seen from explicit local computation. Q.E.D. Now, the last item in the previous paragraph implies that the image of in M does not intersect . Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone. Intrinsic normal cone Intrinsic normal bundle Let be a Deligne–Mumford stack locally of finite type over a field . If denotes the cotangent complex of X relative to , then the intrinsic normal bundle to is the quotient stack which is the stack of fppf -torsors on . A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack . Properties of intrinsic normal bundle More concretely, suppose there is an étale morphism from an affine finite-type -scheme together with a locally closed immersion into a smooth affine finite-type -scheme . Then one can show meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence to be exact on the right hand side. Moreover, for special cases discussed below, we are now considering the quotient as a continuation of the previous sequence as a triangle in some triangulated category. This is because the local stack quotient can be interpreted as in certain cases. Normal cone The intrinsic normal cone to , denoted as , is then defined by replacing the normal bundle with the normal cone ; i.e., Example: One has that is a local complete intersection if and only if . In particular, if is smooth, then is the classifying stack of the tangent bundle , which is a commutative group scheme over . More generally, let is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then is characterised as the closed substack such that, for any étale map for which factors through some smooth map (e.g., ), the pullback is: See also Abelian cone Segre class Residual intersection Virtual fundamental class Notes References External linkes Fibers of the normal cone Algebraic geometry Intersection theory