Infinitesimal derived foliations

1. Problem Statement

The paper addresses the limitations of the existing theory of derived foliations (introduced in Toë20 and TV), particularly regarding their behavior in arbitrary characteristics (non-zero characteristic) and their formal integrability properties.

• The Gap: Classical derived foliations are related to de Rham cohomology and involve actions of the group stack $BKer$ (related to the circle group $S^1$ in the graded setting). However, these structures do not always satisfy strong formal integrability results outside of characteristic zero.
• The Goal: The authors aim to introduce a new notion, infinitesimal derived foliations, which are better suited for arbitrary characteristics. These objects are designed to relate to infinitesimal cohomology (rather than de Rham cohomology) and to satisfy formal integrability theorems analogous to those found in Ekedahl's work on restricted foliations in characteristic $p$.

2. Methodology and Framework

A. Derived Affine Stacks and Cosimplicial-Simplicial Models

The authors establish a robust algebraic foundation using cosimplicial-simplicial commutative rings ($csCR$).

• They define a model category structure on $csCR$ where weak equivalences are completed quasi-isomorphisms (morphisms inducing quasi-isomorphisms on the total complex $Tot_\pi$).
• They construct the $\infty$-category of derived affine stacks ($dAff$) as the opposite of this model category.
• They introduce graded derived affine stacks ($dChAff_k^{gr}$) using graded versions of these rings, allowing for actions of the multiplicative group $\mathbb{G}_m$.

B. The Group Schemes $G_0$ and $H_\pi$

A central technical innovation is the shift from the group $BKer$ (used in previous derived foliation theories) to a new group scheme:

• $G_0$: Defined as the derived loop scheme $\Omega_0 \mathbb{G}_a$ based at the origin. It is equivalent to $\text{Spec}(k[\epsilon^{-1}])$, where $\epsilon^{-1}$ is a generator in homotopical degree 1.
• $H_\pi$: The semi-direct product $\mathbb{G}_m \ltimes G_0$.
• Representation Theory: Representations of $H_\pi$ correspond to graded mixed complexes where the mixed structure has cohomological degree 1 (denoted $+_\epsilon\text{-dg}^{gr}_k$). This contrasts with the standard degree $-1$ mixed complexes used in previous works.

C. The Red-Shift Functor ($RS$)

The authors utilize a "red-shift" equivalence $RS: \epsilon\text{-dg}^{gr}_k \xrightarrow{\sim} +_\epsilon\text{-dg}^{gr}_k$.

• This functor shifts the cohomological degree of the mixed structure from $-1$ to $1$.
• Crucially, on the level of group objects, $RS$ maps $BKer$ to $G_0$.
• The authors construct a geometric realization of this shift on derived affine stacks, though they note it is not a perfect equivalence on the level of stacks due to subtleties with free objects and divided powers.

3. Key Definitions

Infinitesimal Derived Foliations (Definition 2.1):
Let $X$ be a derived affine scheme. An infinitesimal derived foliation on $X$ consists of:

1. A perfect complex $E$ (the shifted cotangent complex).
2. A $H_\pi$-equivariant derived stack $F$ over $X$.
3. The condition that $F$, as a $\mathbb{G}_m$-equivariant stack, is a linear derived affine stack of the form $V(E[-1])$.
4. The action of $G_0$ on $V(E[-1])$ must be compatible with the natural gradings.

Cotangent Complex:
For an infinitesimal derived foliation $F$, the cotangent complex $L_{F/k}$ is defined such that $F \simeq V(L_{F/k}[-1])$.

4. Key Results and Contributions

A. Relation to Infinitesimal Cohomology (Section 3)

The authors prove that the cohomology theory associated with infinitesimal derived foliations recovers the classical infinitesimal cohomology.

• Theorem 3.2: For a smooth affine scheme $X$, the infinitesimal cohomology of the "tautological" foliation $\ast_{X/k}$ (the final object in the category of foliations) is equivalent to the completed derived infinitesimal cohomology of $X$:
  $$\hat{C}^*_{inf}(\ast_{X/k}, \mathcal{O}_X) \simeq R\hat{C}^*(X_{inf}, \mathcal{O})$$
• This establishes that infinitesimal derived foliations are the correct derived geometric objects to study infinitesimal cohomology, just as derived foliations relate to de Rham cohomology.

B. Formal Integrability (Section 4)

This is the most significant technical result.

• Corollary 4.2: The category of smooth infinitesimal derived foliations on a smooth scheme $X$ is equivalent to the category of formally smooth formal groupoids over $X$.
• Mechanism: Any smooth infinitesimal derived foliation $F$ can be formally integrated into a formal groupoid $G_F$. The construction involves taking the loop space $\Omega_X F$ and realizing it as a groupoid in formal schemes.
• Contrast: This result does not hold for classical derived foliations. In classical theory, a Lie algebra defines a derived foliation, but it does not necessarily integrate to a formal groupoid (especially in characteristic $p$). The infinitesimal version forces this integrability.

C. Comparison with Classical Derived Foliations (Section 5)

The authors propose a relationship between the two notions via the Red-Shift construction:

• They suggest a "forgetful" functor from infinitesimal derived foliations to derived foliations.
• Hypothesis: An infinitesimal structure on a derived foliation $F$ (an action of $G_0$ on $V(L_F[-1])$) can be obtained by applying $RS$ to the action of $BKer$ on $V(L_F[1])$.
• Conjecture: Infinitesimal derived foliations may be equivalent to partition Lie algebroids (in the sense of BCN21 and Fu), which explains their strong integrability properties.

5. Significance and Implications

1. Characteristic Independence: The theory works uniformly in arbitrary characteristics, unlike the previous derived foliation theory which relied heavily on characteristic zero properties (e.g., the equivalence between de Rham and infinitesimal cohomology).
2. Formal Integrability: The paper resolves the issue of integrability for a specific class of derived geometric objects. By linking infinitesimal derived foliations to formal groupoids, it provides a derived-geometric generalization of Ekedahl's results on restricted foliations in characteristic $p$.
3. Connection to Partition Lie Algebras: The work bridges the gap between derived algebraic geometry and the theory of partition Lie algebras (via J. Fu's work), suggesting that infinitesimal derived foliations are the geometric avatars of partition Lie algebroids.
4. New Cohomological Tools: It provides a derived framework for computing infinitesimal cohomology using the language of stacks and equivariant derived geometry, offering a new perspective on crystalline and infinitesimal cohomology theories.

Conclusion

Toën and Vezzosi successfully define infinitesimal derived foliations as a refinement of derived foliations that incorporates the structure of the derived loop group $G_0$. This new framework restores formal integrability in all characteristics and establishes a direct link to infinitesimal cohomology. While the full comparison with classical derived foliations via the red-shift functor remains partially conjectural (specifically regarding the coherence of the functor on equivariant objects), the paper lays the groundwork for a unified theory of foliations in derived algebraic geometry.