Almost ring theory

By Ofer Gabber, Lorenzo Ramero. 2000

247 citations recorded by the source corpus.

3.1 Flat, unramified and etale morphisms 3.2 Nilpotent deformations of almost algebras and modules 3.3 Nilpotent deformations of torsors 3.4 Descent 3.5 Behaviour of etale morphisms under Frobenius

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fedfc09edb6a6df02b3e0d30de431e91196b1027

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Derived Functors of Inverse Limits Revisited (2006, methodology): “In Section 4 we also recall and use that we proved in 1965 that the categories studied by Gabber–Ramero in [ 4 ] (as generalizations of examples studied by Faltings [3]) are exactly those abelian categories satisfying the axioms AB4* and AB6 and having a system of generators.”
Perfectoid Spaces: A survey (2013, methodology): “This argument can be made precise by using Faltings’s almost mathematics, as developed systematically by Gabber-Ramero, [16].”
The Galois group of a stable homotopy theory (2014, methodology): “In [29], an approach to Galois theory (in the connected case) for almost rings is given that does not assume a priori the existence of a fiber functor.”
Perfectoid Spaces (2011, methodology): “We will use the book of Gabber-Ramero, [14], as our basic reference.”
The arc-topology (2018, background): “ tft K-algebra. Let f ∈R. This gives an evident inductive sequence of tft K-algebras {Rhf/πri}with transition maps Rhf/πri→R f/πr+1 →···. For any torsion sheaf Fon Spec(R)´et, there is an equivalence (18) lim −→ r RΓ(Spec(Rhf/πri),F) ≃RΓ(Spec(R/f),F). Proof. Let R0 ⊂R be an open bounded OK-subalgebr”
On torsion in the cohomology of locally symmetric varieties (2013, background): “[34] and, for a very brief summary, [53, Section 4].”
Virtual resolutions for a product of projective spaces (2017, background): “homology groups are supported on the irrelevant ideal of X. The beneﬁts of allowing a limited amount of homology are already present in other parts of commutative algebra including almost ring theory [GR03], where one accepts homology annihilated by a given idempotent ideal, and phantom homology [HH”
Cohomological quotients and smashing localizations (2003, unknown): “Almost derived categories Almost rings and modules have been introduced by Gabber and Ramero [10].”