English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so‐called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi‐isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi‐isomorphism (and some variants).  Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincaré duality for higher pages of the Frölicher spectral sequence, construct a functorial three‐space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott–Chern and Aeppli cohomology for Calabi–Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non‐Kählerness degrees are not bimeromorphic invariants in dimensions higher than three and that the    ∂   ∂  ¯    ‐lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

On the structure of double complexes