The intrinsic normal cone (Q1359213)

A cone \(C \rightarrow X\) over a scheme \(X\) is defined as to have a section (vertex) \({\mathbf 0}:X \rightarrow C\) and an \({\mathbb A}^1\)-action (multiplicative contraction onto the vertex), that is a morphism \(\gamma : {\mathbb A}^1 \times C \rightarrow C\) such that \(\gamma \circ (1,{\text{id}}_C) = {\text{id}}_C\); \(\;\gamma \circ (0,{\text{id}}_C) = {\mathbf 0}\) ; and \(\gamma \circ ({\text{id}} \times \gamma) = \gamma \circ m\times {\text{id}}_C\) ; where \(m:{\mathbb A}^1 \times {\mathbb A}^1 \rightarrow {\mathbb A}^1\) is multiplication . If \(\mathcal F\) is a coherent \({\mathcal O}_X\)-module, then the associated cone \(C({\mathcal F}) = {\text{Spec\;Sym}}({\mathcal F})\) is called an abelian cone; and if \(X \rightarrow Y\) is an immersion with ideal sheaf \(\mathcal I\), then \(\bigoplus_{n\geq 0} {\mathcal I}^n/{\mathcal I}^{n+1}\) is a sheaf of \({\mathcal O}_X\)-algebras and \(C_{X/Y} = {\text{Spec}}\bigoplus_{n\geq 0} {\mathcal I}^n/{\mathcal I}^{n+1}\) is called the normal cone of \(X\) in \(Y\). The associated abelian cone \(N_{X/Y} = {\text{Spec\;Sym}}{\mathcal I}/{\mathcal I}^2\) is called the normal sheaf of \(X\) in \(Y\). The authors define a cone for a Deligne-Mumford stack \(X\). Also , the notion of cone stack is given as a generalization of cone. This satisfies similar commutation relations but the equalities above are changed by 2-isomorphisms. For a complex \(E^{\bullet}\) in the derived category \(D({\mathcal O}_{X_{et}})\) satisfying the assumptions: \[ 1.\quad h^i(E^{\bullet})=0,\;i>0;\qquad 2.\quad h^i(E^{\bullet}) \text{ is coherent for }i=0,\;i=-1,\tag \(*\) \] the authors associate the cone stack \(h^1/h^0((E^\bullet)^\vee)\) , where \({E^\bullet}^\vee =R{\mathcal Hom}(E^\bullet, {\mathcal O}_{X_{fl}})\) and \(X_{fl}\) the big fppf-site. (Here \(h^1/h^0(E^\bullet)\) is the stack-theoretic quotient of the action of \(E^0\) on \(E^1\) via \(d:E^0 \rightarrow E^1\), which is a Picard stack.) In particular the cotangent complex \(L^\bullet_X\) satisfies the conditions \((*)\) and therefore defines the abelian cone stack \({\mathfrak N}_X := h^1/h^0((L_X^\bullet)^\vee)\) called the intrinsic normal sheaf. The intrinsic normal cone \({\mathfrak C}_X\) is obtained as follows: étale locally on \(X\) embed an open set \(U\) of \(X\) in a smooth scheme \(W\), take the stack quotient of the normal cone \(C_{U/W}\) by the natural action of \(T_W| _U\), and glue these stacks together. The notion of (perfect) obstruction theory for \(X\) is introduced: This is an object \(E^\bullet\) in the derived category satisfying condition \((*)\) together with a morphism \(E^\bullet \rightarrow L_X^\bullet\) and such that the induced map \({\mathfrak N}_X \rightarrow h^1/h^0((E^\bullet)^\vee)\) is a closed immersion. The obstruction theory \(E^\bullet\) is perfect if \(h^1/h^0((E^\bullet)^\vee)\) is smooth over \(X\). It is shown how to construct, given a perfect obstruction theory for \(X\), a pure dimensional virtual fundamental class in the Chow group of \(X\). Some properties of such classes are proven, both in the absolute and relative context. Via a deformation theory interpretation of obstruction theories the authors prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.