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, particularly higher topoi Algebraic \(K\)-theory

Work

• 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