The Künneth Formula of Fundamental Group Schemes

1. Problem Statement

The paper addresses the validity of the Künneth formula for various fundamental group schemes associated with the product of two connected schemes $X$ and $Y$ proper over a field $k$.

In classical algebraic geometry, Grothendieck established the Künneth formula for the étale fundamental group over an algebraically closed field:
$$\pi_1^{\text{ét}}(X \times_k Y, (x, y)) \cong \pi_1^{\text{ét}}(X, x) \times_k \pi_1^{\text{ét}}(Y, y)$$
However, for other fundamental group schemes (such as Nori, S, F, EN, etc.), the Künneth formula does not always hold, or the conditions for its validity were previously restricted to specific fields (e.g., algebraically closed or characteristic 0) or specific geometric settings.

The authors aim to:

1. Establish necessary and sufficient conditions for the exactness of the homotopy sequence associated with a morphism $f: X \to S$ for general Tannakian categories.
2. Derive equivalent conditions under which the Künneth formula holds for the product $X \times_k Y$ for a broad class of fundamental group schemes over any field $k$, not just algebraically closed ones.

2. Methodology

The authors utilize the framework of Tannakian categories and Tannaka duality. The fundamental group scheme $\pi(\mathcal{C}, x)$ is defined as the Tannaka group scheme of a neutral Tannakian category $\mathcal{C}$ of vector bundles on a scheme $X$ with a fiber functor at a rational point $x$.

Key Technical Tools:

• Tannakian Categories: The paper studies categories like $\mathcal{C}_{\text{NF}}$ (numerically flat), $\mathcal{C}_{\text{N}}$ (essentially finite), $\mathcal{C}_{\text{EN}}$ (semi-essentially finite), $\mathcal{C}_{\text{F}}$ (Frobenius finite), etc.
• Observable Functors: A central concept is the observable functor. A tensor functor $\Phi: \mathcal{C} \to \mathcal{D}$ is observable if every rank 1 subobject in the image comes from a subobject in the source (up to tensor powers). This relates to the surjectivity of the induced homomorphism of group schemes.
• Base Change Properties: The authors rigorously analyze the "base change at $s$" property for vector bundles, ensuring that the formation of cohomology commutes with restriction to fibers.
• Principal Bundles and Colimits: They utilize the correspondence between Tannakian categories and principal bundles. They express the structure sheaf of the principal bundle as a filtered colimit of sub-bundles to analyze the behavior of pushforwards ($f_*$) and pullbacks ($f^*$).
• Homotopy Sequences: The core of the proof involves establishing the exactness of the sequence:
  $$\pi(\mathcal{C}_{X_s}, x) \to \pi(\mathcal{C}_X, x) \to \pi(\mathcal{C}_S, s) \to 1$$
  where $X_s$ is the fiber over $s \in S$.

3. Key Contributions

A. General Homotopy Sequence Theorem (Theorem 3.8)

The authors provide a complete characterization for the exactness of the homotopy sequence for a morphism $f: X \to S$ of connected schemes proper over $k$ with $f_* \mathcal{O}_X \cong \mathcal{O}_S$.
They prove that the sequence $\pi(\mathcal{C}_{X_s}, x) \to \pi(\mathcal{C}_X, x) \to \pi(\mathcal{C}_S, s) \to 1$ is exact if and only if:

1. The restriction functor $|_{X_s}: \mathcal{C}_X \to \mathcal{C}_{X_s}$ is observable.
2. For any object $E \in \mathcal{C}_X$, the pushforward $f_* E$ lies in $\mathcal{C}_S$, and $E$ satisfies base change at $s$.

They further show that if $S$ is integral and $f$ is flat, the exactness of the sequence implies these conditions.

B. The Künneth Formula Theorem (Theorem 4.2)

For the product $X \times_k Y$, the authors establish the equivalence of five conditions:

1. For any $E \in \mathcal{C}_{X \times Y}$, the projection $p_{X*} E$ lies in $\mathcal{C}_X$ and $E$ satisfies base change at $x_0$.
2. (Symmetric condition for $Y$).
3. The homotopy sequence $1 \to \pi(\mathcal{C}_Y, y_0) \to \pi(\mathcal{C}_{X \times Y}, (x_0, y_0)) \to \pi(\mathcal{C}_X, x_0) \to 1$ is exact.
4. (Symmetric exactness condition).
5. The natural homomorphism $\pi(\mathcal{C}_{X \times Y}, (x_0, y_0)) \to \pi(\mathcal{C}_X, x_0) \times_k \pi(\mathcal{C}_Y, y_0)$ is an isomorphism.

This theorem reduces the problem of proving the Künneth formula to verifying the stability of the category under pushforward and the base change property.

C. Extension to Subcategories (Proposition 4.3)

The authors prove that if the Künneth formula holds for a "larger" Tannakian category $\mathcal{C}'$, it automatically holds for any "smaller" Tannakian subcategory $\mathcal{C} \subseteq \mathcal{C}'$ (provided the pullback functors are compatible). This allows them to deduce results for specific fundamental groups from results for broader ones (like the S-fundamental group).

4. Main Results and Applications

The paper applies the general theorems to a wide range of fundamental group schemes over any field $k$ (removing the restriction to algebraically closed fields found in previous literature).

Proposition 5.8 states that for geometrically reduced connected schemes $X, Y$ proper over $k$, the Künneth formula holds for:

1. S-fundamental group scheme ($\pi_S$): Objects are numerically flat bundles.
2. Nori fundamental group scheme ($\pi_N$): Objects are essentially finite bundles.
3. EN-fundamental group scheme ($\pi_{EN}$): Objects are semi-essentially finite bundles.
4. F-fundamental group scheme ($\pi_F$): Objects are essentially Frobenius finite bundles (char $p > 0$).
5. Local fundamental group scheme ($\pi_{Loc}$): Objects are Frobenius trivial bundles.
6. Extended Local fundamental group scheme ($\pi_{ELoc}$).
7. Étale fundamental group scheme ($\pi_{\text{ét}}$).
8. Unipotent fundamental group scheme ($\pi_{\text{uni}}$).

Key Technical Step in Application:
To prove these, the authors first establish that for the S-fundamental group (numerically flat bundles), the pushforward of a numerically flat bundle on $X \times Y$ is numerically flat on $X$, and the base change property holds. They then use Proposition 4.3 to extend this result to all the other fundamental group schemes, as they are all subcategories of the S-fundamental group category (or related via Frobenius structures in characteristic $p$).

5. Significance

• Generalization of Field Conditions: Previous results for the Künneth formula often required $k$ to be algebraically closed (e.g., for Nori and S-fundamental groups) or of characteristic 0 (for EN). This paper unifies these results, proving the formula holds over any field.
• Unified Framework: Instead of treating each fundamental group scheme individually, the authors provide a unified Tannakian framework. By identifying the necessary and sufficient conditions (observable functors and base change), they create a "black box" method to verify the Künneth formula for any new fundamental group scheme defined by a Tannakian category.
• Resolution of Homotopy Sequences: The paper provides a rigorous characterization of when the homotopy sequence of a fibration is exact for general Tannakian categories, a result of independent interest in the study of fundamental group schemes.
• Technical Depth: The use of filtered colimits to analyze the structure of principal bundles and the careful handling of base change properties in the context of Tannakian categories represent a significant methodological advancement in the field.

In summary, this paper successfully generalizes the Künneth formula for fundamental group schemes to arbitrary fields by establishing precise Tannakian criteria for the exactness of homotopy sequences and the stability of vector bundle categories under pushforward.