Algebraization and Tannaka duality. (English) Zbl 1356.14006
The author computes some natural colimits of algebraic spaces using a new Tannaka duality result for algebraic spaces. The paper considers the following three situations: Let \(X\) be any quasi-compact quasi-separated algebraic space then for any I-adically complete ring \(A\) one has \(X(A) \simeq \lim X(A/I^n)\) (“algebraization of jets”), and for any set \(A_i\) of rings one has \(X(\prod A_i) \simeq \prod_i X(A_i)\) (“algebraization of products”).
The third main theorem is a “formal gluing” result. I will just state an important special case here: Given \(X\) as above with a closed subspace \(Z\), complement \(U = X \setminus Z\) and completion \(\hat X\) then X is the pushout of \(\hat X\) and \(U\) along \(V = \hat X \setminus Z\), and similarly D(X) is \(D(\hat X) \times_{D(V)} D(U)\). (Here \(D(X)\) is the infinity category of quasi-coherent complexes on \(X\).)
The reader will note that the assumptions are very mild.
The author provides a large number of instructive examples throughout the text, including proofs of some (easier) special cases of the main results as well as counterexamples showing that most assumptions are sharp. For example, the first two results do not hold for algebraic stacks.
All the proofs proceed via the following Tannaka duality result, which is of independent interest: Given quasi-compact quasi-separated algebraic spaces \(X\) and \(S\) one has that \(\operatorname{Hom}(S,X)\) is equivalent via pullback to exact monoidal functors between the infinity categories of perfect complexes on X and S. Thus the proofs of the main results can be reduced to showing equivalences of categories of perfect complexes. One has to work with (some version of) infinity categories here, considering derived categories would not suffice.
Bharghav’s proof of Tannaka duality makes use of Jacob Lurie’s work in derived algebraic geometry. In fact, various Tannaka duality results predate this work, for example Lurie’s Tannaka duality for derived stacks. The author gives references in the introduction. But when specialising to algebraic spaces previous results required stricter assumptions on the algebraic spaces \(S\) and \(X\).
The third main theorem is a “formal gluing” result. I will just state an important special case here: Given \(X\) as above with a closed subspace \(Z\), complement \(U = X \setminus Z\) and completion \(\hat X\) then X is the pushout of \(\hat X\) and \(U\) along \(V = \hat X \setminus Z\), and similarly D(X) is \(D(\hat X) \times_{D(V)} D(U)\). (Here \(D(X)\) is the infinity category of quasi-coherent complexes on \(X\).)
The reader will note that the assumptions are very mild.
The author provides a large number of instructive examples throughout the text, including proofs of some (easier) special cases of the main results as well as counterexamples showing that most assumptions are sharp. For example, the first two results do not hold for algebraic stacks.
All the proofs proceed via the following Tannaka duality result, which is of independent interest: Given quasi-compact quasi-separated algebraic spaces \(X\) and \(S\) one has that \(\operatorname{Hom}(S,X)\) is equivalent via pullback to exact monoidal functors between the infinity categories of perfect complexes on X and S. Thus the proofs of the main results can be reduced to showing equivalences of categories of perfect complexes. One has to work with (some version of) infinity categories here, considering derived categories would not suffice.
Bharghav’s proof of Tannaka duality makes use of Jacob Lurie’s work in derived algebraic geometry. In fact, various Tannaka duality results predate this work, for example Lurie’s Tannaka duality for derived stacks. The author gives references in the introduction. But when specialising to algebraic spaces previous results required stricter assumptions on the algebraic spaces \(S\) and \(X\).
Reviewer: Julian Holstein (Lancaster)
MSC:
| 14A20 | Generalizations (algebraic spaces, stacks) |
