Proof.
Via the identifications \mathop{N\! L}\nolimits _ f = \tau _{\geq -1}L_ f (Lemma 92.22.4) and H^0(L_ f) = \Omega _ f (Lemma 92.22.2) we have seen parts (2) and (3) in Deformation Theory, Lemmas 91.13.1 and 91.13.3.
Proof of (1). To match notation with Deformation Theory, Section 91.13 we will write \mathop{N\! L}\nolimits _ f = \mathop{N\! L}\nolimits _{\mathcal{O}/\mathcal{O}_\mathcal {B}} and L_ f = L_{\mathcal{O}/\mathcal{O}_\mathcal {B}} and similarly for the morphisms t and t \circ f. By Deformation Theory, Lemma 91.13.8 there exists an element
\xi ' \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*L_{\mathcal{O}_\mathcal {B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G})
such that a solution exists if and only if this element is in the image of the map
\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( \mathop{N\! L}\nolimits _{\mathcal{O}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( L_{\mathcal{O}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*L_{\mathcal{O}_\mathcal {B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G})
The distinguished triangle of Lemma 92.22.3 for f and t gives rise to a long exact sequence
\ldots \to \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( L_{\mathcal{O}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*L_{\mathcal{O}_\mathcal {B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( L_{\mathcal{O}/\mathcal{O}_\mathcal {B}}, \mathcal{G})
Hence taking \xi the image of \xi ' works.
\square
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