Definition 90.12.1. Let \mathcal{F} be a predeformation category. The tangent space T \mathcal{F} of \mathcal{F} is the set \overline{\mathcal{F}}(k[\epsilon ]) of isomorphism classes of objects in the fiber category \mathcal F(k[\epsilon ]).
90.12 Tangent spaces of predeformation categories
We will define tangent spaces of predeformation functors using the general Definition 90.11.9. We have spelled this out in Example 90.11.10. It applies to predeformation categories by looking at the associated functor of isomorphism classes.
Thus T \mathcal{F} is nothing but the tangent space of the associated functor \overline{\mathcal{F}}: \mathcal{C}_\Lambda \to \textit{Sets}. It has a natural vector space structure when \mathcal{F} satisfies (S2), or, in fact, as long as \overline{\mathcal{F}} does.
Lemma 90.12.2. Let \mathcal{F} be a predeformation category such that \overline{\mathcal{F}} satisfies (S2)1. Then T \mathcal{F} has a natural k-vector space structure. For any finite dimensional vector space V we have \overline{\mathcal{F}}(k[V]) = T\mathcal{F} \otimes _ k V functorially in V.
Proof. Let us write F = \overline{\mathcal{F}} : \mathcal{C}_\Lambda \to \textit{Sets}. This is a predeformation functor and F satisfies (S2). By Lemma 90.10.4 (and the translation of Remark 90.10.3) we see that
F(A \times _ k k[V]) \longrightarrow F(A) \times F(k[V])
is a bijection for every finite dimensional vector space V and every A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda ). In particular, if A = k[W] then we see that F(k[W] \times _ k k[V]) = F(k[W]) \times F(k[V]). In other words, the hypotheses of Lemma 90.11.8 hold and we see that TF = T \mathcal{F} has a natural k-vector space structure. The final assertion follows from Lemma 90.11.15. \square
A morphism of predeformation categories induces a map on tangent spaces.
Definition 90.12.3. Let \varphi : \mathcal{F} \to \mathcal{G} be a morphism predeformation categories. The differential d \varphi : T \mathcal{F} \to T \mathcal{G} of \varphi is the map obtained by evaluating the morphism of functors \overline{\varphi }: \overline{\mathcal{F}} \to \overline{\mathcal{G}} at A = k[\epsilon ].
Lemma 90.12.4. Let \varphi : \mathcal{F} \to \mathcal{G} be a morphism of predeformation categories. Assume \overline{\mathcal{F}} and \overline{\mathcal{G}} both satisfy (S2). Then d \varphi : T \mathcal{F} \to T \mathcal{G} is k-linear.
Proof. In the proof of Lemma 90.12.2 we have seen that \overline{\mathcal{F}} and \overline{\mathcal{G}} satisfy the hypotheses of Lemma 90.11.8. Hence the lemma follows from Lemma 90.11.13. \square
Remark 90.12.5. We can globalize the notions of tangent space and differential to arbitrary categories cofibered in groupoids as follows. Let \mathcal{F} be a category cofibered in groupoids over \mathcal{C}_\Lambda , and let x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k)). As in Remark 90.6.4, we get a predeformation category \mathcal{F}_ x. We define
T_ x\mathcal{F} = T\mathcal{F}_ x
to be the tangent space of \mathcal{F} at x. If \varphi : \mathcal{F} \to \mathcal{G} is a morphism of categories cofibered in groupoids over \mathcal{C}_\Lambda and x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k)), then there is an induced morphism \varphi _ x: \mathcal{F}_ x \to \mathcal{G}_{\varphi (x)}. We define the differential d_ x \varphi : T_ x \mathcal{F} \to T_{\varphi (x)} \mathcal{G} of \varphi at x to be the map d \varphi _ x: T \mathcal{F}_ x \to T \mathcal{G}_{\varphi (x)}. If both \mathcal{F} and \mathcal{G} satisfy (S2) then all of these tangent spaces have a natural k-vector space structure and all the differentials d_ x \varphi : T_ x \mathcal{F} \to T_{\varphi (x)} \mathcal{G} are k-linear (use Lemmas 90.10.6 and 90.12.4).
The following observations are uninteresting in the classical case or when k/k' is a separable field extension, because then \text{Der}_\Lambda (k, k) and \text{Der}_\Lambda (k, V) are zero. There is a canonical identification
\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(k, k[\epsilon ]) = \text{Der}_\Lambda (k, k).
Namely, for D \in \text{Der}_\Lambda (k, k) let f_ D : k \to k[\epsilon ] be the map a \mapsto a + D(a)\epsilon . More generally, given a finite dimensional vector space V over k we have
\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(k, k[V]) = \text{Der}_\Lambda (k, V)
and we will use the same notation f_ D for the map associated to the derivation D. We also have
\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(k[W], k[V]) = \mathop{\mathrm{Hom}}\nolimits _ k(V, W) \oplus \text{Der}_\Lambda (k, V)
where (\varphi , D) corresponds to the map f_{\varphi , D} : a + w \mapsto a + \varphi (w) + D(a). We will sometimes write f_{1, D} : a + v \to a + v + D(a) for the automorphism of k[V] determined by the derivation D : k \to V. Note that f_{1, D} \circ f_{1, D'} = f_{1, D + D'}.
Let \mathcal{F} be a predeformation category over \mathcal{C}_\Lambda . Let x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k)). By the above there is a canonical map
\gamma _ V : \text{Der}_\Lambda (k, V) \longrightarrow \overline{\mathcal{F}}(k[V])
defined by D \mapsto f_{D, *}(x_0). Moreover, there is an action
a_ V : \text{Der}_\Lambda (k, V) \times \overline{\mathcal{F}}(k[V]) \longrightarrow \overline{\mathcal{F}}(k[V])
defined by (D, x) \mapsto f_{1, D, *}(x). These two maps are compatible, i.e., f_{1, D, *}f_{D', *}x_0 = f_{D + D', *}x_0 as follows from a computation of the compositions of these maps. Note that the maps \gamma _ V and a_ V are independent of the choice of x_0 as there is a unique x_0 up to isomorphism.
Lemma 90.12.6. Let \mathcal{F} be a predeformation category over \mathcal{C}_\Lambda . If \overline{\mathcal{F}} has (S2) then the maps \gamma _ V are k-linear and we have a_ V(D, x) = x + \gamma _ V(D).
Proof. In the proof of Lemma 90.12.2 we have seen that the functor V \mapsto \overline{\mathcal{F}}(k[V]) transforms 0 to a singleton and products to products. The same is true of the functor V \mapsto \text{Der}_\Lambda (k, V). Hence \gamma _ V is linear by Lemma 90.11.5. Let D : k \to V be a \Lambda -derivation. Set D_1 : k \to V^{\oplus 2} equal to a \mapsto (D(a), 0). Then
\xymatrix{ k[V \times V] \ar[r]_{+} \ar[d]^{f_{1, D_1}} & k[V] \ar[d]^{f_{1, D}} \\ k[V \times V] \ar[r]^{+} & k[V] }
commutes. Unwinding the definitions and using that \overline{F}(V \times V) = \overline{F}(V) \times \overline{F}(V) this means that a_ D(x_1) + x_2 = a_ D(x_1 + x_2) for all x_1, x_2 \in \overline{F}(V). Thus it suffices to show that a_ V(D, 0) = 0 + \gamma _ V(D) where 0 \in \overline{F}(V) is the zero vector. By definition this is the element f_{0, *}(x_0). Since f_ D = f_{1, D} \circ f_0 the desired result follows. \square
A special case of the constructions above are the map
90.12.6.1
\begin{equation} \label{formal-defos-equation-map} \gamma : \text{Der}_\Lambda (k, k) \longrightarrow T\mathcal{F} \end{equation}
and the action
90.12.6.2
\begin{equation} \label{formal-defos-equation-action} a : \text{Der}_\Lambda (k, k) \times T\mathcal{F} \longrightarrow T\mathcal{F} \end{equation}
defined for any predeformation category \mathcal{F}. Note that if \varphi : \mathcal{F} \to \mathcal{G} is a morphism of predeformation categories, then we get commutative diagrams
\vcenter { \xymatrix{ \text{Der}_\Lambda (k, k) \ar[r]_-\gamma \ar[rd]_\gamma & T\mathcal{F} \ar[d]_{d\varphi } \\ & T\mathcal{G} } } \quad \text{and}\quad \vcenter { \xymatrix{ \text{Der}_\Lambda (k, k) \times T\mathcal{F} \ar[r]_-a \ar[d]_{1 \times d\varphi } & T\mathcal{F} \ar[d]^{d\varphi } \\ \text{Der}_\Lambda (k, k) \times T\mathcal{G} \ar[r]^-a & T\mathcal{G} } }
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