descent (mathematics)

The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start.

Suppose {{math|X}} is a topological space covered by open sets {{math|X_i}}. Let {{math|Y}} be the disjoint union of the {{math|X_i}}, so that there is a natural mapping

:$p:\; Y\; \backslash rightarrow\; X.$

We think of {{math|Y}} as 'above' {{math|X}}, with the {{math|X_i}} projection 'down' onto {{math|X}}. With this language, descent implies a vector bundle on {{math|Y }}(so, a bundle given on each {{math|X_i}}), and our concern is to 'glue' those bundles {{math|V_i}}, to make a single bundle {{math|V}} on {{math|X}}. What we mean is that {{math|V}} should, when restricted to {{math|X_i}}, give back {{math|V_i}}, up to a bundle isomorphism.

The data needed is then this: on each overlap

:$X\_\{ij\},$

intersection of {{math|X_i}} and {{math|X_j}}, we'll require mappings

:$f\_\{ij\}:\; V\_i\; \backslash rightarrow\; V\_j$

to use to identify {{math|V_i}} and {{math|V_j}} there, fiber by fiber. Further the {{math|f_ij}} must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation (gluing conditions). For example, the composition

:$f\_\{jk\}\; \backslash circ\; f\_\{ij\}\; =\; f\_\{ik\}$

for transitivity (and choosing apt notation). The {{math|f_ii}} should be identity maps and hence symmetry becomes $f\_\{ij\}=f^\{-1\}\_\{ji\}$ (so that it is fiberwise an isomorphism).

These are indeed standard conditions in fiber bundle theory (see transition map). One important application to note is change of fiber : if the {{math|f_ij}} are all you need to make a bundle, then there are many ways to make an associated bundle. That is, we can take essentially same {{math|f_ij}}, acting on various fibers.

Another major point is the relation with the chain rule: the discussion of the way there of constructing tensor fields can be summed up as "once you learn to descend the tangent bundle, for which transitivity is the Jacobian chain rule, the rest is just 'naturality of tensor constructions'".

To move closer towards the abstract theory we need to interpret the disjoint union of the

:$X\_\{ij\}$

now as

:$Y\; \backslash times\_X\; Y,$

the fiber product (here an equalizer) of two copies of the projection {{math|p}}. The bundles on the {{math|X_ij}} that we must control are {{math|V_i}} and {{math|V_j}}, the pullbacks to the fiber of {{math|V}} via the two different projection maps to {{math|X}}.

Therefore, by going to a more abstract level one can eliminate the combinatorial side (that is, leave out the indices) and get something that makes sense for {{math|p}} not of the special form of covering with which we began. This then allows a category theory approach: what remains to do is to re-express the gluing conditions.

The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem.

The difficulties of algebraic geometry with passage to the quotient are acute. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.

Let $p:X\text{'}\; \backslash to\; X$. Each sheaf F on X gives rise to a descent datum

:$(F\text{'}\; =\; p^*\; F,\; \backslash alpha:\; p\_0^*\; F\text{'}\; \backslash simeq\; p\_1^*\; F\text{'}),\; \backslash ,\; p\_i:\; X\text{'}\text{'}\; =\; X\text{'}\; \backslash times\_X\; X\text{'}\; \backslash to\; X\text{'}$,

where $\backslash alpha$ satisfies the cocycle condition{{Citation | title= Descent data for quasi-coherent sheaves, Stacks Project | url=https://stacks.math.columbia.edu/tag/023A}}

:$p\_\{02\}^*\; \backslash alpha\; =\; p\_\{12\}^*\; \backslash alpha\; \backslash circ\; p\_\{01\}^*\; \backslash alpha,\; \backslash ,\; p\_\{ij\}:\; X\text{'}\; \backslash times\_\{X\}\; X\text{'}\; \backslash times\_\{X\}\; X\text{'}\; \backslash to\; X\text{'}\; \backslash times\_\{X\}\; X\text{'}$.

The fully faithful descent says: The functor $F\; \backslash mapsto\; (F\text{'},\; \backslash alpha)$ is fully faithful. Descent theory tells conditions for which there is a fully faithful descent, and when this functor is an equivalence of categories.

• Grothendieck connection
• Stack (mathematics)
• Galois descent
• Grothendieck topology
• Fibered category
• Beck's monadicity theorem
• Cohomological descent
• Faithfully flat descent

{{reflist}}

• SGA 1, Ch VIII – this is the main reference
• {{cite book | title=Néron Models | series=Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge | volume=21 |author1=Siegfried Bosch |author2=Werner Lütkebohmert |author3=Michel Raynaud | publisher=Springer-Verlag | year=1990 | isbn=3540505873 }} A chapter on the descent theory is more accessible than SGA.
• {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=Cambridge University Press | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}

Other possible sources include:

• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory {{arXiv|math.AG/0412512}}
• Mattieu Romagny, [http://perso.univ-rennes1.fr/matthieu.romagny/notes/stacks.pdf A straight way to algebraic stacks]

• [https://mathoverflow.net/q/22032 What is descent theory?]

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Category:Topology

Category:Category theory

Category:Algebraic geometry