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Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar

Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 12
April 26, 2007
In all of the following problems, X is a stratified space with stratification S, and X has a unique open
stratum U. All perverse sheaves are with respect to the middle perversity.
1. Let f : Y X be a semismall resolution. Show that RfC is a perverse sheaf.
2. Let f : Y X be a small resolution. Show that RfC IC(X, C).
3. Suppose X has exactly two strata, U and Z (so of course U is open and Z is closed). Suppose
p(U) > p(Z) (in particular, this is not a Goresky­MacPherson perversity). Show that all perverse
sheaves are of the form j!E[p(U)]iF[p(Z)], where E and F are local systems on U and Z, respectively,
and j : U X and i : Z X are the inclusion maps. Thus, there is an equivalence of categories

{representations of 1(U) × 1(Z)}.
Thus, perverse sheaves with respect to a non-Goresky­MacPherson perversity are not that interesting--
they do not encode topological information about the singularities of X.
4. Let F be a perverse sheaf, and let S be a stratum that is open in the support of F. Show that
(F|S)[-p(S)] is a local system (in particular, you must show that it is an ordinary sheaf, not a
complex of sheaves). Next, let E be that local sytem. Show that IC( ŻS, E) occurs as a quotient in the


Source: Achar, Pramod - Department of Mathematics, Louisiana State University


Collections: Mathematics