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Derived Artin stacks are closed under homotopy fiber products

Statement

Let $F \to H \leftarrow G$ be a diagram of derived Artin stacks over a characteristic-zero field $k$ . Assume the three stacks are $n$ -geometric for one finite integer $n \geq - 1$ , with the derived-smooth atlas convention of Derived Artin stack and cotangent complex. Then the homotopy fiber product $P = F \times_{H}^{h} G$ is derived Artin (indeed $n$ -geometric).

Statement references

Derived Artin stack and cotangent complex

Derived schemes and the cotangent complex of a morphism

Facts & Assumptions

Given: The diagram of $n$ -geometric derived stacks and the specified derived-affine site.

[F1]

Derived Artin stack and cotangent complex fixes the connective derived-affine site, derived-étale topology, affine smooth class $P$ , and the recursive $n$ -representable, $n$ - $P$ , and $n$ -geometric definitions. It does not assert the stability facts used below.

[F2]

In [F1]'s derived-affine site, a homotopy pullback of affines is the opposite of a homotopy pushout of simplicial commutative $k$ -algebras. HAG II, §2.2.1 specializes its HA context to simplicial $k$ -modules and simplicial commutative $k$ -algebras; Remark 1.1.0.9 states that homotopy coproducts of commutative monoids in that context are given by derived tensor products. Therefore this affine homotopy pullback is represented by the corresponding derived tensor product. Derived schemes and the cotangent complex of a morphism supplies the derived-scheme and cotangent context used here; it does not state this homotopy-pullback formula.

[F3]

Toën--Vezzosi, Homotopical Algebraic Geometry II, §2.2.2 defines the connective simplicial-algebra derived-étale topology and its affine smooth class; Lemma 2.2.2.13 verifies the topology hypotheses of Definition 1.3.2.13, and Lemma 2.2.3.1 verifies the smooth-class hypotheses there. Those class conditions are: covering maps lie in $P$ ; $P$ is closed under equivalences, composition, and homotopy pullbacks; membership in $P$ descends along derived-étale covers of the source; and the canonical maps from either summand to a homotopy coproduct over a common target lie in $P$ . Under this context, Proposition 1.3.3.3(3)--(4) proves stability of $n$ -representable and $n$ - $P$ morphisms under homotopy pullback and composition, and Corollary 1.3.3.5 states that the $n$ -geometric stacks are closed under homotopy pullbacks. These are published inputs with the stated HAG-context hypotheses, not consequences of [F1] alone.

[A1]

The three stacks admit a common finite geometricity bound $n$ .

Proof

1.1F1F3given

The selected derived-affine site and $P$ form the connective HAG context specified in [F3]: Lemma 2.2.2.13 verifies the derived-étale topology conditions, and Lemma 2.2.3.1 verifies the smooth-class conditions. The recursive notions of representability, atlas, and $n$ -geometricity in [F1] are the HAG definitions 1.3.3.1.

2.1F1F2F3A1given∎

Apply HAG II Corollary 1.3.3.5 in this context. It states that, for each fixed $n \geq - 1$ , the full subcategory of $n$ -geometric derived stacks is stable under homotopy pullbacks. Since $F$ , $H$ , and $G$ are all $n$ -geometric in the same context, their homotopy pullback $F \times_{H}^{h} G$ is $n$ -geometric and hence derived Artin. Proposition 1.3.3.3 supplies the representability and $n$ - $P$ stability used by that corollary; it is not inferred from the recursive definition alone.

Remarks

The common geometricity bound is explicit so the direct application does not silently use a separate theorem that raises geometricity ranks. The statement is a closure result for homotopy fiber products; it supplies no cotangent perfectness beyond the hypotheses in the derived-Artin definition.

Derived Artin stacks are closed under homotopy fiber products

Statement

Statement references

Facts & Assumptions

Proof

Remarks

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