$\mathbb{A}^1$-homotopy theory of log schemes

Technical Summary: A¹-Homotopy Theory of Log Schemes

Problem Statement
The paper addresses the construction of a stable motivic homotopy category for fine and saturated (fs) log schemes that satisfies the Grothendieck six-functor formalism. Previous work by Park, Ayoub, and others established a framework for log motives (denoted $\log DM(k)$ and $\log SH(S)$) where the interval $\square := (\mathbb{P}^1, \infty)$ is inverted instead of the affine line $\mathbb{A}^1$. While this non-$\mathbb{A}^1$-invariant setting successfully incorporates cohomology theories like Hodge cohomology and logarithmic topological Hochschild homology, it fails to satisfy the localization property. Specifically, for a strict closed immersion $i$ and its open complement $j$, the sequence $j_\sharp j^* \to \text{id} \to i_* i_*$ is not a cofiber sequence, and $i_*$ is not fully faithful. This failure prevents the derivation of the six-functor formalism, which is essential for constructing motivic nearby cycles functors and computing cohomology of boundaries.

Methodology
The author constructs a new stable motivic $\infty$-category, denoted $SH(X)$, for fs log schemes $X$ by inverting $\mathbb{A}^1$ while retaining specific log-geometric features. The construction proceeds as follows:

1. Restriction to Zariski Log Structures: The theory is developed for fs log schemes with Zariski log structures (where the monoid sheaf is a Zariski sheaf). The author notes that this is not a loss of generality due to Niziol's result, which guarantees that any fs log scheme admits a log blow-up with a Zariski log structure.
2. Vertical Boundaries: A central concept introduced is the "vertical boundary" $\partial_S X$ of a log smooth morphism $f: X \to S$. This is the set of points where the map of monoids $M_{S,f(x)} \to M_{X,x}$ is not vertical. The complement $X - \partial_S X$ represents the "vertical" part of the log scheme.
3. Localization Functors: The category $SH(X)$ is constructed by starting with the $\infty$-category of dividing Nisnevich sheaves on log smooth schemes over $X$ ($lSm/X$) and applying a double localization:
   • $\mathbb{A}^1$-localization: Inverting projections $X \times \mathbb{A}^1 \to X$.
   • Vertical Localization ($ver$): Inverting open immersions $j: U \to V$ in $lSm/X$ such that the induced map on the complements of vertical boundaries, $U - \partial_X U \to V - \partial_X V$, is an isomorphism.
4. Comparison with Classical Theory: For a scheme $S$ with the trivial log structure, the author proves that the new category $SH(S)$ is equivalent to the original Morel-Voevodsky construction. This is achieved by showing that in the presence of $\mathbb{A}^1$-invariance, inverting the class $ver$ is equivalent to inverting the class of open immersions $U - \partial U \to U$.

Key Contributions and Results

• Construction of $SH(X)$: The paper defines the stable motivic $\infty$-category $SH(X)$ and its abelian variants $DA(X, \Lambda)$ for fs log schemes. It establishes that for schemes with trivial log structures, this recovers the classical Morel-Voevodsky categories.
• The Localization Property: The primary technical result is the proof that the constructed categories satisfy the localization property. For a strict closed immersion $i: Z \to S$ with open complement $j: U \to S$, the sequence
  $$j_\sharp j^* F \xrightarrow{ad'} F \xrightarrow{ad} i_* i_* F$$
  is a cofiber sequence in $SH(S)$ (and related categories), and $i_*$ is fully faithful.
• Six-Functor Formalism: As a direct consequence of the localization property, the paper establishes the Grothendieck six-functor formalism ($f^*, f_*, f_!, f^!, \otimes, \underline{Hom}$) for strict morphisms of fs log schemes, relying on the foundational work of Ayoub and Cisinski-Déglise.
• Cohomology of Boundaries: The author derives a canonical long exact sequence for the cohomology of an fs log scheme $X$ log smooth over a base $B$:
  $$\cdots \to E^{p,q}(X) \to E^{p,q}(X - \partial X) \oplus E^{p,q}(\partial X) \to E^{p,q}(\partial X) \to E^{p+1,q}(X) \to \cdots$$
  This expresses the cohomology of the boundary $\partial X$ in terms of the cohomology of the underlying scheme $X - \partial X$.
• Extension of Cohomology Theories: The framework allows for the definition of $\mathbb{A}^1$-invariant cohomology theories for fs log schemes, including:
  • Motivic cohomology $H^p_M(X, \Lambda(q))$
  • Homotopy K-theory $KHP(X)$
  • Algebraic cobordism $MGL^{p,q}(X)$
• Explicit Computations: The paper provides explicit computations for the cohomology of the standard log point $pt_{\mathbb{N}, B}$ and relates the cohomology of $\partial X$ to the stable homotopy type at infinity of $X - \partial X$, connecting the theory to work by Dubouloz-Déglise-Østvær and Levine.

Significance
The paper claims to initiate the study of the Grothendieck six-functor formalism for fs log schemes within an $\mathbb{A}^1$-invariant setting. By inverting $\mathbb{A}^1$ and the specific class of morphisms $ver$, the author resolves the obstruction (the failure of the localization property) present in the non-$\mathbb{A}^1$-invariant log motivic theory. This enables the application of powerful motivic tools to log geometry, specifically facilitating the construction of motivic nearby cycles functors that behave well in positive and mixed characteristics. The work serves as a "reboot" of the author's thesis, refining the inversion strategy to ensure computability while retaining the necessary formalism. The results lay the groundwork for subsequent papers in the series that further develop the six-functor formalism and apply it to specific geometric problems.