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The Stacks project

Definition 98.23.5. Let S be a locally Noetherian base. Let \mathcal{X} be a category fibred in groupoids over (\mathit{Sch}/S)_{fppf}. Assume that \mathcal{X} satisfies (RS*). A naive obstruction theory is given by the following data

  1. for every S-algebra A such that \mathop{\mathrm{Spec}}(A) \to S maps into an affine open \mathop{\mathrm{Spec}}(\Lambda ) \subset S and every object x of \mathcal{X} over \mathop{\mathrm{Spec}}(A) we are given an object E_ x \in D^-(A) and a map \xi _ x : E \to \mathop{N\! L}\nolimits _{A/\Lambda },

  2. given (x, A) as in (1) there are transformations of functors

    \text{Inf}_ x( - ) \to \mathop{\mathrm{Ext}}\nolimits ^{-1}_ A(E_ x, -) \quad \text{and}\quad T_ x(-) \to \mathop{\mathrm{Ext}}\nolimits ^0_ A(E_ x, -)
  3. for (x, A) as in (1) and a ring map A \to B setting y = x|_{\mathop{\mathrm{Spec}}(B)} there is a functoriality map E_ x \to E_ y in D(A).

These data are subject to the following conditions

  1. in the situation of (3) the diagram

    \xymatrix{ E_ y \ar[r]_{\xi _ y} & \mathop{N\! L}\nolimits _{B/\Lambda } \\ E_ x \ar[u] \ar[r]^{\xi _ x} & \mathop{N\! L}\nolimits _{A/\Lambda } \ar[u] }

    is commutative in D(A),

  2. given (x, A) as in (1) and A \to B \to C setting y = x|_{\mathop{\mathrm{Spec}}(B)} and z = x|_{\mathop{\mathrm{Spec}}(C)} the composition of the functoriality maps E_ x \to E_ y and E_ y \to E_ z is the functoriality map E_ x \to E_ z,

  3. the maps of (2) are isomorphisms compatible with the functoriality maps and the maps of Remark 98.21.3,

  4. the composition E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega _{A/\Lambda } corresponds to the canonical element of T_ x(\Omega _{A/\Lambda }) = \mathop{\mathrm{Ext}}\nolimits ^0(E_ x, \Omega _{A/\Lambda }), see Remark 98.21.8,

  5. given a deformation situation (x, A' \to A) with I = \mathop{\mathrm{Ker}}(A' \to A) the composition E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'} is zero in

    \mathop{\mathrm{Hom}}\nolimits _ A(E_ x, \mathop{N\! L}\nolimits _{A/\Lambda }) = \mathop{\mathrm{Ext}}\nolimits ^0_ A(E_ x, \mathop{N\! L}\nolimits _{A/A'}) = \mathop{\mathrm{Ext}}\nolimits ^1_ A(E_ x, I)

    if and only if x lifts to A'.


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