The Laurent cover of a uniform Tate ring #
Let A be a complete Hausdorff Tate ring and f ∈ A. The rational subsets
U₁ = R({f, 1}/1) = {|f| ≤ 1}, U₂ = R({1}/f) = {|f| ≥ 1}, U₁ ∩ U₂ = R({f², f, 1}/(1 · f))
cover Spa(A, A⁺). When A is uniform, that is, when its power-bounded elements form a bounded
set, the augmented Čech sequence of this cover,
A → A⟨U₁⟩ × A⟨U₂⟩ → A⟨U₁ ∩ U₂⟩, a ↦ (a, a), (x, y) ↦ x|U₁∩U₂ - y|U₁∩U₂,
is exact, and its first map is a closed embedding. These are the injectivity and the exactness in
the middle in Corollary 4 of Buzzard–Verberkmoes, the case of a Laurent cover in their theorem that
stably uniform Tate rings are sheafy (their Theorem 7); Corollary 4 also asserts that the second
map is surjective, which is not part of the results here. The statements have the same form as
laurentCover_exact and isClosedEmbedding_laurentCover, which assume strong noetherianness
instead of uniformity.
Main results #
TauCeti.ValuationSpectrum.isPowerBounded_of_algebraMap_mem_locSubring_laurentCover: an element ofA₀[f]whose image inA[1/f]lies inA₀[1/f]is power-bounded.TauCeti.ValuationSpectrum.isClosedEmbedding_laurentCover_of_isUniform: for uniformAthe mapA → A⟨U₁⟩ × A⟨U₂⟩is a closed embedding.TauCeti.ValuationSpectrum.laurentCover_exact_of_isUniform: for uniformAthe kernel of the difference of restrictionsA⟨U₁⟩ × A⟨U₂⟩ → A⟨U₁ ∩ U₂⟩is the image ofA.
Implementation notes #
The rings of definition of the three localisations are D₁ = A₀[f], D₂ = A₀[1/f] and
D₁₂ = A₀[f, 1/f]. The first result is Buzzard–Verberkmoes's Lemma 3 for the cover {1, f}: an
element of D₁ ∩ D₂ is, up to a power of f, a polynomial in f of bounded degree on both sides,
so its powers all lie in one finitely generated A₀-submodule. With uniformity it bounds the
elements of A lying in ϖⁿ D₁ and in ϖⁿ D₂ by ϖⁿ A°, so A carries the topology induced
from A⟨U₁⟩ × A⟨U₂⟩, and the image of the complete ring A is closed. Since D₁₂ = D₁ + D₂, the
difference map is strict before completion, which makes every pair agreeing on U₁ ∩ U₂ a limit
of pairs coming from A. Buzzard–Verberkmoes's Lemma 2 obtains this step from the general fact
that completion preserves exact sequences of strict maps; here it is proved directly for this
sequence.
References #
- K. Buzzard, A. Verberkmoes, Stably uniform affinoids are sheafy, J. reine angew. Math. 740 (2018), 25–39 (arXiv:1404.7020), Lemmas 2 and 3 and Corollary 4.
Polynomials of bounded degree in one element #
Power-bounded elements on the two Laurent pieces #
Buzzard–Verberkmoes, Lemma 3, for the Laurent cover of f. Let A₀ be the ring of
definition of a pair of definition of A, and f ∈ A. If a ∈ A lies in A₀[f], the ring of
definition of R({f, 1}/1) = {|f| ≤ 1}, and its image in A[1/f] lies in A₀[1/f], the ring of
definition of R({1}/f) = {|f| ≥ 1}, then a is power-bounded in A. Compare
isPowerBounded_of_mem_locSubring, which concerns power-boundedness in a localisation Aₛ.
The overlap ring of definition is the sum of those of the two pieces #
Approximation on the Laurent cover #
The Laurent cover of a uniform Tate ring #
Buzzard–Verberkmoes, Lemma 2 and Corollary 4, strictness. Let A be a complete Hausdorff
uniform Tate ring and f ∈ A. The map A → A⟨U₁⟩ × A⟨U₂⟩, a ↦ (a, a), into the coordinate
rings of U₁ = R({f, 1}/1) = {|f| ≤ 1} and U₂ = R({1}/f) = {|f| ≥ 1} is a closed embedding:
it is injective, its image is closed, and the topology of A is the one induced from the
product. This is isClosedEmbedding_laurentCover with uniformity in place of strong
noetherianness.
Buzzard–Verberkmoes, Corollary 4, exactness in the middle. Let A be a complete Hausdorff
uniform Tate ring and f ∈ A, and let U₁ = R({f, 1}/1), U₂ = R({1}/f) and
U₁ ∩ U₂ = R({f², f, 1}/(1 · f)). In
A → A⟨U₁⟩ × A⟨U₂⟩ → A⟨U₁ ∩ U₂⟩, a ↦ (a, a), (x, y) ↦ x|U₁∩U₂ - y|U₁∩U₂,
the kernel of the second map is the image of the first. The restriction maps are those of the
refinements with cofactors f and 1. The first map is injective, and even a closed embedding
(isClosedEmbedding_laurentCover_of_isUniform). Only U₂ comes with a standing hypothesis: the
one for U₁ is automatic at the denominator 1
(TauCeti.Huber.PairOfDefinition.hasDenominatorPower_denom_one), and the one for U₁ ∩ U₂ is
built from those two by TauCeti.Huber.PairOfDefinition.hasDenominatorPower_mul. This is
laurentCover_exact with uniformity in place of strong noetherianness.