On the group of extensions Ext1($\mathscr{G}$(λ0), $\mathscr{E}$(λ1, ..., λ n )) over a discrete valuation ring

Abstract

For given group schemes $\mathscr{G}$^{(λ _i )} (i = 1, 2, ...) deforming the additive group scheme G _a to the multiplicative group scheme G _m , T. Sekiguchi and N. Suwa constructed extensions: 0 → $\mathscr{G}$^(λ₂) → $\mathscr{E}$^{(λ₁, λ₂)} → $\mathscr{G}$^(λ₁) → 0, … 0 → $\mathscr{G}$^{(λ _n )} → $\mathscr{E}$^{(λ₁, ..., λ _n )} → $\mathscr{E}$^{(λ₁, ..., λ_n-1)} → 0, … inductively, by calculating the group of extensions Ext¹($\mathscr{E}$^{(λ₁, ..., λ_n-1)}, $\mathscr{G}$^{(λ _n )}). Here changing the group schemes, we treat the group Ext¹($\mathscr{G}$^(λ₀), $\mathscr{E}$^{(λ₁, ..., λ _n )}) of extensions for any positive integers n. The case of n = 2, 3 were studied by D. Horikawa and T. Kondo.