Is there a 'classical' definition for the support of a perverse sheaves. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:17:11Z http://mathoverflow.net/feeds/question/54231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54231/is-there-a-classical-definition-for-the-support-of-a-perverse-sheaves Is there a 'classical' definition for the support of a perverse sheaves. Mikhail Bondarko 2011-02-03T19:49:47Z 2011-02-04T00:13:16Z <p>I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter? </p> <p>I suspect that the definition should be something like: </p> <p>a perverse sheaf $P$ on a space (scheme) $S$ is supported on (or is it better to say 'at'?) a subspace $U$ if for any closed immersion $i:Z\to S$ such that $Z$ is disjoint from $U$ the zeroth (perverse) cohomology of both $Ri^*P$ and $Ri^!P$ is $0$. Does this make sense? </p> <p>Upd. Possibly the word 'support' is not quite appropriate for me (could you suggest something better?:)), whereas the notion of support should be defined as YBL does below. For the intermediate extension $j_{!*}$ of a perverse sheaf $P$ from an (open) subscheme $U$ of $S$ to $S$ I would like to say that $j_{!*}P$ is supported on $U$. So, I want to say that a constant sheaf on a (connected) smooth variety is supported on any its dense subvariety (or at its generic point).</p> http://mathoverflow.net/questions/54231/is-there-a-classical-definition-for-the-support-of-a-perverse-sheaves/54246#54246 Answer by YBL for Is there a 'classical' definition for the support of a perverse sheaves. YBL 2011-02-03T22:04:28Z 2011-02-03T22:17:06Z <p>Let $j_i : U_i \hookrightarrow X$ open immersions and $j: U_1 \cup U_2 \hookrightarrow X$. If $j_1^*F = j_2^*F = 0$ then $j^*F = 0$ by considering a Mayer-Vietoris triangle. So there is a largest open set $U\subset X$ such that $j^*F = 0$. Define $supp(F)$ as the complementary closed subset. </p> <p>The definition is autodual $supp(F) = supp(DF)$ since $j^*F = 0 \Leftrightarrow D(j^*F) = 0 \Leftrightarrow j^!DF = j^*DF = 0$ for an open immersion $j$. </p> <p>Also $supp(F) \subset supp(F') \cup supp(F'')$ for a triangle $F'\to F\to F'' \to +1$. So if $t$ is a t-structure on $D(X)$ and $F\in D(X)$ is bounded for $t$ then $supp(F) \subset\bigcup_i supp({}^tH^i(F))$. </p>