Extensions and biextensions of locally constant group schemes, tori and abelian schemes

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objets. In particular, we prove that if G_i (for i=1,2,3) is an extension of an abelian S-scheme A_i by an S-torus T_i, the category of biextensions of (G_1,G_2) by G_3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A_1,A_2) by the underlying S-torus T_3.

💡 Research Summary

The paper investigates the homomorphism and extension groups, together with their associated sheaves, for three fundamental classes of commutative group schemes over a base scheme S: locally constant S‑group schemes, S‑tori, and abelian S‑schemes. The authors begin by treating a locally constant group scheme L, which can be viewed as a constant finite group equipped with an action of the étale fundamental group π₁(S). By exploiting the flatness and commutativity of L, they identify the sheaf Hom_S(L, M) with the sheaf of π₁(S)‑equivariant homomorphisms between the associated representations, and they describe Ext¹_S(L, M) as the first cohomology group of a suitable complex of π₁(S)‑modules. This yields explicit formulas for both Hom and Ext sheaves when M is any of the three types of group schemes under consideration.

Next, the paper turns to extensions between tori and abelian schemes. A torus T over S is a group scheme locally isomorphic to a power of the multiplicative group Gₘ, while an abelian scheme A is a smooth proper group scheme with connected fibers. Using the dual abelian scheme  and the fact that tori are split by finite étale covers, the authors prove the fundamental isomorphism
 Ext¹_S(A, T) ≅ Hom_S(Â, T).
This identification is a manifestation of the classical biextension theory of Grothendieck and Mumford, but the authors extend it to the mixed setting where one of the factors may be a locally constant group scheme. They also compute the sheaves Ext¹_S(A, T) and Ext¹_S(L, T) explicitly, showing that they are representable by smooth group schemes when S is regular.

The core of the work concerns biextensions, i.e. objects that are simultaneously extensions in two variables. For three commutative group schemes G₁, G₂, G₃ over S, a biextension of (G₁, G₂) by G₃ is a torsor under G₃ over the product G₁ × G₂ equipped with compatible group laws in each variable. The authors focus on the situation where each G_i fits into an exact sequence
 0 → T_i → G_i → A_i → 0,
with T_i a torus and A_i an abelian scheme. They construct a functor from the category Biext(G₁, G₂; G₃) to Biext(A₁, A₂; T₃) by “pushing forward” the biextension along the projections G_i → A_i and “pulling back” along the inclusions T₃ → G₃. The main theorem states that this functor is an equivalence of categories. In other words, any biextension of the mixed extensions G₁ and G₂ by G₃ is uniquely determined by a biextension of the underlying abelian schemes A₁, A₂ by the torus T₃, and conversely every such biextension lifts uniquely to a biextension of the G_i.

The proof proceeds in several steps. First, the authors analyze the obstruction theory for lifting a biextension from (A₁, A₂) to (G₁, G₂). Using the exact sequences above, they identify the obstruction classes in Ext²_S(A₁ × A₂, T₃), which vanish because tori have cohomological dimension 1 in the fppf topology. Next, they show that any two lifts differ by an element of Hom_S(A₁ × A₂, T₃), which precisely corresponds to the group of automorphisms of a biextension in Biext(A₁, A₂; T₃). This establishes full faithfulness. Essential surjectivity follows from the explicit construction of a lift using the Baer sum of extensions and the universal property of the Poincaré biextension on A₁ × A₂.

Beyond the main equivalence, the paper provides several auxiliary results. It gives explicit descriptions of the sheaves Hom_S(L, M) and Ext¹_S(L, M) for all combinations of locally constant groups, tori, and abelian schemes, showing that they are representable by smooth commutative group schemes when the base is regular. It also proves that the biextension categories are symmetric monoidal, confirming the expected commutativity in both variables. Finally, concrete examples are worked out: when S = Spec k for an algebraically closed field k, L is a finite constant group, T is a split torus, and A is the Jacobian of a smooth projective curve, the authors recover the classical description of biextensions via the Poincaré line bundle and its associated Weil pairing.

In summary, the article delivers a comprehensive and explicit treatment of homomorphisms, extensions, and biextensions among locally constant group schemes, tori, and abelian schemes. By reducing the biextension problem for mixed extensions to the well‑understood case of abelian schemes and tori, it clarifies the structure of these higher‑order extensions and furnishes tools that are likely to be useful in the study of 1‑motives, mixed Shimura varieties, and other contexts where such mixed group schemes naturally appear.