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**Equivariant Tamagawa numbers and Galois module theory. I.**
*(English)*
Zbl 1014.11070

In this important paper the author defines an element \(T\Omega(L/K,0)\) attached to a Galois extension \(L/K\) and provides links with many other constructions and conjectures. The Galois group \(G\) of \(L/K\) is permitted to be non-Abelian, and \(T\Omega(L/K,0)\) is an element of the relative K-group K\(^0(\mathbb Z[G], \mathbb R)\). It is conjectured that \(T\Omega(L/K,0)\) is always zero. The author proves the following:

(1) The class \(T\Omega(L/K,0)\) comes from K\(^0(\mathbb Z[G], \mathbb Q)\) iff the Stark conjecture holds for \(L/K\), and if this case occurs, \(T\Omega(L/K,0)\) is zero iff the Strong Stark conjecture holds.

(2) The nullity of \(T\Omega(L/K,0)\) is equivalent to the Lifted Root Number Conjecture of K. W. Gruenberg, J. Ritter and A. Weiss [cf. Proc. Lond. Math. Soc. (3) 79, 47–80 (1999; Zbl 1041.11075); Jahresber. Dtsch. Math.-Ver. 100, 36–44 (1998; Zbl 0929.11054)]; more precisely, the element \(\omega_{\varphi_S}\) defined by the latter authors agrees with \(T\Omega(L/K,0)\) up to a canonical involution.

(3) The nullity of \(T\Omega(L/K,0)\) implies Chinburg’s \(\Omega(3)\)-conjecture for \(L/K\).

(4) The element \(T\Omega(L/K,0)\) (which was constructed globally, that is, without using any localization result for relative K-groups) agrees with an element, or rather, a family of elements indexed by all prime numbers \(p\) that were defined in an earlier preprint of the author and called equivariant Tamagawa numbers.

(Comment: One now speaks of the conjecture that \(T\Omega(L/K,0)\) vanishes as “the Equivariant Tamagawa number conjecture” for a certain motive.)

The construction of \(T\Omega(L/K,0)\) fits into a larger framework; actually \(T\Omega(L/K,0)=\break T\Omega(L/K,M)\) where the motive \(M\) is specialised to \(\mathbb Q(0)_L\). The main ingredients are reduced determinants and refined Euler characteristics of perfect complexes. To define the latter one needs also to fix a so-called “trivialisation fibre”, that is, a class of isomorphisms between the odd-degree cohomology and the even-degree cohomology of the given complex. The complexes that are fed into the machinery arise from earlier work of the author with M. Flach [Am. J. Math. 120, 1343–1397 (1998; Zbl 0929.11050)] and can be thought of as a kind of Tate four term sequence, and the trivialisation fibre comes from regulator maps.

Two small remarks: The objects of \({\mathcal M}_K(A)\) are \(K\)-motives endowed with \(A\)-action; of course there is a forgetful functor \({\mathcal M}_K(A) \to{\mathcal M}_K \) but perhaps \({\mathcal M}_K(A)\) is not a subcategory of \({\mathcal M}_K \) in the obvious way. On p. 217, if \(R\) is allowed to be any commutative ring, \(R[G]\) will not quite be Gorenstein, but the used properties of \(R\)-duals certainly hold true.

There is a lot of recent work on the equivariant Tamagawa number conjecture, much of it by the author and his coworkers. The article under review is fundamental for all this.

(1) The class \(T\Omega(L/K,0)\) comes from K\(^0(\mathbb Z[G], \mathbb Q)\) iff the Stark conjecture holds for \(L/K\), and if this case occurs, \(T\Omega(L/K,0)\) is zero iff the Strong Stark conjecture holds.

(2) The nullity of \(T\Omega(L/K,0)\) is equivalent to the Lifted Root Number Conjecture of K. W. Gruenberg, J. Ritter and A. Weiss [cf. Proc. Lond. Math. Soc. (3) 79, 47–80 (1999; Zbl 1041.11075); Jahresber. Dtsch. Math.-Ver. 100, 36–44 (1998; Zbl 0929.11054)]; more precisely, the element \(\omega_{\varphi_S}\) defined by the latter authors agrees with \(T\Omega(L/K,0)\) up to a canonical involution.

(3) The nullity of \(T\Omega(L/K,0)\) implies Chinburg’s \(\Omega(3)\)-conjecture for \(L/K\).

(4) The element \(T\Omega(L/K,0)\) (which was constructed globally, that is, without using any localization result for relative K-groups) agrees with an element, or rather, a family of elements indexed by all prime numbers \(p\) that were defined in an earlier preprint of the author and called equivariant Tamagawa numbers.

(Comment: One now speaks of the conjecture that \(T\Omega(L/K,0)\) vanishes as “the Equivariant Tamagawa number conjecture” for a certain motive.)

The construction of \(T\Omega(L/K,0)\) fits into a larger framework; actually \(T\Omega(L/K,0)=\break T\Omega(L/K,M)\) where the motive \(M\) is specialised to \(\mathbb Q(0)_L\). The main ingredients are reduced determinants and refined Euler characteristics of perfect complexes. To define the latter one needs also to fix a so-called “trivialisation fibre”, that is, a class of isomorphisms between the odd-degree cohomology and the even-degree cohomology of the given complex. The complexes that are fed into the machinery arise from earlier work of the author with M. Flach [Am. J. Math. 120, 1343–1397 (1998; Zbl 0929.11050)] and can be thought of as a kind of Tate four term sequence, and the trivialisation fibre comes from regulator maps.

Two small remarks: The objects of \({\mathcal M}_K(A)\) are \(K\)-motives endowed with \(A\)-action; of course there is a forgetful functor \({\mathcal M}_K(A) \to{\mathcal M}_K \) but perhaps \({\mathcal M}_K(A)\) is not a subcategory of \({\mathcal M}_K \) in the obvious way. On p. 217, if \(R\) is allowed to be any commutative ring, \(R[G]\) will not quite be Gorenstein, but the used properties of \(R\)-duals certainly hold true.

There is a lot of recent work on the equivariant Tamagawa number conjecture, much of it by the author and his coworkers. The article under review is fundamental for all this.

Reviewer: Cornelius Greither (Neubiberg)