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Doosung Park

Publications and source records attributed to Doosung Park.

At least 19 recordsLinked to original sources
arXiv2026

Six-functor formalism for Kummer \'etale cohomology of log schemes

We establish a Grothendieck six-functor formalism for Kummer \'etale cohomology including Poincar\'e duality for every separated vertical exact log smooth morphism of noetherian fs log schemes $f\colon X\rightarrow S$ when the coefficient ring $\Lambda$ is killed by an integer invertible on $S$. This is done via log \'etale rigidity \[\mathrm{D}_{\mathrm{l\acute{e}t}}(S,\Lambda)\simeq \mathrm{DA}_{\mathrm{l\acute{e}t}}(S,\Lambda).\] To achieve this, we also prove that Kummer \'etale cohomology satisfies $\mathbb{A}^1$-invariance, invariance under virtual isomorphisms, log cdh-descent, and invariance under verticalization.

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arXiv2026

The logarithmic $h$- and $v$-topologies

We introduce the $h$- and $v$-topologies in the context of logarithmic geometry and discuss their applications to log \'etale cohomology, log differential forms, and log motives.

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arXiv2026

Poincar\'e duality in logarithmic motivic homotopy theory

By adapting arguments of Annala-Hoyois-Iwasa in the log setting, we prove Poincar\'e duality for smooth projective morphisms in logarithmic motivic homotopy theory. As an application, we show that the crystalline cohomology of a log compactification is independent of the choice.

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arXiv2026

On the $p$-adic deformation problem for the $K$-theory of semistable schemes

We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the $p$-adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem.

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arXiv2025

Log syntomic cohomology of truncated polynomials and coordinate axes

We study the logarithmic syntomic cohomology of fine and saturated log schemes and its realization in the logarithmic motivic stable homotopy category $\mathrm{logSH}(\mathrm{pt}_\mathbb{N})$ of a log point. We prove that logarithmic prismatic and syntomic cohomology satisfy saturated descent under the sole assumption that the log structure is free, and that the presheaves $\mathrm{logTHH}$, $\mathrm{logTC}$, $\widehat{\mathbf{\Delta}}$, and $\mathbb{Z}_p^\mathrm{syn}(i)$ are representable and $\square$-invariant in $\mathrm{logSH}_{\mathrm{k\acute{e}t}}^{\mathrm{eff}}(\mathrm{pt}_\mathrm{N})$. As an application, we compute $\mathbb{Z}_p^\mathrm{syn}(i)$ for the projective log coordinate axes $D$ in $\mathbb{P}^2$, obtaining \[ \mathbb{Z}_p^\mathrm{syn}(i)(D) \simeq \mathbb{Z}_p^\mathrm{syn}(i)(k,\mathbb{N})\oplus \mathbb{Z}_p^\mathrm{syn}(i-1)(k,\mathbb{N})[-2] \] Moreover, we determine logarithmic topological cyclic homology for truncated polynomial and semistable examples, directly from the syntomic calculations.

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arXiv2025

Construction of logarithmic cohomology theories I

We propose a method for constructing cohomology theories of logarithmic schemes with strict normal crossing boundaries by employing techniques from logarithmic motivic homotopy theory over $\mathbb{F}_1$. This method recovers the K-theory of the open complement of a strict normal crossing divisor from the K-theory of schemes as well as logarithmic topological Hochschild homology from the topological Hochschild homology of schemes. In our applications, we establish that the K-theory of non-regular schemes is representable in the logarithmic motivic homotopy category, and we introduce the logarithmic cyclotomic trace for the regular log regular case.

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arXiv2024

Logarithmic TC via the Infinite Root Stack and the Beilinson Fiber Square

We apply our previous results on ``saturated descent'' to express a wide range of logarithmic cohomology theories in terms of the infinite root stack. Examples include the log cotangent complex, Rognes' log topological cyclic homology, and Nygaard-complete log prismatic cohomology. As applications, we show that the Nygaard-completion of the site-theoretic log prismatic cohomology coincides with the definition arising from log ${\rm TC}$, and we establish a log version of the ${\rm TC}$-variant of the Beilinson fiber square of Antieau--Mathew--Morrow--Nikolaus.

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arXiv2024

Log motivic nearby cycles

We define the log motivic nearby cycles functor. We show that this sends the motive of a proper smooth scheme over the fraction field of a DVR to the motive of the boundary of a log smooth model assuming absolute purity, which is unconditional in the equal characteristic case. In characteristic $0$, we show that the $\infty$-categories of motives over the standard log point and rigid analytic motives are equivalent, and we relate log motivic nearby cycles functor with Ayoub's motivic nearby cycles functor.

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arXiv2024

Logarithmic Gysin sequences for regular immersions

For a regular immersion of schemes $Z\to X$ and a cohomology theory of fs log schemes, we formulate the logarithmic Gysin sequence using the "logarithmic compactification" $(\mathrm{Bl}_Z X,E)$ instead of the open complement $X-Z$, where $E$ is the exceptional divisor. We show that all $\mathbb{A}^1$-invariant cohomology theories produced from motivic spectra and various non $\mathbb{A}^1$-invariant cohomology theories like Nygaard completed prismatic cohomology admit logarithmic Gysin sequences.

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arXiv2024

Motivic six-functor formalism for log schemes

We establish the motivic six-functor formalism for fs log schemes. In particular, we prove the exact base change property, projection formula, and Poincar\'e duality. We also define Borel-Moore motivic homology, G-theory, and Chow homology of fs log schemes and the category of Chow motives over fs log schemes.

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arXiv2024

Log motivic exceptional direct image functors

In this paper, we construct the motivic exceptional direct image functors for fs log schemes. This construction is a part of the motivic six-functor formalism for fs log schemes.

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arXiv2024

On the logarithmic slice filtration

We consider slice filtrations in logarithmic motivic homotopy theory. Our main results establish conjectured compatibilities with the Beilinson, BMS, and HKR filtrations on (topological, log) Hochschild homology and related invariants. In the case of perfect fields admitting resolution of singularities, we show that the slice filtration realizes the BMS filtration on the $p$-completed topological cyclic homology. Furthermore, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic slice spectral sequence to the BMS spectral sequence. Finally, we consider the Kummer \'etale hypersheafification of logarithmic $K$-theory and show that its very effective slices compute Lichtenbaum \'etale motivic cohomology.

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arXiv2023

Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizio\l~ on log $K$-theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for $A_{\inf}$-cohomology.

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arXiv2023

Syntomic cohomology and real topological cyclic homology

We define the motivic filtrations on real topological Hochschild homology and its companions. In particular, we prove that real topological cyclic homology admits a natural complete filtration whose graded pieces are equivariant suspensions of syntomic cohomology. As an application, assuming a real refinement of the Dundas--Goodwillie--McCarthy theorem, we compute the the $RO(\mathbb{Z}/2)$-graded homotopy groups of $\mathrm{KR}(\mathbb{F}_p[x]/x^e;\mathbb{Z}_p)$, and we compute the equivariant slices of $\Sigma^2 \tau_{\geq 1}\mathrm{KR}(\mathbb{Z}/p^n;\mathbb{Z}_p)$.

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arXiv2023

Real topological Hochschild homology of schemes

We prove that real topological Hochschild homology THR for schemes with involution satisfies base change and descent for the Z/2-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher dimensional projective spaces.

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arXiv2023

Real Topological Hochschild Homology of Perfectoid Rings

We refine several results of Bhatt-Morrow-Scholze on THH to THR. In particular, we compute THR of perfectoid rings. This will be useful for establishing motivic filtrations on real topological Hochschild and cyclic homology of quasisyntomic rings. We also establish a real refinement of the Hochschild-Kostant-Rosenberg theorem.

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arXiv2023

A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

This paper incorporates the theory of Hochschild homology into our program on log motives. We discuss a geometric definition of logarithmic Hochschild homology of derived pre-log rings and construct an André-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem and that logarithmic Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes.

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