Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and Toen-Vaquie.

On differential graded categories

Projective modules and vector bundles The Grothendieck group $K_0$ $K_1$ and $K_2$ of a ring Definitions of higher $K$-theory The fundamental theorems of higher $K$-theory The higher $K$-theory of fields Nomenclature Bibliography Index

The K-Book: An Introduction to Algebraic K-theory

We provide a patch to complete the proof of the Voevodsky-Rost Theorem, that the norm residue map is an isomorphism. (This settles the motivic Bloch-Kato conjecture.)

The norm residue isomorphism theorem

Having determined E*(E) for E = MU, MSp, K and KO we turn now to a determination of E*(E)and E*(E) for E = H(ℤ2). Although historically H*(H(ℤ2); ℤ2) and H*(H(ℤ2); ℤ2) were known as much as fifteen years before KO*(KO), for example, we shall see that the calculation of these Hopf algebras is at least as difficult as the calculation we just did for KO. In theory we could proceed as follows: for each n we have a fibration H(ℤ2,n-1) → PH(ℤ2, n) → H(ℤ2, n), and the total space PH(ℤ2,n)is contractible (the path space PY is contractible for any Y). Thus in the Serre spectral sequence for this fibration E∞pq= 0 if (p,q) ≠ (0,0). With a little luck this should enable us to determine H*(H (ℤ2, n); ℤ2) from a knowledge of H*(H(ℤ2,n - 1); ℤ2). The process could begin at n=1 since H*(H(ℤ2,1); ℤ2) ≅ H*(RP∞; ℤ), which is a ℤ2-vector space with basis {xi:i⩾1}, xi∈ Hi(RP∞; ℤ2). Then we would have Hq(H(ℤ2); ℤ2)≅ dirlimn Hq+n(H(ℤ2,n); ℤ2). This is precisely how Cartan did determine this algebra (see [28]) using some heavy guns from homological algebra. We shall take a different approach, however; we shall construct some specific cohomology operations—the Steenrod squares Sqi—and show that they generate the algebra A* = H*(H(ℤ2); ℤ2). It will then not be too difficult to determine the dual A* = H*(H(ℤ2); ℤ2). We shall also indicate the results for H(ℤp),p an odd prime.

The Steenrod Algebra and its Dual

Notes on crystalline cohomology

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n2-p04.pdf

Cyclic homology, cdh-cohomology and negative K-theory

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic $K$-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

In this paper, we show that the widely held expectation that Weibel's homotopy K-theory satisfies cdh-descent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequences are derived. Finally, some evidence for a conjecture of Weibel concerning negative K-theory is given.

Descent Properties of Homotopy K-Theory

We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.

/pdf/toric-varieties-monoid-schemes-and-cdh-descent-2d2r3k6qqf.pdf

Christian Haesemeyer

Papers

Cyclic homology, cdh-cohomology and negative K-theory

Cyclic homology, cdh-cohomology and negative K-theory

Descent Properties of Homotopy K-Theory

Toric varieties, monoid schemes and cdh descent

The Norm Residue Theorem in Motivic Cohomology