Properties of The Complex IC(X)
The complex ICp(X) has the following properties
- On the complement of some closed set of codimension 2, we have
- is 0 for i+m≠ 0, and for i=−m the groups form the constant local system C
- is 0 for i + m < 0
- If i > 0 then is zero except on a set of codimension at least a for the smallest a with p(a) ≥ m − i
- If i>0 then is zero except on a set of codimension at least a for the smallest a with q(a) ≥ (i)
As usual, q is the complementary perversity to p. Moreover the complex is uniquely characterized by these conditions, up to isomorphism in the derived category. The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either.
Verdier duality takes ICp to ICq shifted by n = dim(X) in the derived category.
Read more about this topic: Intersection Homology
Famous quotes containing the words properties and/or complex:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
—John Dewey (1859–1952)