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.

*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:**- Daniel Quillen, Higher algebraic K-theory: I, [ocr.djvu version], in book: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pages 85-147.
- Chapter 2 of course notes for an algebraic K-theory course, by Daniel R. Grayson, Spring, 2003, UIUC.
- Daniel G. Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. Amer. Math. Soc., 19, 1968, 1499-1500.

**Exercises:**- Show explicitly that a product of two simplices can be constructed combinatorially.
- 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.
- 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.)

*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 K_{0}, the K-theory gambit (K-理论的把戏), S-construction, K_{n}, some known K-groups (for low degree, for finite fields, Z, Q, number fields, alebraically closed fields) Q-construction, S^{-1}S-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:***Higher algebraic K-theory*, by Daniel Quillen, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, 171-176, Canad. Math. Congress, Montreal, Que., 1975.*The K-book: An introduction to algebraic K-theory*(a graduate textbook in progress), by Charles Weibel, http://math.rutgers.edu/~weibel/Kbook.html.*Higher algebraic K-theory II [after Daniel Quillen]*, [ocr.djvu], Algebraic K-theory, Evanston 1976, Springer, Berlin, Heidelberg, New York, Lecture Notes in Mathematics, volume 551, 1976, pages 217-240, reviewed in MR 58 #28137 and ZBL 362.18015.- by Henri Gillet
and Daniel R. Grayson,
*The loop space of the Q-construction*, Illinois Journal of Mathematics, 31 (1987) 574-597, reviewed in MR 89h:18012 and ZBL 628.55011. Also:*Erratum to "The loop space of the Q-construction"*, Illinois Journal of Mathematics, 47 (2003) 745-748, reviewed in MR2007234 (2004h:18012), dvi ps pdf, *Exact sequences in algebraic K-theory*, [ocr.djvu], Illinois Journal of Mathematics, volume 31, 1987, pages 598-617, reviewed in MR 89c:18011 and ZBL 629.18010.

**Exercises:**- 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. - Show that the Grothendieck group of an exact category
, as a set, is isomorphic to the quotient of the set of pairs**M***(M,N)*of objects ofby the equivalence relation generated by equating**M***(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.

*Weight filtrations***Topics:**- exterior and symmetric power operations on K-theory, Adams operations, their action on units, symbols, and modules defined by a regular sequence, weights, weight decomposition of K-groups rationally, Chern classes with values in the Chow ring, Chern character.
- filtrations give spectral sequences in homological algebra as well as in homotopy theory, trivial Postnikov filtration of a space, filtration of topological K-theory by skeleta, Atiyah-Hirzebruch spectral sequence, filtration by codimension of support, Gersten-Quillen complex, Gersten's conjecture (still open in general), Chow groups, , intersection of cycles, the moving lemma, linear equivalence, the Chow ring, the Grassmannian of lines in 3-space and its cellular structure and its Chow ring, rational equivalence, Bloch's formula and application to products and functoriality.
- filtration by commuting automorphsims: a combinatorial approximation to the loop space applied 1 or 2 times to the direct sum version of the S-construction, S-inverse-S-construction, over the standard simplicial affine polynomial ring exact sequences split up to homotopy and permutations are homotopic to their sign, K(X,Y) with Y = affine line or punctured affine line, the direct sum version of that, correspondences based on that, contractibility of the space of stable automorphisms via the switch swindle, identification of the obstruction that arises with motivic cohomology, Voevodsky cancellation, Suslin's result.

**References:***Weight filtrations via commuting automorphisms*, K-theory, volume 9, 1995, pages 139-172, reviewed in MR 96h:19001 and ZBL 826.19003. (See also*On the Grayson Spectral Sequence*, by Andrei Suslin, which does what I couldn't do to complete the story, at least over a field.)*Exterior power operations on algebraic K-theory*, pdf ps, by Daniel R. Grayson, K-theory, volume 3, 1989, pages 247-260, reviewed in MR 91h:19005 and ZBL 701.18007.*Adams operations on higher K-theory*, by Daniel R. Grayson, K-theory, volume 6, 1992, pages 97-111, reviewed in MR 94g:19005 and ZBL 776.19001, [pdf version].

**Open problems:**- Find a universal proof for the formula in K
_{0}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: &Lambda^{2}composed with Λ^{2}.

*Motivic cohomology***References:****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).*The Milnor Conjecture*, by Vladimir Voevodsky*A*, by Fabien Morel and Vladimir Voevodsky.^{1}-homotopy theory of schemes*Bloch-Kato conjecture and motivic cohomology with finite coefficients*, by Andrei Suslin and Vladimir Voevodsky.*Cycles, Transfers and Motivic Homology Theories*, by Vladimir Voevodsky, Eric. M. Friedlander, and Andrei Suslin, Annals of Math Studies 143 (2000), Princeton University Press.*On motivic cohomology with Z/l-coefficients*., by Vladimir Voevodsky.*Patching the Norm Residue Isomorphism Theorem*, by Charles A. Weibel*A Spectral Sequence for Motivic Cohomology*, by Spencer Bloch and Steve Lichtenbaum.*The homotopy coniveau filtration*, by Marc Levine.*Motivic Cohomology, K-theory and Topological Cyclic Homology*, by Thomas Geisser, in*Handbook of K-theory*, ISBN-10 3-540-23019-X, ISBN-13 978-3-540-23019-9, Springer, Berlin, Heidelberg, New York, 1163+xiv pages, 2005, in volume 1, pages 193-234.*The motivic spectral sequence*, [dvi ps pdf], in*Handbook of K-theory*, ISBN-10 3-540-23019-X, ISBN-13 978-3-540-23019-9, Springer, Berlin, Heidelberg, New York, 1163+xiv pages, 2005, in volume 1, pages 39-69.

*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:***Finite generation of the groups K*, [ocr.djvu], by Daniel Quillen, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 179-198. Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973._{i}of rings of algebraic integers*Finite Generation of higher K-groups of a curve over a finite field [after Daniel Quillen]*, [ocr.djvu], Algebraic K-theory, Oberwolfach 1980, Part I, Lecture Notes in Mathematics 966, Springer, Berlin, Heidelberg, New York, 1982, pages 69-90, reviewed in MR 84f:14018 and ZBL 502.14004.*Reduction theory using semistability*, [ocr.djvu], Commentarii Mathematicae Helvetici, volume 59, 1984, pages 600-634, reviewed in MR 86h:22018 and ZBL 564.20027. See also the exposition in Stability of lattices and the partition of arithmetic quotients, by Bill Casselman, Asian J. Math. 8 (2004), no. 4, 607-637.

**Open problems:**- Prove Bass' conjecture, that the K-groups of a regular finitely generated commutative ring are finitely generated. Do the same for the motivic cohomology.

*Additional references for further reading:***Topics:****References:**- K-theory Preprint Archives
*Volumes of symmetric spaces via lattice points*, by Henri Gillet and Daniel R. Grayson, preprint: February, 2004, math.NT/0402085; published paper: Documenta Mathematica 11 (2006) 425-447.- Daniel Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2), 96, 1972, 552-586.
*On the K-theory of fields*, by Daniel R. Grayson, Proceedings of a Conference on K-theory and Geometry, Contemporary Mathematics, volume 83, 1989, pages 31-55, reviewed in MR 90c:18010 and ZBL 705.19004.