Categorical Milnor squares and K-theory of algebraic stacks

Abstract

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the K-theory of algebraic stacks with affine diagonal and nice stabilizers. This yields a generalization of Weibel’s conjecture on the vanishing of negative K-groups for this class of stacks.

Appendices

Appendix A: Algebraic stacks

1.1 ANS stacks

Definition A.1.1

Let G be an affine fppf group scheme over an affine scheme S.

• (i) We say that G is linearly reductive if direct image along the morphism $BG \rightarrow S$ is t-exact (i.e., it is cohomologically affine).

• (ii) We say that G is nice if it is an extension of a finite étale group scheme, of order prime to the characteristics of S, by a group scheme of multiplicative type.

• (iii) We say that G is embeddable if it is a closed subgroup of $\mathrm {GL} _{S}(\mathcal {E})$ for some finite locally free sheaf $\mathcal {E}$ on S.

Nice group schemes are linearly reductive by [2, Rem. 2.2].

Definition A.1.2

A derived algebraic stack $\mathcal {X}$ is called ANS if it has affine diagonal and nice stabilizers.

Example A.1.3

In characteristic zero any reductive group G (such as $\mathrm {GL} _{n,S}$) is linearly reductive. In characteristic $p>0$, any linearly reductive group is nice [2, Thm. 18.9].

Example A.1.4

Let G be a finite étale group scheme over a field k. If G has order prime to the characteristic of k, then G is nice and embeddable. It follows that any separated Deligne–Mumford stack over k is ANS as long as it is tame (i.e., has all stabilizers of order prime to the characteristic).

Example A.1.5

Any algebraic stack with affine diagonal that is tame in the sense of [4, Def. 3.1] is ANS. This generalizes Example A.1.4.

Example A.1.6

Tori are embeddable group schemes of multiplicative type (hence nice). Thus if T is a torus over an affine scheme S acting on an algebraic space X over S with affine diagonal, then the quotient [X/T] is ANS. (However, it is typically not tame in the sense of [4].)

Lemma A.1.7

Let $\mathcal {X}$ be an ANS derived stack. Let $f : \mathcal {X} ' \rightarrow \mathcal {X}$ be a representable morphism with affine diagonal. Then $\mathcal {X} '$ is ANS.

Proof

Since f is representable, the stabilizers of $\mathcal {X} '$ are subgroups of those of $\mathcal {X}$. $\square$

The following is the main result of [1] in the classical case. The generalization to derived stacks is immediate.

Theorem A.1.8

(Alper–Hall–Halpern-Leistner–Rydh) Let $\mathcal {X}$ be an ANS derived stack. Then there exists a finite sequence of open immersions

$$\begin{aligned} \varnothing = \mathcal {U} _0 \hookrightarrow \mathcal {U} _1 \hookrightarrow \cdots \hookrightarrow \mathcal {U} _n = \mathcal {X}, \end{aligned}$$

an embeddable nice group scheme G over an affine scheme S, and Nisnevich squares

where $\mathcal {V} _i$ is étale and affine over $\mathcal {U} _i$ and quasi-affine over BG.

Proof

This follows by combining [1, Thm. 6.3]⁶ with [18, Prop. 2.9]. See [31, Thm. 2.12] for details. $\square$

Proposition A.1.9

Let $\mathcal {X} = [X/G]$ be the quotient of a quasi-compact separated derived algebraic space X with action of a nice group scheme G over an affine scheme S. Then $\mathcal {X}$ admits a scallop decomposition of the form $(\mathcal {U} _i, \mathcal {V} _i, u_i)_i$, where $\mathcal {V} _i$ is of the form $[V_i/G]$ for some affine derived schemes $V_i$ over S with G-action, and $u_i$ is an affine morphism for each i.

Proof

By generalized Sumihiro (see [31, Theorem 2.14]), $\mathcal {X}$ admits an affine Nisnevich cover $u: \mathcal {V} \twoheadrightarrow \mathcal {X}$ where $\mathcal {V}$ is of the form [V/G] with V an affine scheme over S with G-action. The desired scallop decomposition is obtained by a G-equivariant version of the construction in the proof of [43, Lem. 5.7.5] or [38, Prop. 3.2.2.4], which goes through mutatis mutandis:

For every $i\ge 0$, define $\mathcal {U} ^i \subseteq \mathcal {X}$ as the substack of points where the fibre of u has $\ge i$ geometric points. We have $\mathcal {U} ^1 = \mathcal {X}$ (since u is surjective) and $\mathcal {U} ^{n+1} = \varnothing$ for some large enough n (since $\mathcal {X}$ is quasi-compact). This gives a finite filtration of $\mathcal {X}$ by quasi-compact opens $\mathcal {U} _i := \mathcal {U} ^{n+1-i}$.

Consider the fibre powers $V^i$ of V over X and $\mathcal {V} ^i = [V^i/G]$ of $\mathcal {V}$ over $\mathcal {X}$, respectively. Since u is affine, so is each $V^i$. Since $V \rightarrow X$ is affine and étale, the “big diagonal” $\Delta ^i \subseteq V^i$ is an open and closed subscheme. The permutation action of the symmetric group $\Sigma _i$ on $V^i$ is free away from $\Delta ^i$, and commutes with the factorwise G-action on $V^i$. Thus we can write

$$\begin{aligned} \mathcal {V} _i := [(V^i \setminus \Delta ^i) / G \times \Sigma _i] \simeq [W_i/G], \end{aligned}$$

where $W_i = [(V^i \setminus \Delta ^i)/ \Sigma _i]$, as a quotient of an affine scheme by a free action of a finite group, is an affine scheme. Now one checks, exactly as in [38, Prop. 3.2.2.4], that the canonical morphisms $\mathcal {V} _i \rightarrow \mathcal {X}$ factor through affine étale morphisms $u_i : \mathcal {V} _i \rightarrow \mathcal {U} _i$, and that the resulting construction $(\mathcal {U} _i, \mathcal {V} _i, u_i)_i$ is indeed a scallop decomposition. $\square$

1.2 The resolution property

Definition A.2.1

Let $\mathcal {X}$ be a derived algebraic stack. We say that $\mathcal {X}$ has the resolution property if for every discrete coherent sheaf $\mathcal {F}$ of finite type on $\mathcal {X}$, there exists a finite locally free sheaf $\mathcal {E}$ and a surjection $\mathcal {E} \twoheadrightarrow \mathcal {F}$.

The following construction is one of the pleasant consequences of the resolution property.

Construction A.2.2

Let $i : \mathcal {Z} \rightarrow \mathcal {X}$ be a closed immersion of derived stacks. If i is almost of finite presentation (e.g. $\mathcal {X}$ is noetherian), then the ideal $\mathcal {I} \subseteq \pi _0(\mathcal {O} _\mathcal {X})$ defining $\mathcal {Z} _{\mathrm {cl}}$ in $\mathcal {X} _{\mathrm {cl}}$ is of finite type. Thus if $\mathcal {X}$ admits the resolution property, there exists a surjection $\mathcal {E} \twoheadrightarrow \mathcal {I}$ from a finite locally free sheaf $\mathcal {E}$ on $\mathcal {X}$. The induced morphism $s: \mathcal {E} \rightarrow \mathcal {I} \rightarrow \mathcal {O} _\mathcal {X}$ can be viewed as a section of the vector bundle

$$\begin{aligned} \mathbf {V} _\mathcal {X} (\mathcal {E}) = {\text {Spec}}_\mathcal {X} ({{\,\mathrm{{\text {Sym}}}\,}}_{\mathcal {O} _\mathcal {X}}(\mathcal {E})), \end{aligned}$$

and its derived zero locus defines a quasi-smooth closed immersion ${\widetilde{i}} : {\widetilde{\mathcal {Z}}} \rightarrow \mathcal {X}$ whose 0-truncation is $i : \mathcal {Z} \rightarrow \mathcal {X}$ and which fits in a homotopy cartesian square

Repeating this construction with the section $s^{\otimes n}$, for any $n> 0$, gives a tower of infinitesimal thickenings

$$\begin{aligned} \mathcal {Z} \hookrightarrow {\widetilde{\mathcal {Z}}} = {\widetilde{\mathcal {Z}}}(1) \hookrightarrow {\widetilde{\mathcal {Z}}}(2) \hookrightarrow \cdots . \end{aligned}$$

Remark A.2.3

If $\mathcal {X}$ is a derived stack with the resolution property, then any quasi-smooth closed immersion $i : \mathcal {Z} \rightarrow \mathcal {X}$ fits in a homotopy cartesian square

where $\mathcal {E}$ is a finite locally free sheaf on $\mathcal {X}$. This follows by a variant of the proof of [29, Prop. 2.3.8].

We discuss some examples of derived stacks with the resolution property. First, recall that in the classical setting, the property is stable under affine morphisms:

Lemma A.2.4

Let $f : \mathcal {X} \rightarrow \mathcal {Y}$ be a quasi-affine morphism of (classical) algebraic stacks. If $\mathcal {Y}$ has the resolution property, then so does $\mathcal {X}$.

Proof

See [22, Lem. 7.1] or [17, Prop. 1.8(v)]. $\square$

The classifying stack BG has the resolution property for embeddable linearly reductive group schemes G, so one finds that affine quotient stacks [X/G] also admit the resolution property (see [2, Rmk. 2.5]). We prove the following derived generalization of this statement:

Proposition A.2.5

Let S be an affine scheme, G an embeddable linearly reductive group scheme over S, and X a derived affine S-scheme with G-action. Then the derived stack $\mathcal {X} = [X/G]$ admits the resolution property.

The proof of Proposition A.2.5 will require the following lemma. We write ${\mathbf {D}}_{\mathrm {lfr}}(\mathcal {Y})$ for the full subcategory of ${\mathbf {D}}(\mathcal {Y})$ spanned by the finite locally free sheaves, for any derived algebraic stack $\mathcal {Y}$.

Lemma A.2.6

Let the notation be as in Proposition A.2.5. For any integer $n\ge 0$, let $i : \tau _{\le n}(\mathcal {X}) \rightarrow \mathcal {X}$ be the inclusion of the n-truncation. Then we have:

• (i) For any finite locally free $\mathcal {E} \in {\mathbf {D}}_{\mathrm {lfr}}(\mathcal {X})$ and any $\mathcal {F} \in {\mathbf {D}}(\mathcal {X})$, the canonical map of n-truncated spaces

  $$\begin{aligned} \tau _{\le n}{{\,\mathrm{{\text {Maps}}}\,}}(\mathcal {E}, \mathcal {F}) \rightarrow {{\,\mathrm{{\text {Maps}}}\,}}(\tau _{\le n}(\mathcal {E}), \tau _{\le n}(\mathcal {F})) \end{aligned}$$

  is invertible.

• (ii) The induced functor of $(n+1)$-categories

  $$\begin{aligned} i^* : \tau _{\le n+1}{\mathbf {D}}_{\mathrm {lfr}}(\mathcal {X}) \rightarrow {\mathbf {D}}_{\mathrm {lfr}}(\tau _{\le n}(\mathcal {X})) \end{aligned}$$

  is an equivalence.

In particular, the functor of ordinary categories

$$\begin{aligned} {\text {h}}{\mathbf {D}}_{\mathrm {lfr}}(\mathcal {X}) \rightarrow {\mathbf {D}}_{\mathrm {lfr}}(\mathcal {X} _{\mathrm {cl}}) \end{aligned}$$

is an equivalence (where ${\text {h}}$ denotes the homotopy category).

Proof

Note that the map in the first claim is induced by the morphism in ${\mathbf {D}}(\mathcal {X})$

$$\begin{aligned} \tau _{\le n}\underline{\smash {{{\,\mathrm{{\text {Hom}}}\,}}}}_{\mathcal {O} _\mathcal {X}}(\mathcal {E}, \mathcal {F}) \rightarrow i_*\underline{\smash {{{\,\mathrm{{\text {Hom}}}\,}}}}_{\mathcal {O} _{\tau _{\le n}(\mathcal {X})}}(\tau _{\le n}(\mathcal {E}), \tau _{\le n}(\mathcal {F})), \end{aligned}$$

(A.2.a)

where $\underline{\smash {{{\,\mathrm{{\text {Hom}}}\,}}}}$ denotes the internal Hom, by applying in succession the functors of direct image along $f : \mathcal {X} \rightarrow BG$ (which is t-exact since f is affine), direct image along $BG \rightarrow S$ (which is t-exact since G is linearly reductive), and (derived) global sections (which is t-exact since S is affine). Therefore it will suffice to show that (A.2.a) is invertible. By fpqc descent, this can be checked after inverse image along the smooth surjection $X \rightarrow \mathcal {X}$. Since i is representable, $i_*$ satisfies base change and we are thus reduced to the affine case, which is well-known (see e.g. [24, Claim 4.3]).

Consider now claim (ii). By (i) the functor in question is fully faithful (on finite locally frees). For essentially surjectivity, we may assume $n=0$ (so that $\tau _{\le n}(\mathcal {X}) = \mathcal {X} _{\mathrm {cl}}$). Since BG has the resolution property [2, Rmk. 2.5], Lemma A.2.4 implies that for every finite locally free sheaf $\mathcal {E} \in {\mathbf {D}}(\mathcal {X} _{\mathrm {cl}})$, there exists a finite locally free sheaf $\mathcal {F} \in {\mathbf {D}}(BG)$ and a surjection $g^*(\mathcal {F}) \twoheadrightarrow \mathcal {E}$ where $g : \mathcal {X} _{\mathrm {cl}}\rightarrow BG$. Certainly $g^* (\mathcal {F}) \in {\mathbf {D}}(\mathcal {X} _{\mathrm {cl}})$ lifts to $f^*(\mathcal {F}) \in {\mathbf {D}}(\mathcal {X})$, so we are reduced to show that if $\mathcal {E} \twoheadrightarrow \mathcal {F}$ is surjection of locally free sheaves on $\mathcal {X} _{{\mathrm {cl}}}$ and $\mathcal {E} = i^*{\widetilde{\mathcal {E}}}$ for some locally free ${\widetilde{\mathcal {E}}} \in {\mathbf {D}}(\mathcal {X})$, then $\mathcal {F}$ also extends to a locally free sheaf ${\widetilde{\mathcal {F}}}$ on $\mathcal {X}$. Since G is linearly reductive, $\mathcal {F}$ is projective by [20, Lem. 2.17] and the surjection $\mathcal {E} \twoheadrightarrow \mathcal {F}$ splits. The resulting map $e_0 : \mathcal {E} \twoheadrightarrow \mathcal {F} \rightarrow \mathcal {E}$ is an idempotent with image $\mathcal {F}$. By claim (i), we can extend $e_0$ to an idempotent endomorphism e of ${\widetilde{\mathcal {E}}}$. If we set

$$\begin{aligned} {\widetilde{\mathcal {F}}}&:= \varinjlim ({\widetilde{\mathcal {E}}} \xrightarrow {e} {\widetilde{\mathcal {E}}} \xrightarrow {e} \cdots ),\\ {\widetilde{\mathcal {F}}}_1&:= \varinjlim ({\widetilde{\mathcal {E}}} \xrightarrow {\mathrm {id}-e} {\widetilde{\mathcal {E}}} \xrightarrow {\mathrm {id}-e} \cdots ), \end{aligned}$$

then the induced morphism $\phi : {\widetilde{\mathcal {E}}} \rightarrow {\widetilde{\mathcal {F}}} \oplus {\widetilde{\mathcal {F}}}_1$ is invertible. Indeed, by fpqc descent we may pull back to X and thereby assume $\mathcal {X}$ is affine, in which case it is clear that $\phi$ is an isomorphism on homotopy groups. In particular, it follows that ${\widetilde{\mathcal {F}}}$ is locally free, and clearly $i^* ({\widetilde{\mathcal {F}}}) \simeq \mathcal {F}$ by construction. $\square$

Proof of Proposition A.2.5

By [2, Rmk. 2.5], the classical truncation $[X_{\mathrm {cl}}/G]$ has the resolution property. Hence the claim follows from Lemma A.2.6. $\square$

1.3 Compact generation

Definition A.3.1

A derived algebraic stack $\mathcal {X}$ is perfect if the stable $\infty$-category ${\mathbf {D}}(\mathcal {X})$ is compactly generated by its full subcategory ${\text {Perf}}(\mathcal {X})$ of perfect complexes.

In this subsection we prove the following result, which is a derived generalization of [2, Prop. 14.1].

Theorem A.3.2

Let $\mathcal {X}$ be an ANS derived stack. Then $\mathcal {X}$ is perfect.

Lemma A.3.3

Let G be an embeddable linearly reductive group scheme over an affine scheme S. Then the classifying stack BG is perfect. Moreover, ${\mathbf {D}}(BG)$ is compactly generated by finite representations of G (i.e., finite projectives).

Proof

Since G is affine, BG has affine diagonal. By Proposition A.2.5, BG has the resolution property. Hence the claim follows from [22, Prop. 8.4]. $\square$

The following shows that, for nice enough quotients of affine derived schemes, the derived $\infty$-category is not only compactly generated, but projectively generated in the sense of Definition 3.5.6.

Proposition A.3.4

Let G be an embeddable linearly reductive group scheme over an affine scheme S. Then for any affine derived scheme X with G-action, the quotient stack [X/G] is perfect and even crisp in the sense of [22]. Moreover, if $p : [X/G] \rightarrow BG$ is the projection, then the collection of objects $\{p^*(\mathcal {E})\}$ forms a small set of compact projective generators for ${\mathbf {D}}([X/G])_{\ge 0}$, as $\mathcal {E} \in {\mathbf {D}}(BG)$ varies over finite locally free G-modules on S.

Proof

Since p is affine, the functor $p^* : {\mathbf {D}}(BG) \rightarrow {\mathbf {D}}([X/G])$ is compact and generates its codomain under colimits (2.2.1). This already implies that ${\mathbf {D}}([X/G])$ is compactly generated by the objects of the stated form.

We now demonstrate the stronger property of crispness. By Proposition A.2.5, [X/G] admits the resolution property. Since [X/G] has affine diagonal, [22, Prop. 8.4] then implies that [X/G] is crisp. Note that loc. cit. only discusses classical stacks, but we only need the argument from the last paragraph of the proof, which immediately generalizes to the derived setting.

Finally let us show that the object $p^*(\mathcal {E}) \in {\mathbf {D}}([X/G])_{\ge 0}$ is projective for every finite representation $\mathcal {E} \in {\mathbf {D}}(BG)$. By [37, Lem. 7.2.2.6] it will suffice to show that the functor ${\mathcal {M}} aps (p^*(\mathcal {E}), -) : {\mathbf {D}}(BG) \rightarrow \mathrm {Spt}$ is t-exact, where ${\mathcal {M}} aps (-,-)$ denotes the mapping spectrum functor in the stable $\infty$-category ${\mathbf {D}}(BG)$. We have a canonical isomorphism

$$\begin{aligned} {\mathcal {M}} aps (p^*(\mathcal {E}), -) \simeq \Gamma \left( S, (p_*\underline{\smash {{{\,\mathrm{{\text {Hom}}}\,}}}}(\mathcal {E}, -))^G\right) . \end{aligned}$$

Note that $\underline{\smash {{{\,\mathrm{{\text {Hom}}}\,}}}}(\mathcal {E}, -)$ is t-exact because $\mathcal {E}$ is projective in ${\mathbf {D}}(S)$, $p_*$ is t-exact since p is affine, the G-invariants functor $(-)^G$ is identified with the direct image functor ${\mathbf {D}}(BG) \rightarrow {\mathbf {D}}(S)$ and hence is t-exact because G is linearly reductive, and the (derived) global sections functor $\Gamma (S, -)$ is t-exact since S is affine. $\square$

Proof of Theorem A.3.2

By Theorem A.1.8, we can in particular find an affine étale surjection onto $\mathcal {X}$ from a finite coproduct of quotient stacks of the form [X/G], where G is a nice group scheme over an affine scheme S and X is an affine derived S-scheme with G-action. It will now suffice to show that the property of crispness can be detected by affine étale surjections. Indeed, we note that the proof given in [22, Thm. C] for the classical case generalizes to our setting, following Example 9.4 of loc. cit. $\square$

1.4 Dimension of algebraic stacks

Recall several useful notions of dimension from [18].

Definition A.4.1

Let $\mathcal {X}$ be a noetherian stack.

• (i) The Krull dimension $\dim (\mathcal {X})$ of $\mathcal {X}$ is the dimension of the underlying topological space $|\mathcal {X} |$.

• (ii) The blow-up dimension ${{\,\mathrm{{\text {bl}}{\text {dim}}}\,}}(\mathcal {X})$ is the supremum over integers $n\ge 0$ for which there exists a sequence of maps

  $$\begin{aligned} \mathcal {X} _n \rightarrow \cdots \rightarrow \mathcal {X} _0 = \mathcal {X} \end{aligned}$$

  such that $\mathcal {X} _i$ is a nonempty nowhere dense closed substack in an iterated blow-up of $\mathcal {X} _{i-1}$ for all $i>0$. If $\mathcal {X}$ is empty, then ${{\,\mathrm{{\text {bl}}{\text {dim}}}\,}}(\mathcal {X}) = -1$ by convention.

• (iii) The fppf-covering dimension ${{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {fppf}}}\,}}(\mathcal {X})$ is the minimal integer $-1 \le n \le \infty$ such that there exists an fppf morphism $X \rightarrow \mathcal {X}$ where X is a noetherian scheme of Krull dimension n.

• (iv) The covering dimension ${{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {sm}}}\,}}(\mathcal {X})$ (more precisely, smooth-covering dimension) is the minimal integer $-1 \le n \le \infty$ such that there exists a smooth surjection $X \rightarrow \mathcal {X}$ where X is a noetherian scheme of Krull dimension n.

Remark A.4.2

In general, one has the inequalities $\dim (\mathcal {X}) \le {{\,\mathrm{{\text {bl}}{\text {dim}}}\,}}(\mathcal {X}) \le {{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {fppf}}}\,}}(\mathcal {X}) \le {{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {sm}}}\,}}(\mathcal {X})$. For quasi-Deligne–Mumford stacks these are all equal to the usual dimension as defined in [51, Tag 0AFL]. See [18, Lemma 7.8].

Example A.4.3

Let G be an fppf group scheme over an algebraic space S. If G acts on an algebraic space X over S, then the quotient stack $\mathcal {X} = [X/G]$ is of fppf-covering dimension $\le \dim (X)$.

In this subsection we will show that the Nisnevich cohomological dimension of an algebraic stack $\mathcal {X}$, which we denote by ${\text {cd}}_{\mathrm {Nis}}(\mathcal {X})$, is bounded by its fppf-covering dimension:

Proposition A.4.4

Let $\mathcal {X}$ be a noetherian stack and let $\mathcal {F}$ be a sheaf of abelian groups on the Nisnevich site of $\mathcal {X}$. Then

$$\begin{aligned} {\text {H}}^i_{\mathrm {Nis}}(\mathcal {X}, \mathcal {F}) = 0 \end{aligned}$$

for all $i > {{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {fppf}}}\,}}(\mathcal {X})$. In other words, ${\text {cd}}_{\mathrm {Nis}} (\mathcal {X}) \le {{\,\mathrm{{\text {cov}}{\text {dim}}_{\mathrm {fppf}}}\,}}(\mathcal {X})$.

Recall that the Nisnevich topology on the category of noetherian algebraic stacks is generated by a cd-structure [18, Sect. 2C], which is clearly complete and regular. To establish Proposition A.4.4 we will show that this cd-structure is bounded with respect to a density structure.

Construction A.4.5

For any noetherian algebraic stack $\mathcal {X}$, let $\mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$ be the category of algebraic stacks $\mathcal {Y}$ over $\mathcal {X}$ for which the structural morphism $\mathcal {Y} \rightarrow \mathcal {X}$ is representable by Deligne–Mumford stacks. Choose an fppf covering $\mathcal {S} \rightarrow \mathcal {X}$ from a Deligne–Mumford stack $\mathcal {S}$. We define a density structure $D^{\mathcal {S}}_i(-)$ in the sense of [54, Definition 2.20] on the category $\mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$. For $\mathcal {Y} \in \mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$, let $q: \mathcal {S} _{\mathcal {Y}} \rightarrow \mathcal {Y}$ be the fppf covering of $\mathcal {Y}$ given by the base change of $\mathcal {S} \rightarrow \mathcal {X}$ along $\mathcal {Y}$. For $i \ge 0$, let $D^{\mathcal {S}}_i(\mathcal {Y})$ denote the class of open substacks $\mathcal {U} \hookrightarrow \mathcal {Y}$ such that the open substack $\mathcal {U} \times _{\mathcal {Y}} \mathcal {S} _{\mathcal {Y}} \hookrightarrow \mathcal {S} _{\mathcal {Y}}$ defines an element of the class $D_i(\mathcal {S} _{\mathcal {Y}})$, where $D_*(-)$ denotes the density structure on the category of Deligne–Mumford stacks defined in [28, Definition 4.4]. It is easy to check that the classes $D^{\mathcal {S}}_i(-)$ define a density structure which is locally of finite dimension and the dimension of $\mathcal {X}$ with respect to the density structure is equal to the Krull dimension $\dim (\mathcal {S})$.

Recall that for a Deligne–Mumford stack $\mathcal {X}$ and $i \ge 0$, $D_i(\mathcal {X})$ is the collection of open substacks $\mathcal {U} \hookrightarrow \mathcal {X}$ such that for every irreducible, reduced closed substack $\mathcal {Z}$ of $\mathcal {X}$ with $\mathcal {Z} \times _{\mathcal {X}} \mathcal {U}$ the empty stack, there exists a sequence $\mathcal {Z} = \mathcal {Z} _0 \subsetneq \mathcal {Z} _1 \subsetneq \cdots \subsetneq \mathcal {Z} _i$ of irreducible, reduced closed substacks of $\mathcal {X}$.

Lemma A.4.6

Let $\mathcal {X}$ be a Deligne–Mumford stack and $\mathcal {U} \in D_i(\mathcal {X})$. For any open substack $\mathcal {V}$ of $\mathcal {X}$, we have $\mathcal {U} \cap \mathcal {V} \in D_i(\mathcal {V})$.

Proof

By [28, Lemma 4.5] it suffices to show that $|\mathcal {U} \cap \mathcal {V} | \in D_i(|\mathcal {V} |)$. Now we can use the same argument as in the proof of [55, Lemma 2.5]. $\square$

Proposition A.4.7

The Nisnevich cd-structure on the category $\mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$ is bounded with respect to the density structure $D^{\mathcal {S}}_*(-)$.

Proof

We need to show that every Nisnevich square is reducing with respect to the density structure. Consider a Nisnevich square Q in $\mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$:

(A.4.a)

where p is étale, j is an open immersion and the induced morphism $p^{-1} (\mathcal {Y} \setminus \mathcal {U}) \rightarrow (\mathcal {Y} \setminus \mathcal {U})$ is invertible. Choose $\mathcal {W} _0 \in D^S_{i-1}(\mathcal {W})$, $\mathcal {U} _0 \in D^S_i(\mathcal {U})$ and $\mathcal {V} _0 \in D^S_i(\mathcal {V})$. To show that the above square Q is reducing with respect to the density structure, we need to prove that there exists a Nisnevich square $Q'$ in $\mathrm {Stk} ^{\mathrm {DM}}_{/\mathcal {X}}$

(A.4.b)

and a morphism $Q' \rightarrow Q$ such that $\mathcal {W} ' \rightarrow \mathcal {W}$ factors through $\mathcal {W} ' \rightarrow \mathcal {W} _0$, $\mathcal {U} ' \rightarrow \mathcal {U}$ through $\mathcal {U} ' \rightarrow \mathcal {U} _0$, $\mathcal {V} ' \rightarrow \mathcal {V}$ through $\mathcal {V} ' \rightarrow \mathcal {V} _0$, and $\mathcal {Y} ' \in D^S_i(\mathcal {Y})$ (see [54, Definition 2.21]). Applying Lemma A.4.8 to the morphism $j \coprod p$, we can find $\mathcal {Y} _0 \in D^{\mathcal {S}}_i(\mathcal {Y})$ such that $j^{-1}(\mathcal {Y} _0) \subseteq \mathcal {U} _0$ and $p^{-1}(\mathcal {Y} _0) \subseteq \mathcal {V} _0$. Therefore by base changing (A.4.a) along $\mathcal {Y} _0 \hookrightarrow \mathcal {Y}$ and then replacing $\mathcal {Y}$ by $\mathcal {Y} _0$ we are reduced to the case when $\mathcal {U} = \mathcal {U} _0$ and $\mathcal {V} = \mathcal {V} _0$ in (A.4.a). Note that $\mathcal {W} _0\times _{\mathcal {Y}} \mathcal {Y} _0$ is in $D_{i-1}^{\mathcal {S}}(\mathcal {U} _0 \times _{\mathcal {Y}} \mathcal {V} _0)$ by Lemma A.4.6.

Let $\mathcal {Z} = \mathcal {W} \setminus \mathcal {W} _0$ and $\mathcal {C} = \mathcal {Y} \setminus \mathcal {U}$ and set $\mathcal {W} ' = \mathcal {W} _0$, $\mathcal {U} ' = \mathcal {U}$, $\mathcal {V} ' = \mathcal {V} \setminus {\mathrm {cl}}_{\mathcal {V}}(\mathcal {Z})$ and $\mathcal {Y} ' = \mathcal {Y} \setminus (\mathcal {C} \cap {\mathrm {cl}}_{\mathcal {Y}}(p\circ j_V(\mathcal {Z})))$ in (A.4.b) to obtain the Nisnevich square $Q'$ with a natural morphism to Q given by inclusions. Now $\mathcal {Y} ' \times _{\mathcal {Y}} \mathcal {S} _{\mathcal {Y}} \rightarrow \mathcal {S} _{\mathcal {Y}} \in D_i(\mathcal {S} _{\mathcal {Y}})$ by the proof of [28, Proposition 4.9]. Therefore $Q' \rightarrow Q$ satisfies all the required properties. $\square$

Lemma A.4.8

Let $f: \mathcal {W} \rightarrow \mathcal {Y}$ be an étale surjection of noetherian stacks. Then for any $i \ge 0$ and $\mathcal {W} _0 \in D^{\mathcal {S}}_i(\mathcal {W})$ there exists $\mathcal {Y} _0 \in D^{\mathcal {S}}_i(\mathcal {Y})$ such that $f^{-1}(\mathcal {Y} _0) \subset \mathcal {W} _0$.

Proof

Let $\mathcal {S} _{\mathcal {W}} = \mathcal {S} \times _{\mathcal {X}} \mathcal {W}$, $\mathcal {S} _{\mathcal {W} _0} = \mathcal {S} \times _{\mathcal {X}} \mathcal {W} _0$ and let $f_S: \mathcal {S} _{\mathcal {W}} \rightarrow \mathcal {S} _{\mathcal {Y}}$ denote the base change of $f: \mathcal {W} \rightarrow \mathcal {Y}$ along the fppf covering $q: \mathcal {S} _{\mathcal {Y}} \rightarrow \mathcal {Y}$ and ${\widetilde{q}}: \mathcal {S} _{\mathcal {W}} \rightarrow \mathcal {W}$ denote the base change of q along f. Then by [28, Lemma 4.7] applied to $f_S$ there exists ${\widetilde{\mathcal {Y}}} \in D_i(\mathcal {S} _{\mathcal {Y}})$ such that $f_S^{-1}({\widetilde{\mathcal {Y}}}) \subset \mathcal {S} _{\mathcal {W} _0}$. Let $\mathcal {Y} _0 = q({\widetilde{\mathcal {Y}}})$, then $\mathcal {Y} _0 \in D_i(\mathcal {Y})$ since q is open (see [34, Proposition 5.6]), ${\widetilde{\mathcal {Y}}} \subseteq \mathcal {Y} _0 \times _{\mathcal {Y}} \mathcal {S} _{\mathcal {Y}}$ and ${\widetilde{\mathcal {Y}}} \in D_i(\mathcal {S} _{\mathcal {Y}})$. Moreover $f^{-1}(\mathcal {Y} _0) = f^{-1}(q({\widetilde{\mathcal {Y}}})) \subseteq {\widetilde{q}}(f_S^{-1}({\widetilde{\mathcal {Y}}})) \subseteq {\widetilde{q}}(\mathcal {S} _{\mathcal {W} _0}) = \mathcal {W} _0$. $\square$

Appendix B: Formal stacks

1.1 Formal completion

We briefly review some elements of formal derived algebraic geometry. Good references are [19, Sect. 2.1] and [15]. The following definition is [19, Def. 2.1.1].

Definition B.1.1

(Formal completion) Let $i : \mathcal {Z} \rightarrow \mathcal {X}$ be a closed immersion of derived stacks with quasi-compact open complement. The formal completion of $\mathcal {X}$ in $\mathcal {Z}$ is the derived prestack $\mathcal {X} ^\wedge _\mathcal {Z}$ whose R-points, for any derived commutative ring R, are the R-points $x : {\text {Spec}}(R) \rightarrow \mathcal {X}$ which factor set-theoretically through the underlying topological space $|\mathcal {Z} | \subseteq |\mathcal {X} |$. By definition, $i : \mathcal {Z} \rightarrow \mathcal {X}$ factors as

$$\begin{aligned} i : \mathcal {Z} \rightarrow \mathcal {X} ^\wedge _\mathcal {Z} \xrightarrow {{\hat{i}}} \mathcal {X}, \end{aligned}$$

where the first arrow induces an isomorphism on reductions, and the second arrow is a monomorphism. When there is no risk of ambiguity, we will write simply $\mathcal {X} ^\wedge$ for $\mathcal {X} ^\wedge _\mathcal {Z}$.

Remark B.1.2

Note that the formal completion $\mathcal {X} ^\wedge _\mathcal {Z}$ only depends on the underlying topological space $|\mathcal {Z} |$. In particular, if we have a commutative square

where $|\mathcal {Z} '|$ is the set-theoretic inverse image $f^{-1}(|\mathcal {Z} |)$, then the formal completion $\mathcal {X} '^\wedge$ is the derived base change of $\mathcal {X} ^\wedge$. That is, we have a homotopy cartesian square

Remark B.1.3

The formal completion of a derived algebraic stack $\mathcal {X}$ along a closed immersion $i : \mathcal {Z} \rightarrow \mathcal {X}$ is always an ind-algebraic stack. More precisely, one has the following canonical isomorphism (see [15, Prop. 6.5.5]):

$$\begin{aligned} \{{\widetilde{\mathcal {Z}}}\}_{{\widetilde{\mathcal {Z}}} \rightarrow \mathcal {X}} \rightarrow \mathcal {X} ^\wedge _\mathcal {Z}, \end{aligned}$$

where the source is the ind-system indexed by the filtered $\infty$-category of closed immersions ${\widetilde{\mathcal {Z}}} \rightarrow \mathcal {X}$ that induce an isomorphism ${\widetilde{\mathcal {Z}}}_\mathrm {red} \simeq \mathcal {Z} _\mathrm {red}$ on reductions. Note that the transition morphisms are surjective closed immersions.

In the affine case, we can give the following more familiar (but less canonical) description.

Example B.1.4

Let A be a derived commutative ring and $I \subseteq \pi _0(A)$ an ideal, and consider the formal completion of ${\text {Spec}}(A)^\wedge$ in the vanishing locus of I. Choosing generators $f_1,\ldots ,f_m$ for the ideal I, there is an equivalence

$$\begin{aligned} {\text {Spec}}(A)^\wedge \simeq \{{\text {Spec}}(A/\!\!/(f_1^n,\ldots ,f_m^n))\}_n. \end{aligned}$$

See [38, Proof of Prop. 8.1.2.1] or [19, Prop. 2.1.2].

If A is an ordinary commutative ring and is noetherian, then we recover the classical formal completion, i.e., the formal spectrum of A as an I-adic ring:

$$\begin{aligned} {\text {Spec}}(A)^\wedge \simeq \{{\text {Spec}}(A/(f_1^n,\ldots ,f_m^n))\}_n. \end{aligned}$$

See [38, Lem. 17.3.5.7], [19, Prop. 2.1.4], or [15, Proof of Prop. 6.8.2].

More generally in the presence of the resolution property, we have:

Lemma B.1.5

Let $i : \mathcal {Z} \rightarrow \mathcal {X}$ be a closed immersion almost of finite presentation between derived algebraic stacks for which $\mathcal {X}$ has the resolution property. Let ${\widetilde{\mathcal {Z}}}(n)$ be as in Construction A.2.2. Then there is an isomorphism of ind-stacks

$$\begin{aligned} \mathcal {X} ^\wedge _\mathcal {Z} \simeq \{{\widetilde{\mathcal {Z}}}(n)\}. \end{aligned}$$

Proof

This is a simple cofinality argument using the description of the formal completion given in Remark B.1.3. $\square$

Remark B.1.6

If $\mathcal {X}$ is a classical stack, then the formal completion (in any closed substack) in the sense of Definition B.1.1 coincides with the classical formal completion, as long as $\mathcal {X}$ is noetherian. This follows from Example B.1.4, see [19, Cor. 2.1.5].

For the reader’s convenience, we spell out Lemma B.1.5 in the equivariant (quotient stack) case.

Construction B.1.7

Let G be a group scheme over a commutative ring R, and let A be a derived commutative R-algebra with G-action. Let M be a locally free G-equivariant A-module and $s : M \rightarrow A$ a homomorphism of G-equivariant A-modules. The derived quotient of A by s is the G-equivariant A-algebra formed by attaching a cell $s \simeq 0$, i.e., by the cocartesian square in G-equivariant derived commutative rings

where the map 0 is induced by adjunction from the zero map $M \rightarrow A$, and similarly for s.

Similarly, for any $n>0$, we also write $A/\!\!/s^n$ for the same construction where s is replaced by $s^{\otimes n} : M^{\otimes n} \rightarrow A^{\otimes n} \simeq A$ (the n-fold derived tensor product taken over A).

Lemma B.1.8

Let G be a group scheme over a commutative ring R and $A \twoheadrightarrow B$ a homomorphism of G-equivariant derived commutative R-algebras which is surjective on $\pi _0$. Assume that the quotient stack $[{\text {Spec}}(\pi _0(A))/G]$ admits the resolution property, so that there exists a locally free G-equivariant A-module M and $s : M \rightarrow A$ whose image on $\pi _0$ is equal to the kernel of $\pi _0(A) \twoheadrightarrow \pi _0(B)$. Then we have the following presentation of the formal completion:

$$\begin{aligned}{}[{\text {Spec}}(A)/G]^\wedge \simeq \{ [{\text {Spec}}(A/\!\!/s^n)/G] \}_{n>0}. \end{aligned}$$

Lemma B.1.9

Let G be a group scheme over a commutative ring R and $A \twoheadrightarrow B$ a surjective homomorphism of G-equivariant commutative R-algebras with kernel I. Assume that the quotient stack $[{\text {Spec}}(A)/G]$ admits the resolution property, so that there exists a locally free G-equivariant A-module M and $s : M \rightarrow A$ whose image is I. Then we have the following presentation of the formal completion:

$$\begin{aligned}{}[{\text {Spec}}(A)/G]^\wedge \simeq \{ [{\text {Spec}}(A/I^n)/G] \}_{n>0}. \end{aligned}$$

1.2 Quasi-coherent sheaves

For formal stacks (or more generally ind-stacks) such as the formal completion, there is a natural pro-$\infty$-categorical refinement of the stable $\infty$-category of quasi-coherent sheaves.

Construction B.2.1

If $\mathcal {C}$ denotes the $\infty$-category of derived algebraic stacks and representable morphisms, then by Remark 2.2.1 the assignment $\mathcal {X} \mapsto {\mathbf {D}}(\mathcal {X})$ can be regarded as a functor $\mathcal {C} ^\mathrm {op}\rightarrow \mathrm {Pres} _\mathrm {c}$ valued in the $\infty$-category of presentable $\infty$-categories and compact colimit-preserving functors. Now consider its Ind-extension

$$\begin{aligned} {\text {Ind}}(\mathcal {C})^\mathrm {op}\simeq {\text {Pro}}(\mathcal {C} ^\mathrm {op}) \rightarrow {\text {Pro}}(\mathrm {Pres} _\mathrm {c}). \end{aligned}$$

Any derived ind-algebraic stack $\mathcal {X}$ can be regarded an ind-object in $\mathcal {C}$ and hence gives rise to a canonical pro-$\infty$-category which we denote $\widehat{\mathbf {D}}(\mathcal {X})$.

Example B.2.2

Let $\mathcal {X}$ be a derived algebraic stack. Then $\widehat{\mathbf {D}}(\mathcal {X})$ is the constant pro-$\infty$-category $\{{\mathbf {D}}(\mathcal {X})\}$.

Example B.2.3

If $\mathcal {X}$ is a derived ind-algebraic stack represented by a filtered system $\{\mathcal {X} _n\}_n$, then $\widehat{\mathbf {D}}(\mathcal {X})$ is represented by the cofiltered system $\{ {\mathbf {D}}(\mathcal {X} _n) \}_n$.

Example B.2.4

Let A be a derived commutative ring and $I \subseteq \pi _0(A)$ an ideal, and consider the formal completion ${\text {Spec}}(A)^\wedge$ in the vanishing locus of I. Then choosing generators $f_1,\ldots ,f_m$ for the ideal I, we have an equivalence

$$\begin{aligned} \widehat{\mathbf {D}}({\text {Spec}}(A)^\wedge ) \simeq \{{\mathbf {D}}(A/\!\!/(f_1^n,\ldots ,f_m^n))\}_n \end{aligned}$$

by Example B.1.4. If A is a noetherian commutative ring, then we have also

$$\begin{aligned} \widehat{\mathbf {D}}({\text {Spec}}(A)^\wedge ) \simeq \{{\mathbf {D}}(A/(f_1^n,\ldots ,f_m^n))\}_n \end{aligned}$$

again by Example B.1.4.

Appendix C: Weak pro-Milnor squares

In this section we continue to use the language of weight structures as in Sect. 5. For convenience, the term weighted $\infty$-category will refer to an essentially small stable $\infty$-category with a bounded weight structure. We write $\mathcal {C} ^{w=0}$ for the weight-heart of a weighted $\infty$-category $\mathcal {C}$.

1.1 Connected invariants

Definition C.1.1

• (i) We say that a weight-exact functor $f : \mathcal {C} \rightarrow \mathcal {D}$ between weighted $\infty$-categories is thickly surjective if every object $Y \in \mathcal {D} ^{w=0}$ is a direct summand of f(X) for some object $X \in \mathcal {C} ^{w=0}$. In other words, if the induced functor $f^*$ on Ind-completions generates its codomain under colimits.

• (ii) Let $\mathcal {C}$ and $\mathcal {D}$ be weighted $\infty$-categories and $k\ge 0$ an integer. A thickly surjective weight-exact functor $f : \mathcal {C} \rightarrow \mathcal {D}$ is k-connective if it induces k-connective maps

  $$\begin{aligned} {{\,\mathrm{{\text {Maps}}}\,}}_{\mathcal {C} ^{w=0}}(X,Y) \rightarrow {{\,\mathrm{{\text {Maps}}}\,}}_{\mathcal {D} ^{w=0}}(f(X),f(Y)) \end{aligned}$$

  for all X and Y in $\mathcal {C} ^{w=0}$. In other words, if the induced functor of $(k+1)$-categories

  $$\begin{aligned} \tau _{\le k+1}(\mathcal {C}) \rightarrow \tau _{\le k+1}(\mathcal {D}) \end{aligned}$$

  is fully faithful (and hence an equivalence) when restricted to the weight-hearts.

Example C.1.2

If $A \rightarrow B$ is a k-connective map of connective ${\mathcal {E} _\infty }$-rings, then the induced functor ${\text {Perf}}_A \rightarrow {\text {Perf}}_B$ is k-connective with respect to the weight structures of Example 5.1.1.

The following definition is a variant of [35, Def. 2.5].

Definition C.1.3

Let E be a spectrum-valued functor on the $\infty$-category of small stable $\infty$-categories. We say that E is connected if for any k-connective functor $\mathcal {C} \rightarrow \mathcal {D}$ of weighted $\infty$-categories, the induced map of spectra $E(\mathcal {C}) \rightarrow E(\mathcal {D})$ is $(k+1)$-connective.

Example C.1.4

Connective K-theory is an example of a connected invariant. This follows from [13, Cor. 5.16] and the fact that plus-construction sends k-connected maps of spaces into $k+1$-connected maps (cf. [35, Lem. 2.4]). Moreover, nonconnective K-theory is also an example by [46, Thm. 4.3].

Remark C.1.5

Let E be a localizing invariant. If E is connected, then it is 1-connective in the sense of [35, Def. 2.5]. The converse holds if E commutes with filtered colimits. If E does not commute with filtered colimits, one can still show a weaker statement: if E is 2-connective in the sense of [35], then it is connected. This follows from the fact that any weighted $\infty$-category can be functorially realized as the kernel of a weight-exact localization functor ${\text {Perf}}_{A(\mathcal {C})} \rightarrow {\text {Perf}}_{B(\mathcal {C})}$ for some map of connective $\mathcal {E} _1$-rings $A(\mathcal {C}) \rightarrow B(\mathcal {C})$; see the proof of Lemma 3.5.5. (See [7, Sect. 8.1] or [8] for the theory of weight-exact localizations.) Any n-connective functor $\mathcal {C} \rightarrow \mathcal {D}$ clearly induces an n-connective map on $A(\mathcal {C}) \rightarrow A(\mathcal {D})$, and it also induces an n-connective map $B(\mathcal {C}) \rightarrow B(\mathcal {D})$ by the universal properties of localization and of truncation.

1.2 Weak pro-Milnor squares

The starting point for our definition of weak pro-Milnor squares is the following classical definition (see e.g. [3, Sect. 4]):

Definition C.2.1

Let $\{f_n : X_n \rightarrow Y_n\}_n$ be a morphism of cofiltered systems of spectra. We say that f is pro-k-connective if the induced map

$$\begin{aligned} \{\tau _{\le k}(X_n)\}_n \rightarrow \{\tau _{\le k}(Y_n)\}_n \end{aligned}$$

is an isomorphism in ${\text {Pro}}(\mathrm {Spt})$ for every k. It is a weak pro-equivalence if it is pro-k-connective for all k.

Note that the same definition makes sense for the $\infty$-category of $\mathcal {E} _1$-rings, for instance, in place of $\mathrm {Spt}$. The following can be viewed as a many-object generalization (see Example 5.1.1). A weighted pro-$\infty$-category is a pro-object in the $\infty$-category of weighted $\infty$-categories and weight-exact functors.

Definition C.2.2

Let $\{f_n : \mathcal {C} _n \rightarrow \mathcal {D} _n\}_n$ be a cofiltered system of weight-exact functors between weighted $\infty$-categories and $k\ge 0$ an integer. We say that it is pro-k-connective if the induced morphism of pro-$\infty$-categories

$$\begin{aligned} \{\tau _{\le k+1}(\mathcal {C} _{n}^{w=0})\}_n \rightarrow \{\tau _{\le k+1}(\mathcal {D} _{n}^{w=0})\}_n \end{aligned}$$

is invertible. If it is k-connective for all k, then it is called a weak pro-equivalence of weighted pro-$\infty$-categories.

This enables us to define weak pro-Milnor squares of weighted $\infty$-categories.

Definition C.2.3

Let $\{\Delta _n\}_n$ be a cofiltered system of commutative squares of weighted $\infty$-categories and weight-exact functors of the form

For every n, let $\mathcal {A} _n^+ \subseteq \mathcal {A} '_n \times _{\mathcal {B} '_n} \mathcal {B} _n$ denote the full subcategory of the pullback (taken in the $\infty$-category of weighted $\infty$-categories) generated under finite colimits, finite limits, and retracts by the essential image of $\mathcal {A} _n \rightarrow \mathcal {A} '_n \times _{\mathcal {B} '_n} \mathcal {B} _n$. Note that $\mathcal {A} _n^+$ inherits a weight structure from $\mathcal {A} '_n \times _{\mathcal {B} '_n} \mathcal {B} _n$. We say that $\Delta$ is k-pro-precartesian if the functors $\mathcal {A} _n \rightarrow \mathcal {A} _n^+$ induce a pro-k-connective functor on weighted pro-$\infty$-categories. We say it is weakly pro-precartesian if it is k-pro-precartesian for all k.

Definition C.2.4

Let $\{\Delta _n\}_n$ be a cofiltered system of commutative squares of weighted $\infty$-categories and weight-exact functors as above. We say that $\{\Delta _n\}_n$ is k-pro-Milnor if it is pro-precartesian and each of the functors $f_n^*$, $g_n^*$, $p_n^*$, $q_n^*$ is thickly surjective. It is weakly pro-Milnor if it is k-pro-Milnor for all k.

Construction C.2.5

Suppose given a commutative square

of weighted $\infty$-categories. Write ${\widehat{\mathcal {A}}}$, ${\widehat{\mathcal {B}}}$, etc. for the Ind-completions and consider the $\odot$-construction ${\widehat{\mathcal {Q}}} := \widehat{\mathcal {A} '} \odot _{{\widehat{\mathcal {A}}}}^{\widehat{\mathcal {B} '}} {\widehat{\mathcal {B}}}$. By Lemma 3.5.12 there is a weight structure on $\mathcal {Q}:= {\widehat{\mathcal {Q}}}^\omega$ such that all the functors in the induced square

as well as the functor $b : \mathcal {Q} \rightarrow \mathcal {B} '$, are weight-exact. We call $\mathcal {Q}$ the weighted $\odot$-construction, and denote it by

$$\begin{aligned} \mathcal {A} ' \odot _{\mathcal {A}}^{\mathcal {B} '} \mathcal {B}. \end{aligned}$$

Definition C.2.6

Let $\{\Delta _n\}_n$ be a cofiltered system of commutative squares of weighted $\infty$-categories and weight-exact functors of the form

(C.2.a)

If $\Delta$ is a pro-k-Milnor square, then we say it pro-k-base change if the functors $\mathcal {A} '_n \odot _{\mathcal {A} _n}^{\mathcal {B} '_n} \mathcal {B} _n \rightarrow \mathcal {B} '_n$ induce a pro-k-connective functor on weighted pro-$\infty$-categories. If $\Delta$ is a weak pro-Milnor square, then we say it weakly satisfies pro-base change if satisfies pro-k-base change for all k.

1.3 Weak pro-excision

For connected invariants, we have the following analogue of Theorem 3.5.11:

Theorem C.3.1

(Weak pro-excision) Let E be a connected localizing invariant. Suppose given a weak pro-Milnor square of weighted pro-$\infty$-categories of the form (C.2.a) weakly satisfying pro-base change. Then the induced square of pro-spectra

is weakly pro-cartesian. That is, the morphisms $E(\mathcal {A} _n) \rightarrow E(\mathcal {A} '_n) \mathop {\times }\limits _{E(\mathcal {B} '_n)} E(\mathcal {B} _n)$ induce a weak pro-equivalence of pro-spectra.

Lemma C.3.2

Let $\{f_n : \mathcal {C} _n \rightarrow \mathcal {D} _n\}_n$ be a cofiltered system of weight-exact functors between idempotent-complete weighted $\infty$-categories. If it is pro-k-connective, then it is isomorphic to a pro-system of levelwise k-connective functors.

Proof

For every n, consider the commutative square

By assumption, the lower arrow induces an isomorphism of pro-objects as n varies. Since passage to underlying pro-objects commutes with finite limits, it follows that the base changes $\mathcal {E} _n \rightarrow \mathcal {D} _n$ also induce an isomorphism of pro-objects. Thus it will suffice to show that for every n, the induced functor $f'_n: \mathcal {C} _n \rightarrow \mathcal {E} _n$ is k-connective. By construction of $\mathcal {E} _n$, the functor $\mathcal {E} _n \rightarrow \tau _{\le k+1}(\mathcal {C} _n)$ induces an equivalence on homotopy categories of the weight-hearts. Since the same holds for $\mathcal {C} _n \rightarrow \tau _{\le k+1}(\mathcal {C} _n)$, it follows that $\mathcal {C} _n \rightarrow \mathcal {E} _n$ is thickly surjective.

Now for any two objects X and Y in $\mathcal {C} _n^{w=0}$, consider the commutative triangle

Note that the vertical and diagonal maps have fibres

$$\begin{aligned} \tau _{\ge k+1} {{\,\mathrm{{\text {Maps}}}\,}}_{\mathcal {D} _n}(f_n(X), f_n(Y)) ~\text {and}~ \tau _{\ge k+1} {{\,\mathrm{{\text {Maps}}}\,}}_{\mathcal {C} _n}(X, Y), \end{aligned}$$

respectively. These are both $(k+1)$-connective, so it follows from the octahedral axiom that the horizontal map also has k-connective fibre. $\square$

Corollary C.3.3

Let E be a connected invariant. Then E sends pro-k-connective maps of idempotent-complete weighted $\infty$-categories to $(k+1)$-connective maps of pro-spectra. In particular, it sends weak pro-equivalences to weak pro-equivalences.

Proof of Theorem C.3.1

Since $\{\Delta _n\}_n$ is weakly pro-Milnor, it is weakly pro-equivalent to the pro-system induced by the squares

By Theorem 3.4.3, E sends the above square to a cartesian square of spectra for every n. Thus the claim follows from Corollary C.3.3. $\square$