Descent (Mathematics)¶

Recover a global mathematical object from compatible local objects by comparing pullbacks on fiber-product overlaps, enforcing a cocycle, and proving the resulting datum effective.

Version: v3 · 2026-09-06 · History
Domain-specific #: 1648
Origin domain: algebraic geometry
Subdomain: descent theory

Core Idea¶

Descent is the mathematical theory of recognizing when objects given over a cover or a morphism come from one object over the base. Its decisive move is to compare the local objects only after pulling them to a common overlap, require those comparisons to compose coherently on triple overlaps, and then ask whether the resulting descent datum is effective—whether a global object actually exists whose pullbacks reproduce the supplied datum.

Let \(\mathcal C\) be a category with the needed fiber products and let \(\mathcal F\to\mathcal C\) be a fibred category of the objects under study. For a family \(\{u_i:U_i\to U\}\), a descent datum consists of objects \(X_i\in\mathcal F(U_i)\) and isomorphisms

\[ \varphi_{ij}:\operatorname{pr}_0^*X_i\longrightarrow \operatorname{pr}_1^*X_j \]

over \(U_i\times_U U_j\). On the triple fiber product \(U_i\times_U U_j\times_U U_k\), they must satisfy

\[ \operatorname{pr}_{12}^*\varphi_{jk}\circ \operatorname{pr}_{01}^*\varphi_{ij} =\operatorname{pr}_{02}^*\varphi_{ik}. \]

This is the cocycle condition. The identity condition on a diagonal overlap and the inverse relation obtained by swapping two factors follow from it.^[1]

Every global object \(X\in\mathcal F(U)\) has a canonical descent datum on its pullbacks \(u_i^*X\). Hence there is a comparison functor

\[ \mathcal F(U)\longrightarrow DD(\{U_i\to U\}), \]

from global objects to descent data. A datum is effective when it lies in this functor's essential image. Full faithfulness says that morphisms, and therefore uniqueness up to the appropriate isomorphism, can be recovered locally; essential surjectivity says that every coherent local object comes from a global one. When the functor is an equivalence, the chosen cover supports effective descent for that class of objects.^[2]

For a single morphism \(p:Y\to X\), the same structure becomes especially compact: an object \(E\) on \(Y\), an isomorphism between the two pullbacks of \(E\) to \(Y\times_XY\), and a cocycle on \(Y\times_XY\times_XY\). This formulation makes descent useful beyond literal open subsets. It also exposes the main boundary: coherent descent data need not be effective for every object class or every cover. Effectivity is a theorem with hypotheses, not a synonym for the cocycle.^[3]

Structural Signature¶

Sig role-phrases:

the base object — the space, scheme, or categorical base \(U\) on which the sought global object should live
the cover or descent morphism — a family \(U_i\to U\) or one morphism \(Y\to U\) along which objects can be pulled back
the fibred object class — sheaves, modules, bundles, schemes, torsors, or other objects with a coherent pullback operation
the local objects — the objects \(X_i\) over cover members, or the object \(E\) over the covering source
the pair-overlap pullbacks — the two ways of viewing local objects over \(U_i\times_UU_j\) or \(Y\times_UY\)
the overlap isomorphisms — reversible identifications between those pullbacks, rather than mere similarity or equal numerical summaries
the triple-overlap cocycle — the equation making the direct and two-step identifications agree
the canonical descent functor — the map from global objects to the descent data produced by their pullbacks
the effectivity verdict — whether a datum is in the essential image, with full faithfulness separately controlling morphisms and uniqueness
the obstruction or hypothesis boundary — the topology, object class, or geometric condition determining when effectiveness holds or fails

The roles form a recognition test. If local pieces are merely listed, there is no descent datum. If they are compared on overlaps but the comparisons do not satisfy the triple cocycle, there is incompatible gluing data. If the cocycle holds but no global object realizes it, there is a genuine non-effective datum. Only when the datum is induced by a global object has descent succeeded.

This order matters. Pairwise isomorphisms alone do not make a coherent family: the route from \(i\) to \(k\) through \(j\) must equal the direct route on the triple overlap. Conversely, the cocycle does not itself prove existence. The developed abstraction owns both the formation of coherent input and the separate theorem or counterexample governing effectivity.

What It Is Not¶

Not numerical descent. Gradient descent, steepest descent, and related optimization methods generate iterates that lower an objective. They do not use cover pullbacks, overlap isomorphisms, or categorical effectivity.
Not infinite descent. Infinite descent is a proof technique that derives an impossible strictly decreasing chain in a well-founded order.
Not ordinary aggregation. Combining local values into a sum, average, vote, or report lacks the pullback-and-cocycle structure unless those roles are supplied literally.
Not arbitrary gluing. Open-cover gluing is a central effective case, but descent distinguishes the datum from the theorem that the datum comes from a global object and also works along morphisms that are not open inclusions.^[4]
Not the sheaf condition in full generality. A sheaf of sets is the discrete case in which matching sections glue uniquely. Descent also handles category-valued objects that agree only up to isomorphism and can have automorphisms.^[5]
Not descent data alone. Local objects plus cocycle-compatible overlap isomorphisms form the input. The theory also asks about the canonical comparison functor, morphism descent, and effectivity.
Not a claim that every datum is effective. There are étale descent data for projective schemes that fail to be effective in the category of schemes.^[3]
Not Galois descent alone. Semilinear group actions over a Galois extension give one important specialization of the general mechanism.^[6]
Not a Čech cocycle by itself. The cocycle expresses coherence; its interpretation, target category, equivalence relation, and effectivity theorem remain essential.
Not an algebraic stack. A stack is a fibred category satisfying descent axioms for a chosen topology; descent is the mechanism those axioms assert.
Not the live catalog's prime:stack. That node is a last-in/first-out nesting discipline and is only a lexical false friend.
Not descent of a property without qualification. Showing that flatness, finite presentation, or another property is local on the base is related but distinct from reconstructing an object with specified overlap identifications.

Scope of Application¶

Sheaves and modules on open covers. Compatible sheaves or modules on open subsets glue to a global sheaf. This is the most concrete effective case and supplies the equalizer formula behind many constructions.^[4]

Quasi-coherent sheaves in algebraic geometry. Quasi-coherent sheaves satisfy effective descent for fpqc coverings. In affine form, a faithfully flat ring map \(R\to A\) identifies \(R\)-modules with \(A\)-modules carrying descent data.^[7]^[8]

Vector bundles and line bundles. Local trivial bundles are identified by transition isomorphisms on overlaps. The cocycle makes the transition functions coherent; effectivity produces the global bundle. Serre twists \(\mathcal O(n)\) on projective space are standard examples of invertible sheaves obtained from compatible local presentations.^[9]

Schemes and morphisms. One may try to descend a scheme or a morphism along fpqc, fppf, étale, or other covers. The answer depends on the object class: affine morphisms have strong effectiveness results, while unrestricted projective scheme data can fail to descend as schemes.^[10]^[3]

Torsors and forms. A form that becomes standard after a cover is recovered by recording how the local standard object is identified across overlaps. Automorphisms of the standard object turn overlap identifications into cocycles; different cocycle classes can yield different descended forms.

Galois descent. For a finite Galois extension \(L/K\), descent data on an \(L\)-vector space can be expressed as a semilinear action of \(\operatorname{Gal}(L/K)\). The \(K\)-form is recovered from fixed vectors and scalar extension.^[6]

Stacks and moduli. Families of geometric objects often have automorphisms, so a set-valued sheaf condition is too rigid. Stack language separates local objects, isomorphisms between them, and effective object descent while requiring morphism presheaves to be sheaves.^[5]^[11]

Higher and homotopical settings. The same Čech-nerve idea extends beyond ordinary categories, but higher categories require coherent higher homotopies rather than only one strict triple-overlap equation. That extension is related but exceeds this node's classical 1-categorical core.

The scope remains mathematical. “Descending” a policy from branches to headquarters is metaphor unless there is a literal base, pullback-like restriction, overlap comparison by isomorphism, cocycle coherence, and an effectivity question in a specified category.

Clarity¶

Descent becomes clear once three verdicts are kept separate:

Well-formed local data: objects exist over the members of the cover.
Coherent descent data: their pullbacks are identified and the identifications satisfy the cocycle.
Effective descent: the coherent datum is induced by a global object.

The first does not imply the second, and the second does not imply the third. This three-stage distinction prevents the common sentence “the transition maps satisfy the cocycle, so the object glues” from silently importing an effectivity theorem. It is valid for sheaves on open covers and for quasi-coherent sheaves on fpqc covers because the relevant theorems say so; it is not a definition valid for every object class.^[7]^[3]

A second clarification separates existence from uniqueness. Essential surjectivity of the canonical descent functor supplies a global object. Full faithfulness says compatible local morphisms come from a unique global morphism. A stack requires both forms of control: morphism presheaves satisfy the sheaf condition, and descent data for objects are effective.^[5]

Finally, fiber products explain what “the overlap” means when a cover is not a family of subsets. For \(p:Y\to X\), two points or maps in \(Y\) overlap when they have the same image in \(X\), precisely the relation encoded by \(Y\times_XY\). The triple fiber product records three mutually comparable lifts and is the domain on which path independence of the identifications can be tested.

Manages Complexity¶

Descent replaces direct global construction with a controlled local workflow. One chooses a cover on which the object becomes simpler, constructs there, records only the overlap identifications, checks one cocycle pattern, and invokes an effectivity theorem. A difficult global object can therefore be represented by tractable local models plus a small coherence interface.

It also compresses repeated gluing arguments into a categorical question. Rather than rebuilding a quotient or checking independence of representatives for every construction, the analyst studies the functor

\[ \mathcal F(U)\to DD(\mathcal U). \]

Full faithfulness, essential surjectivity, and equivalence name the exact obligations. Once an object class is known to form a stack for a topology, every new local construction can reuse the result instead of reproving gluing from first principles.^[5]^[11]

The method localizes failure. A pairwise comparison that cannot be defined signals incompatible local types. A failed cocycle isolates path-dependent identification on a triple overlap. A coherent but non-effective datum reveals a global representability or object-class obstruction. A failure of full faithfulness shows that morphisms cannot be recovered uniquely from their local restrictions. These are different defects and therefore demand different repairs.

Descent also lets practitioners change the working base. Faithfully flat base change can expose an object in a form where linear algebra or affine methods apply, after which the descent datum records exactly what must survive the return to the original base. The approach is not “solve somewhere easier and hope”: it keeps a formal ledger of the information needed to come back.

Abstract Reasoning¶

Recognition test. Name the base, the cover, the fibred object class, the two pullbacks on pair overlaps, the overlap isomorphism, and the triple cocycle. If any role has no literal referent, the case is generic local-to-global reasoning rather than mathematical descent.

Effectivity test. After verifying the cocycle, ask which theorem places the datum in the essential image of the canonical functor. “The pieces agree” is not a substitute for naming the topology and object class.

Morphisms-versus-objects test. Determine whether the problem is failure to glue maps between already-global objects or failure to produce the objects themselves. The first concerns full faithfulness; the second concerns essential surjectivity.

Refinement move. Pull the datum to a finer cover. A finer cover may simplify local objects or make a theorem applicable, but one must still prove that effectiveness on the refinement returns to the original cover under the relevant hypotheses.

Base-change move. Replace \(U\) by a cover on which the desired object becomes standard, solve locally, and encode the change-of-trivialization information as a cocycle. Predict that changing the local trivializations modifies the cocycle by a coboundary-like conjugation without changing the descended isomorphism class.

Obstruction diagnostic. If the cocycle cannot be solved by choosing compatible trivializations, look for a cohomological or representability obstruction. If the cocycle is valid but no global object exists in the chosen category, consider whether enlarging the category—schemes to algebraic spaces, for example—changes effectivity.

Special-case inference. For a finite Galois cover, translate overlap data into a semilinear group action. Fixed points are the candidate descended object; verify that extending scalars recovers the original object.^[6]

Boundary inference. Do not infer descent of a property from descent of the underlying object, or conversely, without a theorem showing that the property is local for the chosen topology.

Knowledge Transfer¶

Within algebraic geometry, topology, and category theory, descent transfers as a literal method. Sheaves, modules, vector bundles, torsors, forms, schemes, and moduli objects change, but the operative sequence remains recognizable: pull back to a cover, compare on the fiber-product relation, impose cocycle coherence, and test the canonical functor for effectivity. The same proof architecture can move from Zariski covers to fpqc covers or from bundles to quasi-coherent modules while preserving its vocabulary and diagnostics.^[12]^[11]

The method also transfers between concrete and abstract presentations. Transition functions on intersections become an isomorphism on \(Y\times_XY\). A Galois-semilinear action becomes descent data for \(\operatorname{Spec}L\to\operatorname{Spec}K\). A stack axiom becomes the claim that all such data are effective and morphisms glue. These are not metaphors; they are equivalent or specialized encodings of the same category-theoretic mechanism.

Outside mathematics, the honest transfer belongs to prime:local_to_global_aggregation. A federation reconciling regional records may have local pieces, overlap checks, and a global result, but it does not instantiate Descent unless restriction and pullback, isomorphism-valued overlap data, the cocycle, and categorical effectivity are literal. The portable lesson—local witnesses require a verified compatibility and recomposition discipline—travels. The named mathematical apparatus does not.

Examples¶

Canonical¶

Let \(X=\mathbf P^1_k\) with standard affine cover

\[ U_0=D_+(X_0),\qquad U_1=D_+(X_1). \]

On both charts take a free rank-one module, with frames \(e_0\) and \(e_1\). On the overlap, \(t=X_1/X_0\) is invertible. Identify the frames by

\[ e_1=t e_0, \]

with inverse identification \(e_0=t^{-1}e_1\). Together with identity maps on the diagonal overlaps, these transition isomorphisms satisfy the cocycle. Gluing the two trivial local modules produces the invertible sheaf \(\mathcal O_{\mathbf P^1}(1)\). Its restriction to each standard chart is free of rank one even though the global line bundle is not the trivial bundle. This is why descent is more than copying a local object globally: the overlap isomorphism carries the twisting that determines the global form.^[4]^[9]^[13]

Mapped back: \(\mathbf P^1_k\) is the base object; \(U_0,U_1\) are the cover; rank-one modules are the fibred object class; the free modules are the local objects; the common chart is the pair-overlap pullback; multiplication by \(t\) is the overlap isomorphism; identity/inverse compatibility supplies the cocycle; gluing realizes the canonical descent functor in reverse; \(\mathcal O(1)\) is the effective global object; and the open-cover gluing theorem supplies the hypothesis boundary.

Applied / In Practice¶

Consider the faithfully flat finite Galois extension \(\mathbb R\subset\mathbb C\). Let

\[ N=\mathbb C^2, \qquad \tau(z_1,z_2)=(\overline{z_2},\overline{z_1}). \]

The map \(\tau\) is conjugate-semilinear and \(\tau^2=1\), so it is Galois descent data on the complex vector space. Its fixed subspace is

\[ M=N^\tau=\{(z,\overline z):z\in\mathbb C\}, \]

a real vector space with basis \(m_1=(1,1)\) and \(m_2=(i,-i)\). Those two vectors are also a complex basis of \(N\), because the determinant of the matrix with columns \(m_1,m_2\) is \(-2i\). Therefore the natural map

\[ \mathbb C\otimes_{\mathbb R}M\longrightarrow N \]

is an isomorphism. The complex object with semilinear compatibility is effective: it descends to the real vector space \(M\), and complexification recovers exactly the original \(N\).^[6]^[8]

Mapped back: \(\operatorname{Spec}\mathbb R\) is the base; \(\operatorname{Spec}\mathbb C\to\operatorname{Spec}\mathbb R\) is the cover; vector spaces/modules form the object class; \(N\) is the local object; the two embeddings encoded by the Galois involution supply the pair-overlap views; \(\tau\) is the overlap isomorphism; \(\tau^2=1\) is the cocycle in Galois form; scalar extension is the canonical descent functor; the fixed space \(M\) establishes effectivity; and faithful flatness is the theorem boundary.

Structural Tensions¶

T1: Coherent data versus effective data. The cocycle is strong enough to make overlap identifications path-independent, yet weak enough that a global object may still fail to exist in the chosen category. Diagnostic: has only the cocycle been checked, or has an effectivity theorem for this topology and object class been invoked?

T2: Existence versus uniqueness. Essential surjectivity produces an object, while full faithfulness controls whether local morphisms determine a unique global morphism. One can fail without the other. Diagnostic: is the unresolved obligation to construct the object or to show that its realization and maps are unique up to the correct isomorphism?

T3: Local simplicity versus overlap burden. Passing to a cover can trivialize the object, but a finer or more redundant cover enlarges the family of pair and triple overlaps whose compatibility must be controlled. Diagnostic: does the chosen cover reduce construction cost more than it increases coherence cost?

T4: Concrete gluing versus functorial abstraction. Open-set gluing is intuitive and constructive; the fibred-category formulation is less visual but handles faithfully flat and other non-inclusion covers uniformly. Diagnostic: is the open-cover picture sufficient, or is it concealing a morphism-based cover for which fiber products are the real overlaps?

T5: Rigid equality versus isomorphism-valued agreement. Requiring local objects to be literally equal is too strict and coordinate-dependent, while allowing arbitrary isomorphisms creates automorphisms and coherence obligations. Diagnostic: are the local objects being compared after pullback by a specified isomorphism, and has its triple-overlap coherence been checked?

T6: Object descent versus property descent. An underlying object may descend while a chosen property does not, or a property may be local even when a presentation of the object is not. Diagnostic: which theorem concerns reconstruction of the object, and which separately says the property is local on the base?

T7: Classical cocycle versus higher coherence. In an ordinary fibred category a triple-overlap equation captures the needed 1-categorical coherence. Derived and higher settings can demand further coherent homotopies. Diagnostic: is strict equality of the displayed composites meaningful in the target, or must equality itself be replaced by specified higher data?

T8: Autonomy versus reduction. Descent has its own mathematical machinery and failure theory, yet its portable skeleton is Local-to-Global Aggregation. Diagnostic: are pullbacks, fiber-product overlaps, isomorphism cocycles, and effectivity literal? If not, route the case to the parent prime; if so, preserve the Descent node.

Structural–Framed Character¶

Descent is mixed-structural. Its evaluative weight is nil: a datum is coherent or not, effective or not, relative to stated categorical and topological hypotheses. Failure can be mathematically useful as an obstruction rather than a moral or design defect.

It is not human-practice-bound in the relevant sense. Once a category, fibred object class, and cover are fixed, the fiber products, isomorphisms, and equivalence question are formal. Its institutional origin is the Grothendieck school of algebraic geometry, but the resulting definitions are not dependent on an institution's judgment.

Its import-versus-recognize behavior is literal within related mathematical domains: bundle transition maps, faithfully flat module descent, Galois semilinear actions, and stack effectivity instantiate the same pullback-and-cocycle architecture. Beyond mathematics, uses are usually imported analogies whose local pieces and reconciliation rules instantiate only the parent prime.

The limiting criterion is vocabulary travels. Base change, fibred category, fiber product, pullback, isomorphism, cocycle, essential image, and stack are constitutive technical terms. Removing them leaves a general local-to-global discipline but not Descent.

Its character: a formally structural local-to-global mechanism whose operative vocabulary and diagnostics remain pinned to category-theoretic mathematics.

Structural Core vs. Domain Accent¶

What is skeletal. Local information can determine a global object or verdict only when the pieces cover the target, agree on their overlaps, and are recomposed by a justified rule. Failure of recomposition can itself expose an obstruction. This is the portable structure already carried by prime:local_to_global_aggregation.

What is domain-bound. Descent replaces generic pieces with objects in a fibred category; generic overlaps with fiber products; agreement with isomorphisms between two pullbacks; transitive compatibility with a cocycle on a triple fiber product; and recomposition with the canonical functor from global objects to descent data. It further separates full faithfulness, essential surjectivity, and equivalence, and classifies results by topology and object class. These are not optional examples. They are the mechanism by which descent makes predictions, proves constructions, and diagnoses failure.

Why it does not clear the prime bar. The name and full diagnostic interface do not survive unrestricted substrate substitution. An organizational “descent” with regional reports has no literal fiber product or pullback functor unless a mathematical model supplies one, and a metaphorical cocycle is not enough. Once the technical roles are removed, the remaining inference is exactly the parent prime's cover/compatibility/recomposition pattern. Cross-domain reach belongs to Local-to-Global Aggregation; the named node remains a mathematically bounded specialization.

Why it is not a mere composite. Local-to-Global Aggregation, Compatibility, Isomorphism, and Composition do not by conjunction generate the fibred-category bindings, the two projection pullbacks, the precise cocycle, the canonical comparison functor, or the effectivity counterexamples. Descent is a stable theory with its own theorems, variants, and intervention sequence: choose a cover, construct locally, write the datum, test coherence, and prove or refute effectivity.

prime:local_to_global_aggregation — proposed strict subsumption parent. Descent is that prime specialized to category-valued mathematical objects, with pullback isomorphisms and effectivity as its differentia.
prime:compatibility — internal role, no direct edge proposed. The cocycle makes local identifications mutually compatible, but generic Compatibility does not supply the cover, pullbacks, or global realization.
prime:isomorphism — internal arrow type, no direct edge proposed. Descent compares local objects up to isomorphism rather than equality; Isomorphism is a constituent, not the organizing genus.
prime:consistency — related but declined. Cocycle coherence resembles consistency, yet a coherent datum can be non-effective. Joint satisfiability does not entail descent.
prime:aggregation and prime:composition — broader neighbors. They can assemble parts but omit the certified overlap-and-effectivity discipline already captured more sharply by Local-to-Global Aggregation.
prime:stack — lexical false friend. The live node is the LIFO data structure, not a stack in algebraic geometry, and must not receive an edge.

Relationships to Other Abstractions¶

Current abstraction Descent (Mathematics) Domain-specific

Parents (1) — more general patterns this builds on

Descent (Mathematics) is a kind of Local-to-Global Aggregation Prime

prime:local_to_global_aggregation — proposed strict subsumption parent. Descent is that prime specialized to category-valued mathematical objects, with pullback isomorphisms and effectivity as its differentia.

Hierarchy path (1) — routes to 1 parentless root

Descent (Mathematics) → Local-to-Global Aggregation

Neighborhood in Abstraction Space¶

Descent (Mathematics) sits in a sparse region of the domain-specific corpus (72^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Formal Languages, Types & Programs (41 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Ordinary gluing. Gluing on an open cover is a standard effective case; Descent is the functorial theory that also handles non-inclusion covers and separates data from effectivity. Tell: is the problem merely constructing a quotient on open subsets, or testing a canonical descent functor for a specified cover and object class?
The sheaf condition. Matching sections of a set-valued presheaf agree by equality and glue uniquely. General descent allows objects with nontrivial isomorphisms and automorphisms. Tell: are the local things sections in sets, or objects in categories that must be compared up to isomorphism?
Descent data. These are the local objects, overlap isomorphisms, and cocycle—the input to the effectivity question. Tell: has a global realization been established, or only a coherent datum written down?
Effective descent. This is the condition or theorem that the canonical functor is an equivalence, not the neutral name for every datum. Tell: are all coherent data asserted to arise globally under stated hypotheses?
Galois descent. A specialization where a Galois group acts semilinearly and fixed points recover a form over the base field. Tell: does the cover come from a Galois field extension with a group action, or from a general site/morphism?
Faithfully flat or fpqc descent. A major topology-qualified theorem family, especially for modules and quasi-coherent sheaves. Tell: is faithful flatness the stated hypothesis, or is another topology and object class in play?
A Čech cocycle. The overlap equation encoding coherence; it can classify data or obstructions without making every datum effective. Tell: is the cocycle being used as input, or has its realization as a global object been proved?
A stack in algebraic geometry. A fibred category satisfying descent axioms. Tell: is the referent the ambient category that has effective descent, or the descent mechanism itself?
prime:stack. The catalog's last-in/first-out nested container. Tell: are push and pop constrained by recency, or are mathematical objects being pulled back along covers?
Local-to-Global Aggregation. The substrate-neutral parent that covers compatibility and recomposition across many fields. Tell: do fiber products, pullbacks, isomorphism cocycles, and effectivity remain literal? If not, use the prime.
Gradient or steepest descent. An iterative optimization method following decreasing directions. Tell: is there an objective and iterates, or a cover and overlap data?
Infinite descent. A reductio that produces ever-smaller counterexamples in a well-founded order. Tell: is the contradiction a forbidden decreasing chain, or the issue global reconstruction from local data?
Descent set. A combinatorial set of indices at which adjacent values decrease. Tell: is the object a permutation statistic, or a category-valued gluing problem?
Cohomological descent. A related technique for computing or comparing cohomology through a simplicial resolution or hypercover; it is not identical to classical object descent. Tell: is the output an effective global object, or a cohomological equivalence/computation?
Descent of properties. The claim that a property can be checked after a cover. Tell: is an object being reconstructed with specified overlap isomorphisms, or only a predicate being shown local on the base?

References¶

[1] The Stacks Project. “Descent data in fibred categories,” Section 8.3, tag 02ZC. Definition of local objects, overlap isomorphisms, the cocycle, morphisms of descent data, and the identity/inverse consequences. Verified 2026-08-26. registry ↩

[2] The Stacks Project. “Effective descent,” Definition 8.3.5, tag 026E. Defines canonical descent data, the global-to-descent-data functor, and effectivity as membership in its essential image. Verified 2026-08-26. registry ↩

[3] The Stacks Project. “Non-effective descent data for projective schemes,” Lemma 110.68.1, tag 08KF. Constructs projective scheme data along an étale cover that are not effective in schemes. Verified 2026-08-26. registry ↩a ↩b ↩c ↩d

[4] The Stacks Project. “Glueing sheaves,” Section 6.33, tag 00AK. Gives the construction of a global sheaf from compatible sheaves on an open cover and the equivalence with gluing data. Verified 2026-08-26. registry ↩a ↩b ↩c

[5] The Stacks Project. “Stacks,” Section 8.4, tag 0268. Separates the sheaf condition for morphisms from effective descent for objects in the definition of a stack. Verified 2026-08-26. registry ↩a ↩b ↩c ↩d

[6] Keith Conrad. “Galois Descent”. Develops descent of vector spaces through semilinear Galois actions and recovery from fixed points. Verified 2026-08-26. registry ↩a ↩b ↩c ↩d

[7] The Stacks Project. “Faithfully flat descent,” Section 59.16, tag 03O6. Proves effective fpqc descent for quasi-coherent sheaves and states the descent-data/cocycle formulation. Verified 2026-08-26. registry ↩a ↩b

[8] The Stacks Project. “Effective descent for modules along faithfully flat ring maps,” Proposition 35.3.9, tag 023N. Establishes the equivalence between base modules and modules carrying faithfully flat descent data. Verified 2026-08-26. registry ↩a ↩b

[9] The Stacks Project. “Invertible sheaves on Proj,” Lemma 27.10.3, tag 01MT. Establishes the invertible sheaves \(\mathcal O_X(n)\) on the standard-open cover of Proj. Verified 2026-08-26. registry ↩a ↩b

[10] The Stacks Project. “Descending affine morphisms,” Section 35.37, tag 0244. Establishes effectivity for affine morphisms under the stated descent setting. Verified 2026-08-26. registry ↩

[11] Angelo Vistoli. “Notes on Grothendieck topologies, fibered categories and descent theory”. In Fundamental Algebraic Geometry: Grothendieck's FGA Explained, Mathematical Surveys and Monographs 123, AMS, 2005, pp. 1–104. Systematic account of sites, fibred categories, stacks, and descent theory. Verified 2026-08-26. registry ↩a ↩b ↩c

[12] A. Grothendieck and M. Raynaud. Revêtements étales et groupe fondamental (SGA 1), Exposé VIII, “Descente fidèlement plate.” Documents Mathématiques 3, Société Mathématique de France, 2003 [seminar 1960–61; original LNM 224, 1971]. Historical primary treatment of faithfully flat descent. Verified 2026-08-26. registry ↩

[13] The Stacks Project. “More on invertible modules,” Section 31.29, tag 0BD6. Computes the Picard group of projective space over a field and distinguishes the twists \(\mathcal O(n)\), including the nontriviality of \(\mathcal O(1)\) on \(\mathbf P^1\). Verified 2026-08-26. registry ↩