The Stacks project

Definition 90.3.1. Let $\Lambda $ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define $\mathcal{C}_\Lambda $ to be the category with

  1. objects are pairs $(A, \varphi )$ where $A$ is an Artinian local $\Lambda $-algebra and where $\varphi : A/\mathfrak m_ A \to k$ is a $\Lambda $-algebra isomorphism, and

  2. morphisms $f : (B, \psi ) \to (A, \varphi )$ are local $\Lambda $-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi $.

We say we are in the classical case if $\Lambda $ is a Noetherian complete local ring and $k$ is its residue field.

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View Definition 90.3.1 as pdf