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 M(X) {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 Collections: Mathematics