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The intrinsic normal cone. (English) Zbl 0909.14006
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.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 18F99 Categories in geometry and topology
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