Resumen de Dimension injective des produits croisés

Marie-Paule Malliavin

• Let $k$ be a field, $R$ a commutative noetherian Gorenstein $k$-algebra of finite Krull dimension, $\mathfrak{G}$ a finite dimensional Lie algebra over $k$ which operates on $R$ by $k$-derivations and $\sigma$ a 2-cocycle of $\mathfrak{G}$ with coefficients in $R$; we will denote by $R\ast_\sigma\mathfrak{G}$ the crossed product of $R$ by mean of $\sigma$; then we will provethe inequalities: $$\mbox{injdim }R\leq \mbox{injdim } R\ast_\sigma\mathfrak{G}\leq \mbox{injdim }R + \mbox{dim } \mathfrak{G}.$$ We will give also a formula relating injdim $R\ast_\sigma\mathfrak{G}$ with the Ore localisations of this algebra with respect to the prime ideals of $R$.