Lemma 90.10.4. Let \mathcal{F} be a category cofibred in groupoids over \mathcal{C}_\Lambda . If \mathcal{F} satisfies (S2), then the condition of (S2) also holds when k[\epsilon ] is replaced by k[V] for any finite dimensional k-vector space V.
Proof. In the case that \mathcal{F} is cofibred in sets, i.e., corresponds to a functor F : \mathcal{C}_\Lambda \to \textit{Sets} this follows from the description of (S2) for F in Remark 90.10.3 and the fact that k[V] \cong k[\epsilon ] \times _ k \ldots \times _ k k[\epsilon ] with \dim _ k V factors. The case of functors is what we will use in the rest of this chapter.
We prove the general case by induction on \dim (V). If \dim (V) = 1, then k[V] \cong k[\epsilon ] and the result holds by assumption. If \dim (V) > 1 we write V = V' \oplus k\epsilon . Pick a diagram
\vcenter { \xymatrix{ & x_ V \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & k[V] \ar[d] \\ A \ar[r] & k } }
Choose a morphism x_ V \to x_{V'} lying over k[V] \to k[V'] and a morphism x_ V \to x_\epsilon lying over k[V] \to k[\epsilon ]. Note that the morphism x_ V \to x_0 factors as x_ V \to x_{V'} \to x_0 and as x_ V \to x_\epsilon \to x_0. By induction hypothesis we can find a diagram
\vcenter { \xymatrix{ y' \ar[d] \ar[r] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }
This gives us a commutative diagram
\vcenter { \xymatrix{ & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & k[\epsilon ] \ar[d] \\ A \times _ k k[V'] \ar[r] & k } }
Hence by (S2) we get a commutative diagram
\vcenter { \xymatrix{ y \ar[d] \ar[r] & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ (A \times _ k k[V']) \times _ k k[\epsilon ] \ar[d] \ar[r] & k[\epsilon ] \ar[d] \\ A \times _ k k[V'] \ar[r] & k } }
Note that (A \times _ k k[V']) \times _ k k[\epsilon ] = A \times _ k k[V' \oplus k\epsilon ] = A \times _ k k[V]. We claim that y fits into the correct commutative diagram. To see this we let y \to y_ V be a morphism lying over A \times _ k k[V] \to k[V]. We can factor the morphisms y \to y' \to x_{V'} and y \to x_\epsilon through the morphism y \to y_ V (by the axioms of categories cofibred in groupoids). Hence we see that both y_ V and x_ V fit into commutative diagrams
\vcenter { \xymatrix{ y_ V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ x_ V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } }
and hence by the second part of (S2) there exists an isomorphism y_ V \to x_ V compatible with y_ V \to x_{V'} and x_ V \to x_{V'} and in particular compatible with the maps to x_0. The composition y \to y_ V \to x_ V then fits into the required commutative diagram
\vcenter { \xymatrix{ y \ar[r] \ar[d] & x_ V \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }
In this way we see that the first part of (S2) holds with k[\epsilon ] replaced by k[V].
To prove the second part suppose given two commutative diagrams
\vcenter { \xymatrix{ y \ar[r] \ar[d] & x_ V \ar[d] \\ x \ar[r] & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r] \ar[d] & x_ V \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }
We will use the morphisms x_ V \to x_{V'} \to x_0 and x_ V \to x_\epsilon \to x_0 introduced in the first paragraph of the proof. Choose morphisms y \to y_{V'} and y' \to y'_{V'} lying over A \times _ k k[V] \to A \times _ k k[V']. The axioms of a cofibred category imply we can find commutative diagrams
\vcenter { \xymatrix{ y_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y'_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }
By induction hypothesis we obtain an isomorphism b : y_{V'} \to y'_{V'} compatible with the morphisms y_{V'} \to x and y'_{V'} \to x, in particular compatible with the morphisms to x_0. Then we have commutative diagrams
\vcenter { \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[\epsilon ] \ar[d] \ar[r] & k[\epsilon ] \ar[d] \\ A \ar[r] & k } }
where the morphism y \to y'_{V'} is the composition y \to y_{V'} \xrightarrow {b} y'_{V'} and where the morphisms y \to x_\epsilon and y' \to x_\epsilon are the compositions of the maps y \to x_ V and y' \to x_ V with the morphism x_ V \to x_\epsilon . Then the second part of (S2) guarantees the existence of an isomorphism y \to y' compatible with the maps to y'_{V'}, in particular compatible with the maps to x (because b was compatible with the maps to x). \square
Comments (0)