Klaus Mattis

Hi, I am Klaus Mattis, a fourth 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

Publications

• The derived \(\infty\)-category of Frobenius modules (joint with Timo Weiß) (Arxiv) (PDF) (to appear: Journal of Algebra)

  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 arithmetic fracture squares in \(\infty\)-topoi (Arxiv) (PDF) (Journal of Homotopy and Related Structures)

  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 derived \(\infty\)-category of Cartier modules (joint with Timo Weiß) (Arxiv) (PDF) (Journal of Pure and Applied Algebra)

  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 \(F\colon X\to X\).

Preprints

• The \(6\frac{1}{2}\)-functor formalism of Cartier modules (joint with Timo Weiß) (Arxiv) (PDF)

  Let \(p\) be a prime. We construct a coherent formalism of geometric operations on Cartier modules with coefficients in ind-coherent sheaves on noetherian derived \(\mathbb{F}_p\)-schemes. For such a scheme \(X\), we consider the stable presentable category \(\mathrm{Cart}(X)\) of ind-coherent sheaves \(M\) equipped with a structural morphism \(F_{*,X} M \to M\), where \(F_X\) denotes the Frobenius endomorphism. We construct pushforward functors for arbitrary morphisms and exceptional pullback functors for separated morphisms almost of finite type, together with coherent composition and base-change equivalences. We single out conditions under which these functors admit further adjoints. On derived schemes separated and almost of finite type over Fp, we construct compatible closed symmetric monoidal structures given by shriek tensor products, and establish which functors are symmetric monoidal. The constructions arise from a general procedure for forming functor formalisms of Cartier objects, using their description as lax limits and the universal properties of span \(2\)-categories developed by Cnossen–Lenz–Linskens.

• 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. Moreover, we prove an étale version of May’s nilpotence conjecture, that states that \(H\mathbb{Z} \in \mathrm{Sp}\) detects the vanishing of \(\mathbb{E}_\infty\)-rings. We use this to show a version of Nishida’s nilpotence theorem in \(\mathrm{SH}_{\operatorname{\acute{e}t}}(S)\), i.e. that any positive degree self map of the unit is nilpotent.

• 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.

• 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 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

09/26	Unstable \(\eta\)-completion and the motivic Eilenberg-Moore spectral sequence	SLMath, Berkeley
06/26	Étale synthetic deformations	Young topologists meeting, Copenhagen
05/26	Monadic resolutions for generalized spaces	Universität Regensburg
03/26	Cdh-local motivic cohomology	Recent Advances in Motivic Cohomology, Osnabrück
02/26	Monadic resolutions for generalized spaces	Universität Wuppertal
01/26	Monadic resolutions for generalized spaces	University Copenhagen
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 Workshop on motivic loop spaces in Mainz.
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