[Paper Review] Representation of relative sheaf cohomology

Tatsuo Suwa|arXiv (Cornell University)|Oct 15, 2018

Homotopy and Cohomology in Algebraic Topology15 references4 citations

TL;DR

This paper establishes a canonical isomorphism between relative sheaf cohomology and a cohomology theory based on soft or fine resolutions of sheaf complexes, using both Čech and derived category methods. It proves a relative de Rham-type theorem showing that for paracompact spaces and fine resolutions, the cohomology of sections vanishing on an open subset is canonically isomorphic to the relative cohomology of the original sheaf.

ABSTRACT

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Cech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a "relative de Rham type theorem" from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated.

Motivation & Objective

• To provide a concrete, computable representation of relative sheaf cohomology, which is otherwise defined abstractly via flabby resolutions.
• To bridge Čech cohomology and derived category theory by showing that relative cohomology can be computed via complexes of sections over open covers.
• To generalize the de Rham theorem to the relative setting using soft or fine resolutions, particularly in complex geometry.
• To develop a cohomology theory for sheaf morphisms via the co-mapping cone construction in derived categories.
• To establish canonical isomorphisms across multiple cohomological frameworks, ensuring consistency and independence of choices.

Proposed method

• Introduces a cohomology theory $ H^q_{D_{ ilde{K}}}(X,X') $ defined via the complex of triples $ (\xi_0, \xi_1, \xi_{01}) $ over a cover $ \mathcal{V} = \{V_0 = X', V_1\} $, where $ \xi_0 = 0 $ for relative cochains.
• Uses the Čech-type complex $ \mathscr{K}^\bullet(\mathcal{V}, \mathcal{V}') $ of sections over a cover of $ X $ and its open subset $ X' $, with differential defined via alternating sums.
• Establishes canonical isomorphisms between $ H^q_{D_{ ilde{K}}}(X,X') $ and $ H^q(X,X'; \mathscr{S}) $ when $ \mathscr{K}^\bullet $ is a fine resolution of $ \mathscr{S} $, independent of the choice of $ V_1 $.
• Reinterprets the relative cohomology as the cohomology of a 'co-mapping cone' dual to the mapping cone in derived categories, enabling generalization to sheaf morphisms.
• Applies the framework to Dolbeault cohomology, proving a relative Dolbeault theorem via Stein coverings and canonical isomorphisms with Čech-Dolbeault cohomology.
• Considers the complex $ j_!j^{-1}\mathscr{E}^{(\bullet)}_X $ for $ C^\infty $-manifolds, showing that its cohomology computes relative cohomology with coefficients in $ j_!j^{-1}\mathbb{C}_X $.

Experimental results

Research questions

• RQ1 How can relative sheaf cohomology $ H^q(X,X';\mathscr{S}) $ be represented concretely using soft or fine resolutions?
• RQ2 What is the relationship between the Čech cohomology of a sheaf complex over a two-open cover and the relative cohomology of its sections?
• RQ3 Can the relative de Rham theorem be generalized to the setting of soft or fine resolutions, and is the resulting isomorphism canonical?
• RQ4 How does the derived category notion of the co-mapping cone relate to relative cohomology in the context of sheaf complexes?
• RQ5 What are the implications of this framework for relative Dolbeault cohomology and cohomology with compact support in complex geometry?

Key findings

• The cohomology $ H^q_{D_{ ilde{K}}}(X,X') $, defined via a two-open cover $ \mathcal{V} = \{X', V_1\} $, is independent of the choice of $ V_1 $, up to canonical isomorphism.
• For any fine resolution $ 0 \to \mathscr{S} \to \mathscr{K}^\bullet $ with $ \mathscr{K}^\bullet|_{X'} $ fine, there is a canonical isomorphism $ H^q_{D_{ ilde{K}}}(X,X') \simeq H^q(X,X';\mathscr{S}) $, proving the relative de Rham-type theorem.
• The relative Dolbeault cohomology $ H^{p,q}_{\bar{\vartheta}}(X,X') $ is canonically isomorphic to $ H^q(X,X';\mathscr{O}^{(p)}) $, generalizing the classical Dolbeault theorem to the relative case.
• The cohomology $ H^q_{d'}(X) $ of the complex $ j_!j^{-1}\mathscr{E}^{(\bullet)}_X $ is canonically isomorphic to $ H^q(X; j_!j^{-1}\mathbb{C}_X) $, providing a concrete model for relative cohomology with compact support.
• Each class in $ H^q_{D'}(X,X') $ is represented by a pair $ (\xi_1, \xi_{01}) $, where $ \xi_1 $ is a closed $ q $-form on $ X $ with support in $ \Omega $, and $ \xi_{01} $ is a $ (q-1) $-form on $ X' $ with $ d\xi_{01} = \xi_1|_{X'} $.
• The framework allows for a direct, sign-correct construction of the isomorphism between Čech and sheaf cohomology, avoiding the sign ambiguities present in classical Weil lemma-based proofs.