arXiv subjects

Kirill Magidson

Publications and source records attributed to Kirill Magidson.

3 recordsLinked to original sources

arXiv2025

Formal Integration of Derived Foliations

Frobenius' theorem in differential geometry asserts that every involutive subbundle of the tangent bundle of a manifold $M$ integrates to a decomposition of $M$ into smooth leaves. We prove an infinitesimal analogue of this result for locally coherent qcqs schemes $X$ over coherent rings. More precisely, we integrate partition Lie algebroids on $X$ to formal moduli stacks $X \rightarrow S$ where $S$ is the formal leaf space and the fibres of $X \rightarrow S$ are the formal leaves. We deduce that deformations of $X$-families of algebro-geometric objects are controlled by partition Lie algebroids on $X$. Combining our integration equivalence with a result of Fu, we deduce that To\"{e}n-Vezzosi's infinitesimal derived foliations (under suitable finiteness hypotheses) are formally integrable.

math.AG↗

arXiv2024

Witt vectors and $\delta$-Cartier rings

We give a universal property of the construction of the ring of $p$-typical Witt vectors of a commutative ring, endowed with Witt vectors Frobenius and Verschiebung, and generalize this construction to the derived setting. We define an $\infty$-category of $p$-typical derived $\delta$-Cartier rings and show that the derived ring of $p$-typical Witt vectors of a derived ring is naturally an object in this $\infty$-category. Moreover, we show that for any prime $p$, the formation of the derived ring of $p$-typical Witt vectors gives an equivalence between the $\infty$-category of all derived rings and the full subcategory of all derived $p$-typical $\delta$-Cartier rings consisting of $V$-complete objects.

math.KT↗

arXiv2024

Divided Powers and Derived De Rham Cohomology

We develop the formalism of derived divided power algebras, and revisit the theory of derived De Rham and derived crystalline cohomology in this framework. We characterize derived De Rham cohomology of a derived commutative algebra $A$ over a base $R$, together with the Hodge filtration on it, in terms of the universal property as the largest filtered divided power thickening of $A$. We show that our approach recovers the classical De Rham cohomology in the case of a smooth map $R\rightarrow A$, and therefore in general, recovers the derived De Rham cohomology in the sense of Illusie. Along the way, we develop some generalities on square-zero extensions and derivations in derived algebraic geometry and apply them to give the universal property of the first Hodge truncation of the derived De Rham cohomology. Finally, we define derived crystalline cohomology relative to a general divided power base, show that it satisfies the main properties of the crystalline cohomology and coincides with the classical crystalline cohomology in the smooth case.

math.AG↗