Root pairing

Definition

Let $R$ be a commutative ring, and let $M$ and $N$ be modules over $R$. Let $\langle \cdot, \cdot \rangle \colon M \times N \to R$ be a perfect pairing, that is, an $R$-bilinear map with two properties. The map $x \mapsto \langle x, \cdot \rangle$ is an isomorphism from $M$ onto $\Hom_R(N, R)$, and $y \mapsto \langle \cdot, y \rangle$ is an isomorphism from $N$ onto $\Hom_R(M, R)$. Let $I$ be a set. A root pairing indexed by $I$ is a pair of injective maps $i \mapsto \alpha_i$ from $I$ to $M$ and $i \mapsto \alpha_i^\vee$ from $I$ to $N$ such that:

1. $\langle \alpha_i, \alpha_i^\vee \rangle = 2$ for each $i \in I$;
2. for all $i, j \in I$ there is $k \in I$ with $s_i(\alpha_j) = \alpha_k$ and $s_i^\vee(\alpha_j^\vee) = \alpha_k^\vee$.

Here $s_i(x) = x - \langle x, \alpha_i^\vee \rangle \alpha_i$ for $x \in M$ and $s_i^\vee(y) = y - \langle \alpha_i, y \rangle \alpha_i^\vee$ for $y \in N$. The $\alpha_i$ are the roots and the $\alpha_i^\vee$ are the coroots.