Algebraic stack

Builds on Scheme Algebraic space Smooth morphism Fiber product of schemes

Definition

Let $S$ be a scheme, and let $\mathcal{C}$ be the category of schemes over $S$ with the fppf topology. An algebraic stack over $S$ is a category $\mathcal{X}$ with a functor $p \colon \mathcal{X} \to \mathcal{C}$ such that:

$\mathcal{X}$ is a stack in groupoids over $\mathcal{C}$ : every fibre category is a groupoid, and objects and morphisms glue along fppf coverings.
The diagonal $\mathcal{X} \to \mathcal{X} \times_{\mathcal{C}} \mathcal{X}$ is representable by algebraic spaces.
There are an $S$ -scheme $U$ and a 1-morphism from $U$ to $\mathcal{X}$ that is surjective and smooth.

In clause 3, a scheme $U$ is seen as the stack of $S$ -schemes over $U$ . Clause 2 makes this 1-morphism representable by algebraic spaces, so surjective and smooth make sense.

Remarks

Let $k$ be a field and $G$ a finite group, seen as a constant group scheme over $k$ . The classifying stack $BG$ , whose objects over $T$ are $G$ -torsors over $T$ , is an algebraic stack. The map $\operatorname{Spec} k \to BG$ given by the trivial torsor is surjective and étale, hence smooth. Every algebraic space is an algebraic stack whose fibre categories have only identity morphisms. A functor that fails the sheaf condition, such as the constant presheaf with value $\set{0, 1}$ , fails clause 1. If in clause 3 the map can be chosen étale, the stack is called a Deligne–Mumford stack. This is the definition of the Stacks Project; some authors also require the diagonal to be quasi-compact and separated.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Algebraic space concept
Fiber product of schemes concept
Scheme concept
Smooth morphism concept
Algebraic stack definition

Tag	`VP1N9S4J4F`
URI	`m2://VP1N9S4J4F`
Kind	definition
Writing	W0 stub
MSC	14A20, 18F20
Defines	Algebraic stack
Uses	Scheme, Algebraic space, Smooth morphism, Fiber product of schemes
Changed	2 November 2026

Algebraic stack

Definition

Remarks

Reading path

Further reading

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