Local Tate Duality¶
Local Tate duality perfectly pairs complementary-degree Galois cohomology of a finite module and its dual over a p-adic local field.
Core Idea¶
For a finite Galois module \(M\) over a \(p\)-adic local field \(K\), local Tate duality perfectly pairs \(H^r(K,M)\) with \(H^{2-r}(K,M^D)\), where \(M^D\) is the Tate dual and \(r=0,1,2\). Cup product leads to the local Brauer group \(H^2(K,\mathbb G_m)\cong\mathbb Q/\mathbb Z\). The finite coefficient \(H^2(K,\mu_n)\) is not itself all of \(\mathbb Q/\mathbb Z\).[^ref-bf879c79835a]
Scope of Application¶
The stated form concerns finite continuous Galois modules over finite extensions of \(\mathbb Q_p\). The constant module \(\mathbb Z/n\mathbb Z\) paired with \(\mu_n\) is a coefficient-level specialization; degree-one adjoint-representation local conditions can be paired with their annihilators in deformation theory. These are distinct mathematical uses, not a license to conflate related theorems for abelian varieties or \(p\)-adic representations.[ref-bf879c79835a][ref-f341c82e6074]
Clarity¶
Specify the field, coefficient module, Tate dual, cohomology degrees and Brauer invariant target. Perfectness means an actual character-duality isomorphism, not just a bilinear relationship.
Manages Complexity¶
Complementary-degree duality lets one infer the size or structure of some local cohomology groups from their dual partners and define orthogonal degree-one local conditions.
Abstract Reasoning¶
Form the dual coefficient, cup the degree-\(r\) and degree-\(2-r\) classes, map to \(H^2(K,\mathbb G_m)\), and use its local invariant. Do not silently replace the coefficient or generalize an annihilator claim to arbitrary local subgroups.
Knowledge Transfer¶
The pattern informs local-global arithmetic and Selmer conditions. Its exact statement does not transfer unchanged to every duality theorem or representation category. The Local Field relationship records a necessary p-adic carrier, not a taxonomic genus or a claim about all local fields.
[^ref-bf879c79835a]: J. S. Milne, “Arithmetic Duality”, §1 (2026), later expert historical exposition of Tate's local finite-module theorem and its Brauer target, not Tate's original 1957 source. [^ref-f341c82e6074]: Specialist Galois-representation notes: local duality and annihilator conditions, theorem and application section.
Relationships to Other Abstractions¶
Current abstraction Local Tate Duality Domain-specific
Parents (1) — more general patterns this builds on
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Local Tate Duality presupposes Local field Domain-specific
The stated local Galois-cohomology pairing requires a p-adic local field.
Hierarchy paths (6) — routes to 5 parentless roots
- Local Tate Duality → Local field → Locality Of Reference → Recurrence
- Local Tate Duality → Local field → Locality Of Reference → Heavy-Tailed Distributions
- Local Tate Duality → Local field → Locality Of Reference → Spatial Indexing → Search and Retrieval → Trade-offs → Constraint
- Local Tate Duality → Local field → Locality Of Reference → Spatial Indexing → Search and Retrieval → Problem Space → Representation → Abstraction
- Local Tate Duality → Local field → Locality Of Reference → Spatial Indexing → Search and Retrieval → Problem Space → State and State Transition → Phase Space
- Local Tate Duality → Local field → Locality Of Reference → Spatial Indexing → Search and Retrieval → Problem Space → Problem Representation → Representation → Abstraction
Neighborhood in Abstraction Space¶
Local Tate Duality sits in a sparse region of the domain-specific corpus (62nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Sheaves & Birational Geometry (22 abstractions)
Nearest neighbors
- Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain — 0.85
- Zariski Tangent Space — 0.85
- McKay Graph — 0.84
- Cohomological dimension — 0.84
- Sheaf of Modules — 0.84
Computed from structural-signature embeddings · 2026-10-08