Triangulated categories of mixed motives
Monday, 27.10.08, 11:15-12:15, Raum 404, Eckerstr. 1
Lecture 1 will deal with the explicit construction\nof the categories DM(S) for a general scheme S\nand its basic properties.
Localization of mixed motives
Tuesday, 28.10.08, 11:15-12:15, Raum 403, Eckerstr. 1
Lecture 2 will be about localization in DM(S):\nif i : Z --> S is a closed immersion with complementary\nopen immersion j : U --> S, for any object M of DM(S),\nthere is a distinguished triangle of shape\nj!j^*(M) --> M --> ii^(M) --> j_!j^*(M)[1]\n(this holds with rational coefficients in general,\nand with integral coefficients if S and Z are smooth\nover a common base). Using Ayoub's thesis, localization\nallows to produce the six Grothendieck operations\non the categories DM(X) (with rational coefficients).\nThe aim of the talk is to give an idea of the proof\nof localization (which is quite tricky).\nEven if I suspect there won't be enough time to\nmention this during the talk, I mention here some other\nnice consequences: localization property also implies that\nrational motivic cohomology of regular schemes defined\nfrom DM coincide with motivic cohomology in the sense\nof Beilinson (using Adams operations on algebraic K-theory),\nby reducing to the case of a field, which is already known.\nThis and a version of Riemann-Roch in turns implies an absolute\npurity theorem in DM. These consequences have their\nrole to play in lecture 3.
l-adic completion of etale motives
Wednesday, 29.10.08, 14:15-15:15, Raum 403, Eckerstr. 1
Lectures 3 will be about the étale version DMet(X)\nof DM(X), which has nice properties (at least for schemes X\nwhich are of finite type over a an excellent regular scheme\nof Krull dimension less or equal to 1):\n- it coincides with DM(X) up to torsion\n- localization is true in DMet(X) (even with integral\ncoefficients)\n- the absolute purity theorem is true in DMet(X)\n- the torsion part of DMet(X) is essentially\nthe category of torsion l-adic sheaves\n- using the l-adic completion in DMet(X), one constructs\na nice l-adic realization from the rational version of DM(X)\ninto smooth Ql-sheaves which commutes with the\nGrothendieck six operations.