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The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X , the relative Spec C = Spec X R {\displaystyle C=\operatorname {Spec} _{X}R}
From the elementary properties of convex cones, C is the interior of its closure and is a proper cone. The elements in the closure of C are precisely the square of elements in E. C is self-dual. In fact the elements of the closure of C are just set of all squares x 2 in E, the dual cone is given by all a such that (a,x 2) > 0.
One can also define the dual notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a co-cone from F to N is a family of morphisms : for each object X of J, such that for every morphism f : X → Y in J the following diagram commutes: Part of a cone from F to N
An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [10] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.
The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration:
Consider, for example, the case where Y is the disk , and : = is the standard inclusion of the circle as the boundary of . Then the mapping cone C f {\displaystyle C_{f}} is homeomorphic to two disks joined on their boundary, which is topologically the sphere S 2 {\displaystyle S^{2}} .
This can be much easier: for example, if X is regularly embedded in Y then its normal cone is a vector bundle, so we are reduced to the problem of finding the intersection product of a subscheme C W V of a vector bundle C X Y with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to C W V.