Klaus Mattis

Hi, I am Klaus Mattis, a third year PhD-student at JGU Mainz, under the supervision of Tom Bachmann.

For more information, consult my CV.

You can reach me via mail at klaus.mattis@uni-mainz.de.

Research Interests

• Unstable and stable motivic homotopy theory
• Higher categories and \(\infty\)-topoi
• Synthetic deformations of categories
• Algebraic \(K\)-theory
• Characteristic \(p\) algebraic geometry

Work

• Nilpotence of \(\eta\) in étale motivic spectra (joint with Swann Tubach) (Arxiv) (PDF)

  We show that every object of the stable étale motivic homotopy category over any scheme is \(\eta\)-complete. In some cases we show that in fact the fourth power of \(\eta\) is null, whereas the third power of \(\eta\) is always nonvanishing, similar to the situation in topology.

• Monadic resolutions for generalized spaces (joint with Tom Bachmann and Anton Engelmann) (Arxiv) (PDF)

  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.

• The derived \(\infty\)-category of Frobenius modules (joint with Timo Weiß) (Arxiv) (PDF)

  We prove that for \(X\) a quasi-compact \(\mathbb{F}_p\)-scheme with affine diagonal (e.g.\ \(X\) quasi-compact and separated) there is a t-exact equivalence \(\mathcal D(\mathrm{Frob}(\mathrm{QCoh}(X),F_*)) \to \mathrm{Frob}(\mathcal D(\mathrm{QCoh}(X)),\mathcal D(F_*))\) of stable \(\infty\)-categories. Here, \(\mathrm{Frob}(-,-)\) denotes the \(\infty\)-category of generalized Frobenius modules. This generalizes our earlier result, where we proved the above for regular Noetherian \(\mathbb{F}_p\)-schemes. As a byproduct we prove that the derived \(\infty\)-category of Frobenius (and Cartier) modules satisfies Zariski descent.

• Unstable étale motives (Arxiv) (PDF)

  We prove a rigidity result for certain \(p\)-complete étale \(\mathbb{A}^1\)-invariant sheaves of anima over a qcqs finite-dimensional base scheme \(S\) of bounded étale cohomological dimension with \(p\) invertible on \(S\). This generalizes results of Suslin–Voevodsky, Ayoub, Cisinski–Déglise, and Bachmann to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies, in particular the \(p\)-completion of the étale sheafification of any \(2\)-effective \(2\)-connective motivic space, as well as the \(p\)-completion of any \(4\)-connective \(\mathbb{A}^1\)-invariant étale sheaf. We use this rigidity result to prove (a weaker version of) an étale analog of Morel’s theorem stating that for a Nisnevich sheaf of abelian groups, strong \(\mathbb{A}^1\)-invariance implies strict \(\mathbb{A}^1\)-invariance. Moreover, this allows us to construct an unstable étale realization functor on \(2\)-effective \(2\)-connective motivic spaces.

• The derived \(\infty\)-category of Cartier modules (joint with Timo Weiß) (Arxiv) (PDF)

  For an endofunctor \(F\colon\mathcal{C}\to\mathcal{C}\) on an (\(\infty\)-)category \(\mathcal{C}\) we define the \(\infty\)-category \(\operatorname{Cart}(\mathcal{C},F)\) of generalized Cartier modules as the lax equalizer of \(F\) and the identity. This generalizes the notion of Cartier modules on \(\mathbb{F}_p\)-schemes considered in the literature. We show that in favorable cases \(\operatorname{Cart}(\mathcal{C},F)\) is monadic over \(\mathcal{C}\). If \(\mathcal{A}\) is a Grothendieck abelian category and \(F\colon\mathcal{A}\to\mathcal{A}\) is an exact and colimit-preserving endofunctor, we use this fact to construct an equivalence \(\mathcal{D}(\operatorname{Cart}(\mathcal{A},F)) \simeq \operatorname{Cart}(\mathcal{D}(\mathcal{A}),\mathcal{D}(F))\) of stable \(\infty\)-categories. We use this equivalence to give a more conceptual construction of the perverse t-structure on \(\mathcal{D}^b_{\operatorname{coh}}(\operatorname{Cart}(\operatorname{QCoh}(X), F_*))\) for any Noetherian \(\mathbb{F}_p\)-scheme \(X\) with finite absolute Frobenius \(F\colon X\to X\).

• Unstable arithmetic fracture squares in \(\infty\)-topoi (Arxiv) (PDF)

  We show that for a large class of \(\infty\)-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf \(F\) as the pullback of the rationalization of \(F\) with the product of the \(p\)-completions of \(F\) ranging over all primes \(p \in \mathbb{Z}\).

• The pro-Nisnevich topology (Arxiv) (PDF)

  We construct the pro-Nisnevich topology, an analog of the pro-étale topology. We then show that the Nisnevich \(\infty\)-topos embeds into the pro-Nisnevich \(\infty\)-topos, and that the pro-Nisnevich \(\infty\)-topos is locally of homotopy dimension \(0\).

• Unstable \(p\)-completion in motivic homotopy theory (Arxiv) (PDF)

  We define unstable \(p\)-completion in general \(\infty\)-topoi and the unstable motivic homotopy category, and prove that the \(p\)-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the (\(p\)-completed) homotopy groups of the \(p\)-completion of a nilpotent motivic space \(X\) fit into short exact sequences:
  \( 0 \to \mathbb L_0 \pi_n(X) \to \pi_n^p(X^{\wedge}_p) \to \mathbb L_1 \pi_{n-1}(X) \to 0\),
  where the \(\mathbb L_i\) are (versions of) the derived \(p\)-completion functors, analogous to the classical situation.

Talks

07/25	Étale rigidity for motivic spaces	Motives and Arithmetic Geometry, Darmstadt
06/25	Canonical resolutions for motivic spaces	Young topologists meeting, Stockholm
04/25	Étale rigidity for motivic spaces	ENS Lyon
04/25	Étale rigidity for motivic spaces	University of Toronto
06/24	Proof of the Hopkins-Morel-Hoyois Theorem	International Workshop on Algebraic Topology, Shanghai
02/24	Unstable \(p\)-completion in motivic homotopy theory (Slides)	YoungHom

Academic service

I am an organizer of the GAUS Workshop on Motives and Higher Categories in Mainz.
I am an organizer of the GAUS Junior AG on Maps between spherical group rings in Mainz.
I am a co-organizer of the winter school on unstable motivic homotopy theory in Mainz.

Impressum

Angaben gem. § 5 TMG:
Klaus Mattis
Institut für Mathematik
Johannes Gutenberg-Universität
Staudingerweg 9
55128 Mainz