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Cohomology of a Stack

Cohomology of a stack extends sheaf-cohomological constructions to algebraic stacks by choosing an appropriate site and deriving global sections, including equivariant interpretations for quotient stacks.

Version
v1 · 2026-09-28 · History
Domain-specific #
8532
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Algebraic Geometry, Homological Algebra → Mathematics

Core Idea

Cohomology of a stack extends sheaf cohomology from schemes and algebraic spaces to algebraic stacks, whose points may possess automorphisms and whose geometry is presented through a groupoid rather than a single ordinary space. Given a stack, an appropriate site such as its lisse–étale or étale site, and a sheaf or complex of coefficients, cohomology is obtained as the derived functors of global sections or by an equivalent derived-category construction. The resulting graded groups record obstructions to gluing local data while retaining the stack's stabilizer information.

How would you explain it like I'm…

 

No faithful explanation at this level. A five-year-old picture can only show an ordinary collection of points, erasing the automorphisms carried by stack points and collapsing the idea into cohomology of the coarse space, which is exactly what stack cohomology is not.

Symmetry-Aware Gluing Test

In geometry, mathematicians use cohomology to measure how small, local pieces of information fit together into a whole, and where they fail to fit. A stack is a special kind of shape whose points can carry their own symmetries, like a point that can be spun around without changing. Cohomology of a stack does this measuring while keeping track of those symmetries. That's why two stacks that look like they have the same points can still give different answers.

Sheaf Cohomology on Stacks

Cohomology of a stack extends sheaf cohomology, a tool that measures the obstructions to gluing local data into global data, from ordinary spaces to algebraic stacks. A stack is a geometric object whose points may have automorphisms (symmetries), and it is often described using a groupoid, a set of points together with the ways points are identified. To compute its cohomology, you choose a suitable site (a notion of 'open covers') and coefficients, then take derived global sections. The answer must not depend on how the stack is presented. For a quotient stack [X/G] it gives an algebraic version of equivariant cohomology, and for the classifying stack BG it captures information about the group G. Importantly, it is not just the cohomology of the stack's coarse space of points, which can forget the symmetries.

 

Cohomology of a stack extends sheaf cohomology from schemes and algebraic spaces to algebraic stacks, whose points may carry automorphisms and whose geometry is presented by a groupoid rather than a single space. Given a stack, a site such as the lisse-etale or etale site, and a sheaf or complex of coefficients, cohomology is defined as the derived functors of global sections or an equivalent derived-category construction. The graded groups record obstructions to gluing local data while retaining stabilizer information. The construction must be independent of presentation: for a groupoid or simplicial presentation, descent yields a spectral sequence or total complex combining the cohomology of the levels with the groupoid relations. For a quotient stack [X/G] this recovers an algebraic analogue of equivariant cohomology, and the classifying stack BG, the case where X is a point, captures characteristic information about G. The coefficient type (quasi-coherent, etale, l-adic, de Rham, or others) and the topology determine which theorem is being asserted. It is not the cohomology of the coarse moduli space, which can erase isotropy and descent data, and it is not the Chow ring, though cycle class maps can relate them.

Scope of Application

  • Quotient stacks. Presentations [X/G] retain group action and isotropy in the cohomological object.

  • Classifying stacks. BG connects stack cohomology with group, equivariant, and characteristic-class phenomena.

  • Descent and sheaves. Local data on a site are glued with automorphism-compatible conditions.

  • Obstruction theory. Cohomological classes control existence, lifting, and deformation problems in moduli settings.

  • Spectral sequences. Groupoid, simplicial, or atlas presentations provide computable pages when differentials and convergence are tracked.

Clarity

Cohomology of a stack extends sheaf-cohomological gluing and obstruction data to objects presented with groupoid symmetry and stabilizers. Clarity requires the stack, chosen site or topology, coefficient sheaf or complex, and derived construction; saying ‘cohomology’ without these can hide materially different invariants. Presentation independence is load-bearing: a groupoid atlas is computational data, not the invariant itself.

Manages Complexity

Cohomology of a stack compresses local geometric data, gluing obstructions, and stabilizer symmetry into graded groups defined on an appropriate site or through a presentation-independent derived construction. The analyst tracks the stack, topology, coefficients, and descent data rather than recomputing each atlas overlap by hand. Quotient, algebraic, and differentiable-stack branches use related machinery with different sites. Spectral sequences organize contributions by simplicial level or group action.

Abstract Reasoning

Site-selection move. Choose the stack's lisse–étale, étale, or other appropriate site and coefficient object before inferring cohomology. Descent move. From a groupoid or simplicial presentation, build a total complex or spectral sequence and infer global classes from compatible local data. Invariance move. Replace the atlas by an equivalent presentation and require the resulting groups to agree. Stabilizer move. For quotient stacks, reason from group action and equivariant data rather than collapsing immediately to the coarse quotient. Boundary move.

Knowledge Transfer

Within the home domain. Cohomology of a stack transfers across algebraic geometry, moduli problems, orbifolds, and descent settings where objects have automorphisms and local data glue only up to coherent isomorphism. Site, sheaves, derived functors, stabilizers, descent, and spectral sequences retain formal roles. Beyond the home domain (C — invariant machinery). It applies literally to stacks in any compatible geometric category, not metaphorically to ordinary software stacks. Its boundary is theory-dependent: topology, coefficients, and cohomology notion must be specified, and coarse moduli spaces can lose stabilizer information. Similar groups do not imply equivalent stacks or moduli interpretations.

Neighborhood in Abstraction Space

Cohomology of a Stack sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08