The Stacks project

Example 92.7.1 (Graded algebras). Let $\mathcal{F}$ be the category defined as follows

  1. an object is a pair $(A, P)$ consisting of an object $A$ of $\mathcal{C}_\Lambda $ and a graded $A$-algebra $P$ such that $P_ d$ is a finite projective $A$-module for all $d \geq 0$, and

  2. a morphism $(f, g) : (B, Q) \to (A, P)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a map $g : Q \to P$ which is $f$-linear and induces an isomorpism $Q \otimes _{B, f} A \cong P$.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, P)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a graded $k$-algebra $P$ with $\dim _ k(P_ d) < \infty $ for all $d \geq 0$, let $x_0 = (k, P)$ be the corresponding object of $\mathcal{F}(k)$. We set

\[ \mathcal{D}\! \mathit{ef}_ P = \mathcal{F}_{x_0} \]

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