides with unstable semialgebraic topology over $S$ (that is, sheaves of spaces on the real spectrum of $S$). Moreover we show that for pointed connected motivic spaces over $S$, the real étale motivic localization is given by smashing with the telescope of the map $ρ: S^0 \to {\mathbb G}_m$.

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arXiv2024

Strongly A^1-invariant sheaves (after F. Morel)

Strongly (respectively strictly) A1-invariant sheaves are foundational for motivic homotopy theory over fields. They are sheaves of (abelian) groups on the Nisnevich site of smooth varieties over a field k, with the property that their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich cohomology groups) are invariant under replacing a variety X by the affine line over X. A celebrated theorem of Fabien Morel states that if the base field k is perfect, then any strongly A1-invariant sheaf of abelian groups is automatically strictly A1-invariant. The aim of these lecture notes is twofold: (1) provide a complete proof if this result, and (2) outline some of its applications.

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arXiv2024

On the Whitehead theorem for nilpotent motivic spaces

We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov factorizations for nilpotent morphisms of motivic spaces under a mild technical hypothesis and establish an analog of the Whitehead theorem for nilpotent motivic spaces. As an application, we deduce a surprising unstable motivic periodicity result.

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arXiv2024

$\mathbb{E}_{\infty}$-coalgebras and $p$-adic homotopy theory

We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.

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arXiv2023

Notes on motivic infinite loop space theory

In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory, together with their proofs, in the case of infinite perfect fields. These are our extended notes on these talks.

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arXiv2023

On quadratically enriched excess and residual intersections

We use recent duality results of Eisenbud--Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections.

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arXiv2022

Normed motivic spectra and power operations

We use motivic colimits to construct power operations on the homotopy groups of normed motivic spectra admitting a (normed) map from HF_2. We establish enough of their standard properties to prove that the motivic dual Steenrod algebra is generated by one element under ring and power operations, establishing a motivic analog of Steinberger's theorem.

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