arXiv subjects

Can Yaylali

Publications and source records attributed to Can Yaylali.

8 recordsLinked to original sources

arXiv2026

Representability of continuous K-theory in rigid analytic motivic $\mathbb{A}^1$-homotopy theory

We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (\`a la Kerz--Saito--Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is $\mathbb{A}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the $\mathbb{A}^{1}$-homotopy category of rigid spaces (\`a la Dahlhausen--Yaylali). We identifiy the representing object with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is $\mathbb{A}^1$-invariant on local Tate pairs (without any regularity assumption).

math.KT↗

arXiv2025

Truncations and the Motive of the Stack of Local $G$-Shtukas

We compute the rational motive of the stack of local $G$-shtukas, for a split reductive group $G$, representing compactly supported cohomology in terms of the motive of the stack of $G$-zips. This result makes explicit use of the truncated version of local $G$-shtukas established by Viehmann-Wedhorn and the theoretical background on the $*$- and $!$-adjunction for pro-systems of algebraic objects established by the author.

math.AG↗

arXiv2024

Motivic homotopy theory of the classifying stack of finite groups of Lie type

Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the $T$-equivariant Tate motives of the flag variety $G/B$. On the way, we show that for a reductive group $G$ over a field $k$, with maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type $k$-scheme $X$, we have an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of Edidin-Graham to higher equivariant Chow groups in the non-split case. We also extend our main result to reductive group schemes over a regular base that admit maximal tori. Further, we apply our methods to more general quotient stacks. In this way, we are able to compute the motive of the stack of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over fields of positive characteristic.

math.AG↗

arXiv2024

Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces

To any rigid analytic space (in the sense of Fujiwara--Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six-functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we prove mod $p$ rigidity for rigid spaces over $\mathbb{Q}_p$. Moreover, we prove the equivalence of $\mathbb{A}^1$-invariant sheaves and $\mathbb{B}^1$-invariant sheaves in the mod $\ell$ case.

math.AT↗

arXiv2024

$T$-equivariant motives of flag varieties

We use the construction of the stable homotopy category by Khan-Ravi to calculate the integral $T$-equivariant $K$-theory spectrum of a flag variety over an affine scheme, where $T$ is a split torus associated to the flag variety. More precisely, we show that the $T$-equivariant $K$-theory ring spectrum of a flag variety is decomposed into a direct sum of $K$-theory spectra of the classifying stack $\text{B}T$ indexed by the associated Weyl group. We also explain how to relate these results to the motivic world and deduce classical results for $T$-equivariant intersection theory and $K$-theory of flag varieties.\par For this purpose, we analyze the motive of schemes stratified by affine spaces with group action, that preserves these stratifications. We work with cohomology theories, that satisfy certain vanishing conditions, which are satisfied for example by motivic cohomology and $K$-Theory.

math.AG↗

arXiv2024

Rational motives on pro-algebraic stacks

We define an $\infty$-category of rational motives for inverse limits of algebraic stacks, so-called pro-algebraic stacks. We show that it admits a $6$-functor formalism for certain classes of morphisms. On pro-schemes, we show that this $6$-functor formalism is in the sense of Liu-Zheng. This theory yields an approach to the theory of motives for non-representable algebraic stacks and non-finite type morphisms.

math.AG↗

arXiv2022

Notes on derived algebraic geometry

These are notes on derived algebraic geometry in the context of animated rings. More precisely, we recall the proof of Toën-Vaquié that the derived stack of perfect complexes is locally geometric in the language of $\infty$-categories. Along the way, we recall the necessary notions in derived commutative algebra and derived algebraic geometry. We also analyze the deformation theory and quasi-coherent modules over derived stacks.

math.AG↗

arXiv2022

Derived $F$-zips

We define derived versions of $F$-zips and associate a derived $F$-zip to any proper, smooth morphism of schemes in positive characteristic. We analyze the stack of derived $F$-zips and certain substacks. We make a connection to the classical theory and look at problems that arise when trying to generalize the theory to derived $G$-zips and derived $F$-zips associated to lci morphisms. As an application, we look at Enriques-surfaces and analyze the geometry of the moduli stack of Enriques-surfaces via the associated derived $F$-zips. As there are Enriques-surfaces in characteristic $2$ with non-degenerate Hodge-de Rham spectral sequence, this gives a new approach, which could previously not be obtained by the classical theory of $F$-zips.

math.AG↗