Thomas Geisser

Bi-relative algebraic K-theory and topological cyclic homology

It is well-known that algebraic K-theory preserves products of rings. However, in general, algebraic K-theory does not preserve fiber products of rings, and one defines bi-relative algebraic K-theory so as to measure the deviation. It was proved recently by Cortinas [6] that, rationally, bi-relative algebraic K-theory and bi-relative negative cyclic homology agree. In this paper, we show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application, we show that for a possibly singular curve over a perfect field k of positive characteristic p, the cyclotomic trace map induces an isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic homology groups in non-negative degrees. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology. Let F be a functor that to a unital associative ring A associates a symmetric spectrum F (A). If I ⊂ A is a two-sided ideal, the relative term F (A, I) is defined to be the homotopy fiber of the map F (A) → F (A/I) induced from the canonical projection. If further f : A → B is a ring homomorphism such that f : I → f(I) is an isomorphism onto a two-sided ideal of B, there is an induced map F (A, I)→ F (B, f(I)), and the bi-relative term F (A,B, I) is defined to be the homotopy fiber of this map. Then there is a distinguished triangle in the stable homotopy category

On the

We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic (0,p), the p-adic etale K-theory and p-adic topological cyclic homology agree.

K

The purpose of this article is to give a survey of some of the applications of de Jong’s theorem on alterations [dJ]. Most of the applications fall into one of the following two categories:

-theory and topological cyclic homology of smooth schemes over a discrete valuation ring

We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is isomorphic to the sheaf of p-adic vanishing cycles. We use this together with results of the second author and Madsen to evaluate the

Applications of de Jong’s Theorem on Alterations

K

The de Rham-Witt complex and -adic vanishing cycles

-theory with finite coefficients of the quotient field K of the henselian local ring of X at a generic point of Y. The result affirms the Lichtenbaum-Quillen conjecture for the field K.

Arithmetic homology and an integral version of Kato's conjecture

Summary We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.

Arithmetic cohomology over finite fields and special values of

We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))

\zeta

for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaums Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tates conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.

-functions

In this paper we combine ideas of Soulé [23] and Deninger [5, 6] to prove a p-adic analogue of Beilinson’s conjectures for motives associated to Hecke characters of imaginary quadratic fields. Let E be an elliptic curve defined over an imaginary quadratic field K with complex multiplication by the ring of integers of K. In [23], Soulé proved the following theorem: Let p be a prime which splits in K and l ≥ 0 such that p − 1 divides neither l, l+1 nor l+2. Then there exists a Zp-submodule Vl ⊆ K2l+2(E,Zp) and a regulator map rl : K2l+2(E,Zp) → Zp, such that the index of rl(Vl) in Zp equals nl, where nl is the p-adic valuation of the value at s = −l of a p-adic L-series analog to L(E, s). On the other hand let φ be a Hecke character of an imaginary quadratic field K of positive weight w. Then Deninger constructed a motive M in MQ(K), the category of Chow motives overK with coefficients in Q, such that the L-series of M coincides with the L-series of φ. The motive M arises naturally as a factor of the Grothendieck restriction RF/K(h1(E)) for a CM-elliptic curve E of Shimura type over a finite extension F ofK. Then he proved parts of the Beilinson conjectures for M , i.e. he related the leading coefficient of L(M,−l) to a map from H A (M,Q(l+ w + 1)) = K2l+w+1(M) (l+w+1) Q to Deligne cohomology. Here we combine the ideas of both papers to prove a generalization of Soulé’s theorem for motives MΩ attached to Hecke characters of infinity type (a, b) and weight w = a + b > 0 in the category MZp(K) of Chow motives over K with coefficients in Zp. More precisely, we first prove a Grothendieck-Riemann-Roch theorem for K-groups with coefficients Ka(X,Z/p ) and p big enough relative to dimX . Then we show that the functors Ka(−,Z/p) factor through MZp , and finally we prove the following theorem: