Types of Sheaves

Why This Matters

Sheaf theory sits at the crossroads of topology, algebra, and geometry—it's the machinery that lets mathematicians rigorously move between local data and global structure. When you're working through problems in algebraic geometry, cohomology theory, or even modern number theory, you're constantly asking: what can I learn about a whole space from information defined on small pieces? Different types of sheaves answer this question in fundamentally different ways, and understanding which sheaf to use—and why—is what separates surface-level familiarity from genuine mastery.

You're being tested on more than definitions here. Exam questions will ask you to identify why a particular sheaf type has the properties it does, how different sheaves relate to cohomological computations, and when finiteness conditions matter versus when extension properties matter. The categories below are organized by the core principle each sheaf type embodies: constancy and local behavior, extension properties for cohomological computation, and algebraic finiteness conditions. Don't just memorize which sheaf is which—know what structural problem each one solves.

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Sheaves Defined by Constancy Conditions

These sheaves are characterized by how "uniform" their sections are across the space. The key question: does the sheaf assign the same data everywhere, or only locally?

Constant Sheaves

• Assigns a fixed set (or group) $A$ to every open set—the sections over any connected open set are simply the elements of $A$
• Restriction maps are identity maps between copies of $A$, making the sheaf structure as simple as possible
• Foundation for singular cohomology—constant sheaves with coefficients in $\mathbb{Z}$ or a field underlie classical cohomology theories in algebraic topology

Locally Constant Sheaves

• Sections are constant on each connected component—the sheaf "looks constant" in small neighborhoods but can vary globally
• Correspond bijectively to representations of $\pi_1(X)$—this connection to covering spaces and fundamental groups makes them central to monodromy theory
• Define local systems of coefficients for twisted homology and cohomology, generalizing constant-coefficient theories

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Sheaves with Extension Properties (Acyclic Sheaves)

These sheaves are workhorses for computing cohomology. Their defining feature is that sections can be extended from smaller open sets to larger ones, which kills higher cohomology and makes them useful for resolutions.

Flasque (Flabby) Sheaves

• Every section over an open set extends to the whole space—the restriction maps $\mathcal{F}(U) \to \mathcal{F}(V)$ are surjective for $V \subseteq U$
• Acyclic for sheaf cohomology—flasque sheaves have vanishing higher cohomology $H^i(X, \mathcal{F}) = 0$ for $i > 0$
• Used to build flasque resolutions for computing derived functors, though they're often too large to work with explicitly

Soft Sheaves

• Sections extend from closed sets—any section over a closed subset $Z$ extends to the whole space
• Acyclic on paracompact Hausdorff spaces—this weaker extension property still suffices for cohomology computations in nice topological settings
• More common in analysis than flasque sheaves, since sheaves of continuous or smooth functions are typically soft

Fine Sheaves

• Admit partitions of unity—given any open cover, there exist sheaf endomorphisms subordinate to the cover that sum to the identity
• Automatically soft (and hence acyclic on paracompact spaces), but the partition of unity structure gives additional computational power
• Essential for de Rham theory—the sheaf of smooth differential forms $\Omega^k$ on a manifold is fine, which is why de Rham cohomology works

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Sheaves with Algebraic Finiteness Conditions

In algebraic geometry, the key constraint isn't extension properties but finiteness. These sheaves behave well with respect to algebraic operations and have manageable cohomology.

Coherent Sheaves

• Locally finitely presented—over an affine open set, the sheaf corresponds to a finitely generated module with finitely generated relations
• Closed under kernels, cokernels, and extensions—this abelian category structure makes coherent sheaves the "right" objects for homological algebra on schemes
• Cohomology is finite-dimensional on proper schemes over a field, a crucial finiteness theorem (Serre, Grothendieck) that fails for general quasi-coherent sheaves

Quasi-Coherent Sheaves

• Locally isomorphic to $\widetilde{M}$ for some module $M$—no finiteness requirement, so infinite-dimensional and infinitely-generated modules are allowed
• Form the natural category for scheme theory—morphisms of schemes induce pullback and pushforward functors on quasi-coherent sheaves
• Include all coherent sheaves as a full subcategory; quasi-coherent sheaves are the "big" category where constructions like direct limits live

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Sheaves from Geometric Structures

These sheaves arise naturally from geometric objects and encode richer structure than purely algebraic sheaves.

Vector Bundles (as Sheaves of Sections)

• Sheaf of sections $\mathcal{E}$ of a rank-$n$ vector bundle is locally free of rank $n$—over small open sets, $\mathcal{E}|_U \cong \mathcal{O}_U^n$
• Locally free sheaves and vector bundles are equivalent on reasonable spaces, so geometric intuition (fibers, transition functions) translates to algebraic language
• Chern classes and characteristic classes are defined via vector bundles/locally free sheaves, connecting sheaf theory to differential topology

Constructible Sheaves

• Locally constant along strata of a stratification—the space decomposes into pieces where the sheaf is well-behaved
• Finite-dimensional stalks and finitely many strata give strong finiteness, making constructible sheaves computable
• Foundation for perverse sheaves and intersection cohomology—these generalizations handle singular spaces where ordinary cohomology fails

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Sheaves in Arithmetic and Étale Geometry

When classical topology fails (e.g., over finite fields), alternative topologies and their sheaves take over.

Étale Sheaves

• Defined on the étale site rather than the Zariski topology—étale covers are "algebraic local homeomorphisms" that detect finer structure
• Enable $\ell$-adic cohomology—étale sheaves with coefficients in $\mathbb{Z}_\ell$ or $\mathbb{Q}_\ell$ give a Weil cohomology theory for varieties over any field
• Connect to Galois representations—étale fundamental groups and étale sheaves bring Galois theory into geometric settings, crucial for the Langlands program

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Quick Reference Table

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Self-Check Questions

1. What property do flasque, soft, and fine sheaves all share, and why does this make them useful for computing sheaf cohomology?

2. A locally constant sheaf on a path-connected space $X$ is equivalent to what algebraic data involving $\pi_1(X)$? Why does this equivalence fail for constant sheaves?

3. Compare coherent sheaves and quasi-coherent sheaves: which category is closed under infinite direct sums, and which has finite-dimensional cohomology on proper schemes?

4. If you're working on a smooth manifold and need to prove that de Rham cohomology equals sheaf cohomology, which type of sheaf (and which property) makes this comparison work?

5. Explain why étale sheaves are necessary in algebraic geometry over finite fields, where the Zariski topology is too coarse. What cohomology theory do they enable?