André recently gave a beautiful proof of Hochster’s direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem ...(the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where “discriminants” acquire all
p
-power roots. In this paper, we explain a quicker proof of Hochster’s conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze’s Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.
We prove that the Witt vector affine Grassmannian, which parametrizes
W
(
k
)-lattices in
W
(
k
)
1
p
n
for a perfect field
k
of characteristic
p
, is representable by an ind-(perfect scheme) over
...k
. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably
h
-descent results for vector bundles.
We prove two basic structural properties of the algebraic
K
-theory of rings after
K
(1)-localization at an implicit prime
p
. Our first result (also recently obtained by Land–Meier–Tamme by ...different methods) states that
L
K
(
1
)
K
(
R
)
is insensitive to inverting
p
on
R
; we deduce this from recent advances in prismatic cohomology and
TC
. Our second result yields a Künneth formula in
K
(1)-local
K
-theory for adding
p
-power roots of unity to
R
.
By finding a p-adic obstruction, we construct many examples of complete noetherian local normal Fp-algebras R such that no module-finite extension R↪S is Cohen–Macaulay. These examples should be ...contrasted with a result of Hochster–Huneke: the directed union of all such extensions is always Cohen–Macaulay.
Almost direct summands Bhatt, Bhargav
Nagoya mathematical journal,
06/2014, Letnik:
214
Journal Article
Recenzirano
Odprti dostop
We prove new cases of the direct summand conjecture using fundamental theorems in p-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification locus lies entirely in ...characteristic p.
The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply naturally to a ...broader class of schemes that we call ‘F-smooth’. The F-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
We prove that the coherent cohomology of a proper morphism of noetherian schemes can be made arbitrarily $p$-divisible by passage to proper covers (for a fixed prime $p$). Under some extra ...conditions, we also show that $p$-torsion can be killed by passage to proper covers. These results are motivated by the desire to understand rational singularities in mixed characteristic, and have applications in $p$-adic Hodge theory.
Let
X
be a closed equidimensional local complete intersection subscheme of a smooth projective scheme
Y
over a field, and let
X
t
denote the
t
-th thickening of
X
in
Y
. Fix an ample line bundle
O
Y
...(
1
)
on
Y
. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer
c
, such that for all integers
t
⩾
1
, the cohomology group
H
k
(
X
t
,
O
X
t
(
j
)
)
vanishes for
k
<
dim
X
and
j
<
-
c
t
. Note that there are no restrictions on the characteristic of the field, or on the singular locus of
X
. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant
c
is unbounded, even in a fixed dimension.
We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP ...and crystalline-TP spectral sequences need not degenerate.
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