Remark 97.21.8 (Canonical element). Assumptions and notation as in Lemma 97.21.2. Choose an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ such that $\mathop{\mathrm{Spec}}(A) \to S$ corresponds to a ring map $\Lambda \to A$. Consider the ring map

$A \longrightarrow A[\Omega _{A/\Lambda }], \quad a \longmapsto (a, \text{d}_{A/\Lambda }(a))$

Pulling back $x$ along the corresponding morphism $\mathop{\mathrm{Spec}}(A[\Omega _{A/\Lambda }]) \to \mathop{\mathrm{Spec}}(A)$ we obtain a deformation $x_{can}$ of $x$ over $A[\Omega _{A/\Lambda }]$. We call this the canonical element

$x_{can} \in T_ x(\Omega _{A/\Lambda }) = \text{Lift}(x, A[\Omega _{A/\Lambda }]).$

Next, assume that $\Lambda$ is Noetherian and $\Lambda \to A$ is of finite type. Let $k = \kappa (\mathfrak p)$ be a residue field at a finite type point $u_0$ of $U = \mathop{\mathrm{Spec}}(A)$. Let $x_0 = x|_{u_0}$. By (RS*) and the fact that $A[k] = A \times _ k k[k]$ the space $T_ x(k)$ is the tangent space to the deformation functor $\mathcal{F}_{\mathcal{X}, k, x_0}$. Via

$T\mathcal{F}_{U, k, u_0} = \text{Der}_\Lambda (A, k) = \mathop{\mathrm{Hom}}\nolimits _ A(\Omega _{A/\Lambda }, k)$

(see Formal Deformation Theory, Example 89.11.11) and functoriality of $T_ x$ the canonical element produces the map on tangent spaces induced by the object $x$ over $U$. Namely, $\theta \in T\mathcal{F}_{U, k, u_0}$ maps to $T_ x(\theta )(x_{can})$ in $T_ x(k) = T\mathcal{F}_{\mathcal{X}, k, x_0}$.

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