Noetherian approximation of algebraic spaces and stacks
This paper, as its name suggests, deals with the problem of approximating certain structures such as algebraic spaces, schemes and stacks. The fundamental example as to what approximation means here is the following: Every commutative ring \(R\) can be written as a direct limit of its subrings that are finitely generated \(\mathbb Z\)-algebras and every \(R\)-module \(M\) can be written as the direct limit of its finitely generated \(R\)-submodules. This means that every affine scheme \(X\) is an inverse limit of affine schemes of finite type over \(\mathrm{Spec}\mathbb Z\) and every quasi-coherent sheaf on \(X\) is direct limit of quasi-coherent sheaves of finite type. In the paper it is shown for example that every quasi- compact and quasi-separated Deligne-Mumford stack \(X\) is an inverse limit of Deligne-Mumford stacks of finite type over \(\mathrm{Spec}\mathbb Z\). Approximation here is in contrast with standard limit results which are relative: given an inverse limit \(X = \underset\longleftarrow\lim_{\lambda} X_{\lambda}\) describe finitely presented objects over \(X\) in terms of finitely presented objects over \(X_{\lambda}\) for sufficiently large \(\lambda\). The following definitions are fundamental for understanding the paper: An algebraic stack \(X\) is said to have the completeness property if every quasicoherent sheaf on \(X\) is a filtered direct limit of finitely presented sheaves or, equivalently, if the abelian category \(QCoh(X)\) is compactly generated. Such an stack is called \textit{pseudo-noetherian} if it is quasi-compact, quasi-separated and \(X'\) has the completeness property for every finitely presented morphism \(X' \to X\) of algebraic stacks. For \(S\) a pseudo-noetherian stack and \(X \to S\) a morphism of stacks, an approximation of \(X\) over \(S\) is a finitely presented \(S\)-stack \(X_0\) together with an affine \(S\)-morphism \(X \to X_0\). \(X/S\) can be approximated if there is such an approximation over \(S\). A morphism \(f : X \to Y\) of algebraic stacks is of strict approximation type if f can be written as a composition of affine morphisms and finitely presented morphisms. The morphism \(f\) is of approximation type if there exists a surjective representable and finitely presented étale morphism \(p : X' \to X\) such that \(f \circ p\) is of strict approximation type. An algebraic stack \(X\) is said to be of approximation type if \(X \to\mathrm{Spec}\mathbb Z\) is so. There are four main results in the paper dealing with above notions. The first one (Theorem A) asserts that every stack of approximation type is pseudo-notherian. The second main result (Theorem B) shows that when \(X\) is a quasicompact stack with quasi-finite and separated diagonal, then there exists a scheme \(Z\) and a finite, finitely presented and surjective morphism \(Z \to X\) that is flat over a dense quasi-compact open substack \(U \subseteq X\). Another main result of the paper (Theorem C) shows that certain sets of properties of morphisms of algebraic stacks (named PA, PC and PI in the paper) passes to a limit \(X \to S\) if these properties are satisfied by the inverse system \(\{X_{\lambda}\}\) for all \(\lambda\) sufficiently large, i.e. \(\lambda \geq \alpha\) for some \(\alpha\). See page 4 of the article for the statement and also page 30. Note that sometimes it is necessary that the morphisms \(X_\lambda \to S\) and the bonding maps \(X_\lambda \to X_\mu\) satisfy extra conditions such as closed immersion or of finite type. The last main result (Theorem D) deals with the situation that \(S\) is a pseudo-noetherian algebraic stack and \(X \to S\) be a morphism of approximation type. Theorem D then says that there exists a finitely presented morphism \(X_0 \to S\) and an affine \(S\)-morphism \(X \to X_0\). The theorem shows moreover that the sets of properties mentioned earlier are satisfied for \(X_0 \to S\) provided that they are satisfied for \(X \to S\) Many interesting tools and applications of the above results are proved in the paper. For example a version of étale d dévissage is shown in order to prove Theorem B. This is a statement of the form: Given \(X' \to X\) surjective and étale, then \(X\) can be approximated if \(X'\) can be approximated. This is a generalization of the older forms of dévissage because in those treatments it was always assumed that \(X \to X\) is an étale neighborhood and that \(X\) is quasi-affine. The present form of dévissage, however, is also reduced to the case that \(X' \to X\) is an étale neighborhood. As an application of the results, there is a generalization of a theorem of Chevalley. This generalization states that if \(X\) is affine and \(Y\) an algebraic space and \(X \to Y\) integral and surjective, then \(Y\) is also affine. Note that the notherianity assumption in Chevalley's theorem is dropped and also finite morphisms have been replaced by integral morphisms.
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