##
**Categorical Milnor squares and \(K\)-theory of algebraic stacks.**
*(English)*
Zbl 1504.19004

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A Milnor square of rings (Section 2 of [J. W. Milnor, Introduction to algebraic \(K\)-theory. Princeton, NJ: Princeton University Press (1971; Zbl 0237.18005)]) is a cartesian square of rings \[ \begin{tikzcd} A \rar \dar & A/I \dar \\ B \rar & B/J. \end{tikzcd} \] The following result is due to Land and Tamme, building on work of Morrow, Geisser and Hesselholt, and Suslin (see Corollaries 2.10 and 2.33 in [M. Land and G. Tamme, Ann. Math. (2) 190, No. 3, 877–930 (2019; Zbl 1427.19002)], as well as [T. Geisser and L. Hesselholt, Invent. Math. 166, No. 2, 359–395 (2006; Zbl 1107.19002); M. Morrow, J. Reine Angew. Math. 736, 95–139 (2018; Zbl 1393.19003); G. Tamme, Compos. Math. 154, No. 9, 1801–1814 (2018; Zbl 1395.18013)] and [A. A. Suslin, in: Number theory, algebra and algebraic geometry. Collected papers. In honor of the seventieth birthday of Academician Igor Rostislavovich Shafarevich. Moscow: Maik Nauka/Interperiodica Publishing. 255–279 (1995; Zbl 0871.19002); translation from Tr. Mat. Inst. Steklova 208, 290–317 (1995)]): Given a Milnor square of rings in which \(\operatorname{Tor}_i^A(A/I, B) = 0\) for all \(i\) (Tor-independence), the associated square of \(K\)-theory spectra is cartesian.

The first purpose of this paper is to categorify the hypothesis in this theorem. Theorem A, part (i) says the following: Suppose \[ \begin{tikzcd} \mathcal A \rar{f^*} \dar{p^*} & \mathcal B \dar{q^*} \\ \mathcal{A}' \rar{g^*} & \mathcal{B}'\end{tikzcd} \] is a commutative square of presentable stable \(\infty\)-categories and colimit-preserving functors that furthermore satisfies the following:

- 1.
- the canonical functor \(\mathcal A \to \mathcal A' \times_{\mathcal B'} \mathcal B\) is fully faithful (an analogue of being cartesian)
- 2.
- the right adjoints of each of the functors \(f^*\), \(g^*\), \(p^*\) and \(q^*\) preserve filtered colimits (called compactness of the functor here) and these functors are also surjective, in that their images generate the codomains under colimits (these conditions are presumed to be analogues of the surjectivity in the Milnor square)
- 3.
- the base-change transformation \(f^*p_* \to q_*g^*\) is invertible (an analogue of the Tor condition above)

This new theorem may be applied to the derived \(\infty\)-categories of left modules over rings, in which case the old theorem on Milnor squares is recovered. The virtues of the new formalism are twofold. First: the result now applies not only to \(K\)-theory but to any invariant of stable \(\infty\)-categories that sends exact sequences to exact triangles. Second: the result now can be applied in circumstances where one has a category resembling the derived category of modules over a ring. The examples of such categories considered here are the categories of perfect complexes in the derived category of an algebraic stack (with certain further good behaviour imposed: slightly weaker than affine diagonal and tameness in the sense of [D. Abramovich et al., Ann. Inst. Fourier 58, No. 4, 1057–1091 (2008; Zbl 1222.14004)]). The stacks in question are said to be “ANS” stacks.

With the application to stacks in mind, it seems, a variation on Theorem A, part (i) is also considered: Theorem A, part (ii) which is essentially the same as part (i), but applied to pro-objects.

The applications to stacks are two excision theorems: given a cartesian and cocartesian square of noetherian ANS stacks \[ \begin{tikzcd} Z' \rar \dar & X' \dar{f} \\ Z \rar{i} & X\end{tikzcd} \] in which \(i\) is a closed immersion, then there is an induced cartesian square of pro-spectra: \[ \begin{tikzcd} \{K(X)\} \rar \dar & \hat K(X^\wedge_Z) \dar \\ \{K(X')\} \rar & \hat K({X'}^\wedge_{Z'}) \end{tikzcd} \] where \(f\) is either an affine morphism (Theorem B) or a proper representable morphism that is an isomorphism away from \(Z\) (Theorem C). Here \(\hat K(X^\wedge_Z)\) denotes the formal completion of \(X\) along the closed substack \(Z\). In the case of Theorem B, if certain Tor-groups also vanish, the formal completions may be abandoned in favour of \(K(Z)\) and \(K(Z')\).

Theorem C allows the extension of the proof [M. Kerz et al., Invent. Math. 211, No. 2, 523–577 (2018; Zbl 1391.19007) ] of “Weibel’s conjecture” to finite dimensional ANS stacks. This says that the groups \(K_n(X)\) vanishes when \(n\) is less than the negative of the dimension ( [C. A. Weibel, Invent. Math. 61, 177–197 (1980; Zbl 0437.13009)] ). In these dimensions and also when \(n\) is equal to the negative of the dimension, the \(K\)-theory is homotopy invariant.

The main technical innovation in the proofs is an generalization of the \(\odot\) construction of [M. Land and G. Tamme, Ann. Math. (2) 190, No. 3, 877–930 (2019; Zbl 1427.19002)]. Here the construction is generalized from module categories to presentable stable \(\infty\)-categories.

Reviewer: Ben Williams (Vancouver)

### MSC:

19E08 | \(K\)-theory of schemes |

14A20 | Generalizations (algebraic spaces, stacks) |

14F42 | Motivic cohomology; motivic homotopy theory |

19D35 | Negative \(K\)-theory, NK and Nil |

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |

14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |

14D23 | Stacks and moduli problems |

### Citations:

Zbl 0237.18005; Zbl 1427.19002; Zbl 1107.19002; Zbl 1393.19003; Zbl 1395.18013; Zbl 0871.19002; Zbl 1222.14004; Zbl 1391.19007; Zbl 0437.13009
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\textit{T. Bachmann} et al., Sel. Math., New Ser. 28, No. 5, Paper No. 85, 72 p. (2022; Zbl 1504.19004)

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