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

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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 Gᵢ (for i=1,2,3) is an extension of an abelian S-scheme Aᵢ by an S-torus Tᵢ, the category of biextensions of (G₁,G₂) by G₃ is equivalent to the category of biextensions of the underlying abelian S-schemes (A₁,A₂) by the underlying S-torus T₃.