The Stacks project

89.2 Notation and Conventions

A ring is commutative with $1$. The maximal ideal of a local ring $A$ is denoted by $\mathfrak {m}_ A$. The set of positive integers is denoted by $\mathbf{N} = \{ 1, 2, 3, \ldots \} $. If $U$ is an object of a category $\mathcal{C}$, we denote by $\underline{U}$ the functor $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, -): \mathcal{C} \to \textit{Sets}$, see Remarks 89.5.2 (12). Warning: this may conflict with the notation in other chapters where we sometimes use $\underline{U}$ to denote $h_ U(-) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(-, U)$.

Throughout this chapter $\Lambda $ is a Noetherian ring and $\Lambda \to k$ is a finite ring map from $\Lambda $ to a field. The kernel of this map is denoted $\mathfrak m_\Lambda $ and the image $k' \subset k$. It turns out that $\mathfrak m_\Lambda $ is a maximal ideal, $k' = \Lambda /\mathfrak m_\Lambda $ is a field, and the extension $k/k'$ is finite. See discussion surrounding (89.3.3.1).