The Stacks project

Lemma 89.29.2. With notation and assumptions as in Situation 89.29.1.

  1. We have $\overline{\mathcal{F}_{l/k}} = (\overline{\mathcal{F}})_{l/k}$.

  2. If $\mathcal{F}$ is a predeformation category, then $\mathcal{F}_{l/k}$ is a predeformation category.

  3. If $\mathcal{F}$ satisfies (S1), then $\mathcal{F}_{l/k}$ satisfies (S1).

  4. If $\mathcal{F}$ satisfies (S2), then $\mathcal{F}_{l/k}$ satisfies (S2).

  5. If $\mathcal{F}$ satisfies (RS), then $\mathcal{F}_{l/k}$ satisfies (RS).

Proof. Part (1) is immediate from the definitions.

Since $\mathcal{F}_{l/k}(l) = \mathcal{F}(k)$ part (2) follows from the definition, see Definition 89.6.2.

Part (3) follows as the functor ( commutes with fibre products and transforms surjective maps into surjective maps, see Definition 89.10.1.

Part (4). To see this consider a diagram

\[ \xymatrix{ & l[\epsilon ] \ar[d] \\ B \ar[r] & l } \]

in $\mathcal{C}_{\Lambda , l}$ as in Definition 89.10.1. Applying the functor ( we obtain

\[ \xymatrix{ & k[l\epsilon ] \ar[d] \\ B \times _ l k \ar[r] & k } \]

where $l\epsilon $ denotes the finite dimensional $k$-vector space $l\epsilon \subset l[\epsilon ]$. According to Lemma 89.10.4 the condition of (S2) for $\mathcal{F}$ also holds for this diagram. Hence (S2) holds for $\mathcal{F}_{l/k}$.

Part (5) follows from the characterization of (RS) in Lemma 89.16.4 part (2) and the fact that ( commutes with fibre products. $\square$

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