This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these ...conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups.
Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development. They proceed to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations.
Comprehensive and self-contained,The Norm Residue Theorem in Motivic Cohomologyunites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.
Notes on isocrystals Kedlaya, Kiran S.
Journal of number theory,
August 2022, 2022-08-00, Letnik:
237
Journal Article
Recenzirano
Odprti dostop
For varieties over a perfect field of characteristic p, étale cohomology with Qℓ-coefficients is a Weil cohomology theory only when ℓ≠p; the corresponding role for ℓ=p is played by Berthelot's rigid ...cohomology. In that theory, the coefficient objects analogous to lisse ℓ-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with ℓ-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship between the two cases, via the theory of companions, will be treated in sequel papers.
We prove a Poincaré duality theorem for finite groups, where the (co)homologies involved are a variant of classical group cohomology due to Zarelua and Staic, denoted by Hλ⁎. In particular, we show ...that in certain cases (but not always), for a finite group G of order n and a G-module M, we have the isomorphismHλn−i−1(G,M)→≅Hiλ(G,Mtw), where Mtw is a twisting of M defined in Section 5.
A conjecture of Flach and Morin Chiarellotto, Bruno; Mazzari, Nicola; Nakada, Yukihide
Journal of number theory,
November 2024, 2024-11-00, Letnik:
264
Journal Article
Recenzirano
Odprti dostop
A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the p-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined ...over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.
Let
a
be an ideal of Noetherian ring R and M a finitely generated R-module such that
cd
(
a
,
M
)
=
c
. In this paper, we investigate
Att
R
(
H
a
c
(
M
)
)
. Among other things, it is shown that
Max
...{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
}
⊆
Att
R
(
H
a
c
(
M
)
)
. We also show that
Att
R
(
H
a
c
(
M
)
)
=
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
p
=
Ann
R
(
H
a
c
(
R
/
p
)
)
}
and
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
dim
R
/
p
−
1
≤
cd
(
a
,
R
/
p
)
≤
dim
R
/
p
}
⊆
Att
R
(
H
a
c
(
M
)
)
.
Finally, we prove that if
(
R
,
m
)
is a local ring and
dim
R
/
a
=
1
then
Att
R
(
H
a
c
(
M
)
)
=
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
cd
(
a
,
M
)
}
. Then by using this, it is shown that if
(
R
,
m
)
is a local ring then
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
dim
R
/
(
a
+
p
)
=
1
}
⊆
Att
R
(
H
a
c
(
M
)
)
.
Let M2n be a Poisson manifold with Poisson bivector field Π. We say that M is b-Poisson if the map Πn:M→Λ2n(TM) intersects the zero section transversally on a codimension one submanifold Z⊂M. This ...paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M,Π) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.
A note on Deligne's formula Schenzel, Peter
Journal of pure and applied algebra,
December 2024, 2024-12-00, Letnik:
228, Številka:
12
Journal Article
Recenzirano
Odprti dostop
Let R denote a Noetherian ring and an ideal J⊂R with U=SpecR∖V(J). For an R-module M there is an isomorphism Γ(U,M˜)≅lim→HomR(Jn,M) known as Deligne's formula (see 8, p. 217 and Deligne's Appendix in ...7). We extend the isomorphism for any R-module M in the non-Noetherian case of R and J=(x1,…,xk) a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction.
We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects.
The theory is situated in the intersection of the general Čech theory and the theory ...of derived categories.
That is to say, on the one hand the cohomology is described as the relative cohomology of the sections
of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories.
We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter.
Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies
that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology
of sheaf morphisms. Some special cases together with applications are also indicated.
Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction
theory and other problems in algebraic analysis.
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