How can the Albert-Brauer-Hasse-Noether theorem be interpreted topologically via sheaf cohomology? Seeking precise duality dictionary and references

I am looking for a topological or geometric interpretation of the Albert–Brauer–Hasse–Noether (ABHN) theorem, which establishes the local-global principle for central simple algebras over a global field $K$.

The classical exact sequence is given by:

$$0\rightarrow \text{Br}(K)\rightarrow \bigoplus _{v}\text{Br}(K_{v})\xrightarrow{\sum \text{inv}_{v}}\mathbb{Q}/\mathbb{Z}\rightarrow 0$$

Using the identification $\text{Br}(K) \cong H^2(\text{Gal}(K^{\text{sep}}/K), (K^{\text{sep}})^\times)$, this sequence can be viewed through Galois cohomology. Since group cohomology translates topologically via classifying spaces ($BGL_n$ or $BG$), it seems plausible to view this arithmetic duality through a topological lens.

Questions: 1.Topological Analogues: Is there a geometric analogue of the ABHN sequence if we replace $K$ with the function field of a complex curve or Riemann surface $X$? Does it translate into a statement involving the Picard group, sheaf cohomology, or the topology of punctured points on $X$? 2. Classifying Spaces and Gerbes:Since elements of $H^2(X, \mathcal{O}_X^\times)$ classify bundle gerbes or lifting obstructions, can the restriction map $\text{Br}(K) \to \bigoplus_v \text{Br}(K_v)$ be interpreted as restricting a global topological obstruction to local punctured disks or charts? 3. Étale Cohomology and Duality: How does the transition from Galois cohomology to étale cohomology realize the ABHN theorem as a purely topological duality statement (e.g., Poincare Duality or Artin–Verdier Duality)?

I am seeking references, big-picture intuitions, or concrete dictionary-like mappings between these arithmetic entities and topological concepts.