Daniel R. Grayson


This web page describes my six talks (in English) at the conference on Cohomology of groups and algebraic K-theory in Hangzhou, July 2-13, 2007. The old web site is at Cohomology of groups and algebraic K-theory.

  1. Constructing spaces combinatorially
    I intend to start gently. In the first talk I will carefully define simplicial sets and geometric realizations, and state a few preliminary theorems.
    Topics: category, small category, functor, natural transformation, solution sets of equations, imposing equations, limit, colimit, compactly generated Hausdorff space, spaces constructed combinatorially, simplices (as spaces), spheres, products, simplicial set, simplices (of a simplicial set), geometric realization, Yoneda's lemma, nerve of a category, filtering categories are contractible, homotopy groups, homotopies, homological algebra embeds into homotopy theory, producing homotopy equivalences and fibration sequences, bicomplexes row-wise, bisimplicial sets row-wise, Waldhausen's realization lemmas, Quillen's Theorems A and B.
    References: Exercises:
    1. Show explicitly that a product of two simplices can be constructed combinatorially.
    2. Show that each point of the geometric realization of a simplicial set X corresponds uniquely to an interior point attached to a nondegenerate simplex of X.
    3. Show that the opposite of the category of finite ordered nonempty sets is equivalent to the category whose objects are the finite ordered sets with distinct maximal and minimal elements, and whose arrows are the order-preserving maps that preserve minimality and maximality. (This may be useful for defining new simplicial sets.)
  2. 高阶代数K-理论的定义和定理。 Definitions and theorems of higher algebraic K-theory
    Topics: abelian category (阿贝尔范畴), exact category (正合范畴), examples, the Grothendieck group K0, the K-theory gambit (K-理论的把戏), S-construction, Kn, some known K-groups (for low degree, for finite fields, Z, Q, number fields, alebraically closed fields) Q-construction, S-1S-construction, +-construction, G-construction, equivalence of definitions, additivity theorem, localization theorem for abelian categories, localization theorem for projective modules, dévissage theorem, resolution theorem, fundamental theorem, cofinality theorem.
    References: Exercises:
    1. Given an exact category M, let N be the exact category obtained from M by discarding just the exact sequences that do not split. Show N is an exact category.
    2. Show that the Grothendieck group of an exact category M, as a set, is isomorphic to the quotient of the set of pairs (M,N) of objects of M by the equivalence relation generated by equating (M',N') to (M,N) whenever there is a pair of exact sequences 0 --> M' --> M --> X --> 0 and 0 --> N' --> N --> X --> 0, sharing the same third object.
  3. Weight filtrations
    References: Open problems:
    1. Find a universal proof for the formula in K0 that expresses the composite of two exterior power operations as a polynomial in the exterior power operations, and use that to deduce formulas for exterior power operations on higher K-groups of exact categories with functors on them that behave like exterior power functors. [See Akin, Buchsbaum, and Weyman for the only known case: &Lambda2 composed with Λ2.
  4. Motivic cohomology
    Topics: motivic cohomology should play the role in singular cohomology with integer coefficients in the world of algebra, Milnor K-theory, Bloch-Kato conjecture (proof in hand), Bloch's definition of higher Chow groups, commutativity of 1-dimensional Tor with supports, the definition of motivic cohomology, Beilinson-Soule vanishing conjecture, modified Lichtenbaum conjecture, Beilinson-Lichtenbaum conjecture (follows from Block-Kato by Suslin-Voevodsky).
  5. Finite generation of K-groups
    Topics: Bass' conjecture, nonsingular projective algebraic curves, plane curves, projection to the projective line, coherent sheaves, canonical line bundles, global sections, finiteness of cohomology, Euler characteristic, degree, bound for subsheaves, torsion sheaves, Harder-Narasmimhan canonical filtration, the Bruhat-Tits building at infinity, contractibility, cusps, contractibility, Tits building, Solomon-Tits theorem, group homology of the Steinberg module, finite generation of K-groups; number fields, algebraic integers, lattices, negative logarithm of the covolume.
    References: Open problems:
    1. Prove Bass' conjecture, that the K-groups of a regular finitely generated commutative ring are finitely generated. Do the same for the motivic cohomology.
  6. Additional references for further reading: