Definition 89.3.1. 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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