2.1 Simplicial objects

2.2 Sheaves on simplicial spaces and simplicial schemes

For $X$ a topological space, we write $\operatorname {Sh}(X)$ for the category of abelian sheaves on $X$. For $X$ a scheme, we write $\operatorname {Sh}(X)$ for either the category of abelian étale sheaves or the category of pro-étale $\mathcal {O}_L$ or $L$ modules for $L/\mathbb {Q}_\ell$ an algebraic extension (as defined in [Reference Bhatt and ScholzeBS15]).

If $A$ is a simplicial space (which for us means simplicial topological space) or a simplicial scheme, we extend these definitions in the standard way to define $\operatorname {Sh}(A)$: concretely, we define a sheaf on $A$ to be a family of sheaves $\mathcal {F}_n$ on the simplex spaces $A_n$ equipped with a compatible family of maps $A(f)^* \mathcal {F}_n \rightarrow \mathcal {F}_m$ for $f:[n]\rightarrow [m]$ in $\Delta$. It can be shown that these form an abelian category that is naturally identified with the category of sheaves on a site built from $A$, so that, in particular, there are enough injectives and the standard formalism of derived categories of sheaves applies. The case of simplicial spaces is treated concretely in [Sta19, Tag 09VK], or can be considered in parallel with the case of schemes by first passing to the corresponding simplicial site and then applying the formalism of [Sta19, Tag 09WB], Case (A).

Given a simplicial space or simplicial scheme $A$ with an augmentation $\epsilon$ towards $X$, we obtain adjoint pushforward and pullback functors

\[ \epsilon_*: \operatorname{Sh}(A)\rightarrow \operatorname{Sh}(X), \epsilon^*:\operatorname{Sh}(X) \rightarrow \operatorname{Sh}(A) \]

such that $\epsilon ^*\mathcal {G}$ is given by the obvious system of $\epsilon _n^*\mathcal {G}$ on $A_n$ and

\[ \epsilon_*(\mathcal{F})=\mathrm{Eq} ( {\epsilon_{0}}_*\mathcal{F}_0 \rightrightarrows {\epsilon_{1}}_*\mathcal{F}_1 ). \]

For $K \in D^+(A)$, the bounded below derived category of $\operatorname {Sh}(A)$, there is a functorial skeletal spectral sequence [Sta19, Tag 0D7A]

(2.2.0.1)\begin{equation} E_1^{p,q}=R^q {\epsilon_p}_* K \Rightarrow R^{p+q}\epsilon_* K. \end{equation}

such that for each $q$ the cochain complex $E_1^{\bullet,q}$ is the Moore complex of the cosimplicial sheaf $[p] \mapsto R^q {\epsilon _p}_* K$ on $X$ (see § 2.1.6).

Definition 2.2.1 We write $\tilde {C}(A/X, \bullet )$ for the relative reduced cohomology complex functor $D^+(X) \rightarrow D^+(X)$, i.e. the cone of the adjunction unit $u_{A/X}: \operatorname {Id}_{D^+(X)} \rightarrow R\epsilon _* \epsilon ^*$, so that there is a functorial exact triangle for $K \in D^+(X)$

\[ K \rightarrow R\epsilon_* \epsilon^*K\rightarrow \tilde{C}(A/X, K) \rightarrow K[1]. \]

We write its cohomology sheaves as

\[ \tilde{H}^q(A/X, \bullet):=H^q(\tilde{C}(A/X, \bullet)). \]

We say $A/X$ satisfies cohomological descent on a full subcategory $D \subset D^+(X)$ if $u_{A/X}|_D$ is an isomorphism of functors or, equivalently, if $\tilde {C}(A/X, K)\cong 0$ for each $K \in D$, or equivalently if $\tilde {H}^q(A/X,K)=0$ for each $K \in D$ and each $q \in \mathbb {Z}$.

Lemma 2.2.2 If $f: A/X \rightarrow B/X$ is a homotopy equivalence of simplicial spaces or simplicial schemes over $X$, then $f$ induces an isomorphism of functors on $D^+(X)$

\[ R{\epsilon_{A}}_* \epsilon_{A}^* \cong R{\epsilon_{B}}_* \epsilon_{B}^*. \]

In particular, if $A/X$ is contractible, then $A/X$ satisfies cohomological descent.

Proof. We obtain a map $f^*$ between the skeletal spectral sequences (2.2.0.1) for $A/X$ and $B/X$. On the $q$th column $E_1^{\bullet, q}$ this is the map on Moore complexes coming from the map of cosimplicial sheaves

\begin{align*} f^*: ([p]\mapsto R^q {\epsilon_{B_p}}_* \epsilon_{B_p}^* K )\rightarrow ([p]\mapsto R^q {\epsilon_{A_p}}_* \epsilon_{A_p}^* K). \end{align*}

Because $f$ is a homotopy equivalence, so is the induced map on cosimplicial sheaves: it comes from applying the functor on simplicial spaces or simplicial schemes over $X$ induced by the functor on topological spaces or schemes over $X$ sending $\pi : Y \rightarrow X$ to $R^q \pi _* \pi ^* K$. A functor constructed in this way preserves homotopy equivalences (see the second paragraph of § 2.1.3 above). The induced map on complexes is then a homotopy equivalence, thus an isomorphism on cohomology, so that we obtain an isomorphism of spectral sequences starting at the $E_2$ page.

2.3 Filtered derived categories and simplicial filtrations

2.3.1 Filtered derived categories

We follow [Sta19, Tag 05RX, Tag 015O]. To summarize: for an abelian category $\mathcal {A}$, we write $\mathcal {A}^f$ for the exact category of finitely filtered objects in $\mathcal {A}$. It admits exact filtered piece, graded part, and forgetful functors $F^p$, $\operatorname {Gr}^p, \mathrm {Forget}: \mathcal {A}^f \rightarrow \mathcal {A}$. For the category of sheaves on a topological space, scheme, simplicial space, or simplicial scheme $X$, we write $DF^+(X)$ for the bounded below filtered derived category, i.e. the bounded below derived category of $\operatorname {Sh}(X)^f$. The filtered piece, graded and forgetful functors induced triangulated functors $DF^+(X) \rightarrow D^+(X)$. Moreover, if $A/X$ is a simplicial space or simplicial scheme augmented towards $X$, then we have a filtered derived functor $R{\epsilon ^f}_*: DF^+(A) \rightarrow DF^+(X)$ and a canonical isomorphism

\[ \operatorname{Forget} \circ R{\epsilon^f}_* = R\epsilon_* \circ \operatorname{Forget}. \]

2.3.2 Simplicial filtrations

Let $A$ be a simplicial space or simplicial scheme. We say a map of simplicial spaces or simplicial schemes $B \rightarrow A$ is a closed (respectively, open; respectively, clopen) immersion if for all $k \geq 0$, $B_k \rightarrow A_k$ is a closed (respectively, open; respectively, clopen) immersion.

A filtration of $A$ is an increasing sequence of closed sub-simplicial spaces/schemes $\iota _i: F_i A \hookrightarrow A, i \in \mathbb {Z}$. It is finite if $F_i A=\emptyset$ for $i \ll 0$ and $F_i A= A$ for $i \gg 0$. It is split if $\iota _i$ is a clopen immersion for all $i$.

Example 2.3.3 For $A$ a simplicial space or simplicial scheme over $X$, $\mathrm {Cone}(A/X)$ (see Example 2.1.2) has a natural finite split filtration with

\[ F_0 =X_\bullet \quad\textrm{and}\quad F_1 \mathrm{Cone}(A/X)=\mathrm{Cone}(A/X), \]

and similarly for $\mathrm {Cocone}(A/X)$.

Attached to a filtration $F_\bullet$ of $A$ we have filtered and graded piece functors

\[ F^i: \operatorname{Sh}(A) \rightarrow \operatorname{Sh}(A), \mathcal{F} \mapsto F^i\mathcal{F} = \ker ( \mathcal{F} \rightarrow {\iota_{i}}_*{\iota_{i}}^{*}\mathcal{F} ), \]

\[ \operatorname{Gr}^i: \operatorname{Sh}(A) \rightarrow \operatorname{Sh}(A), \mathcal{F} \mapsto \operatorname{Gr}^i \mathcal{F} = F^i \mathcal{F} / F^{i+1} \mathcal{F}. \]

These functors are exact: this can be checked on $m$-simplices, where it immediately reduces to the corresponding assertion for filtrations of spaces/schemes by closed subspaces/subschemes; indeed, if we write $j$ for the locally closed immersion $F_{i+1} A_m \backslash F_i A_{m} \hookrightarrow A_m$, then there is a canonical identification of sheaves on $A_m$

\[ j_! j^* \mathcal{F}_m = (\operatorname{Gr}^i\mathcal{F})_m. \]

We can reinterpret this description of the graded pieces geometrically as the following useful lemma which also describes the simplicial structure.

Lemma 2.3.4 For $F_\bullet A$ a finite filtration, let $\iota _p: F_p A \rightarrow A$. The unit $F^p \mathcal {F} \rightarrow {\iota _{p+1}}_{*}\iota _{p+1}^* F^p \mathcal {F}$ induces an isomorphism $\operatorname {Gr}^p \mathcal {F} ={\iota _{p+1}}_*\iota _{p+1}^* F^p \mathcal {F}$.

As a consequence of exactness, applying $F^i$ or $\operatorname {Gr}^i$ termwise to complexes induces a functor $D^+(A) \rightarrow D^+(A)$. When the filtration $F_\bullet A$ is finite, the functors $F^i$ assemble to a functor

\[ \operatorname{Fil}_{F_\bullet} : \operatorname{Sh}(A) \rightarrow \operatorname{Sh}(A)^f \]

that is exact (i.e. maps quasi-isomorphisms to filtered quasi-isomorphisms; this is equivalent to the assertion above that each filtered or graded piece functor is exact), so that termwise application to complexes induces a functor $\operatorname {Fil}_{F_\bullet }:D^+(A) \rightarrow DF^+(A)$ whose associated filtered and graded piece functors are canonically identified with $F^i$ and $\operatorname {Gr}^i$ as above and such that there is a canonical identification $\operatorname {Forget} \circ \operatorname {Fil} = \operatorname {Id}_{D^+(A)}.$

3.1 Definitions and first properties

For $\mathcal {P}/X$ a pospace or poscheme, we write $\mathord {\geq }_{\mathcal {P}}$ for the closed relation obtained by swapping the coordinates; we have $\Delta _{\mathcal {P}}=\mathord {\geq }_{\mathcal {P}} \cap \mathord {\le }_{\mathcal {P}}$, thus $\Delta _{\mathcal {P}}$ is closed, so $\mathcal {P}/X$ is separated as a map of topological spaces/schemes. We will also write $\mathord {<}_\mathcal {P}=\mathord {\le }_{\mathcal {P}} \backslash \Delta _{\mathcal {P}}$, the complement of the diagonal in $\mathord {\le }_\mathcal {P}$, and similarly for $\mathord {>}_{\mathcal {P}}$.

It will be convenient to use the following relative interval notation.

Definition 3.1.3 (Intervals)

Suppose $\mathcal {P}/X$ is a pospace or poscheme, and suppose given $T_1 \rightarrow \mathcal {P}$ and $T_2 \rightarrow \mathcal {P}$. The open interval $(T_1,T_2)$ is the fiber product

viewed as a pospace or poscheme over $T_1 \times T_2$ with ordering pulled back from the middle coordinate. The closed and mixed intervals $[T_1, T_2]$, $(T_1, T_2]$, $[T_1,T_2)$ over $T_1 \times T_2$ are defined by changing between $\leq$ and $\mathord {<}$ appropriately in the bottom left.

Given $T \rightarrow \mathcal {P}$, we similarly define the intervals $(-\infty, T)$, $(-\infty, T]$, $(T,\infty )$ and $[T, \infty )$ as poschemes or pospaces over $T$. These latter can be interpreted literally in the above notation as intervals in the pospace or poscheme obtained by adding disjoint maximum and minimum sections $\infty$ and $-\infty$ to $\mathcal {P}/X$.

Any pospace or poscheme that admits a ranking is split, and our motivating applications all fall into this category. It is possible to construct examples that are not split, but imposing this condition will simplify some hypotheses later (because then maxima and minima are centers; see Remark 3.1.8).

Definition 3.1.9 The nerve (or order complex) of a pospace (respectively, poscheme) $\mathcal {P}/X$ is the simplicial space (respectively, simplicial scheme) $N\mathcal {P}$, canonically augmented to $X$, whose topological space (respectively, scheme) of $m$-simplices is the topological space (respectively, scheme) of chains of length $m+1$ in $\mathcal {P}$,

\[ N\mathcal{P}_m := \underbrace{\mathord{\le}_\mathcal{P} \times_\mathcal{P} \mathord{\le}_\mathcal{P} \times_\mathcal{P} \dotsm \times_\mathcal{P} \mathord{\le}_\mathcal{P} }_{m-1 \textrm{ terms}} \]

with the obvious transition maps. In other words, it is the topological space (respectively, scheme) over $X$ of pospace (respectively, poscheme) maps from $[m]\times X$ to $\mathcal {P}$. We also write

\[ N\mathcal{P}^\circ_m := \underbrace{\mathord{<}_\mathcal{P} \times_\mathcal{P} \mathord{<}_\mathcal{P} \times_\mathcal{P} \dotsm \times_\mathcal{P} \mathord{<}_\mathcal{P} }_{m-1 \textrm{ terms}}, \]

for the subspace (respectively, subscheme) of $N\mathcal {P}_m$ consisting of non-degenerate $m$-simplices (i.e. strictly ordered chains). We say $\mathcal {P}/X$ is of bounded length if $N\mathcal {P}^\circ _m = \emptyset$ for $m$ sufficiently large, i.e. if there is a bound on the length of chains of proper inequalities in $\mathcal {P}/X$.

Note that a finitely ranked pospace or poscheme is bounded, and that any ranked topologically Noetherian poscheme is finitely ranked.

The following is immediate from the definitions.

3.2 A contractibility criterion

We now show that if a pospace or poscheme $\mathcal {P}/X$ admits a maximum/minimum/center then $N\mathcal {P}/X$ is contractible. This is essentially an immediate consequence of Example 3.1.10(iv): the case of isolated maxima and minima follows from the contractibility of cones and cocones, while the case of a center follows from contracting separately the two cones forming the join. We make this precise as a consequence of the following more general result, which will have further applications later on.

Proof. We treat the first case: let $r$ denote the composition of the maximum, a map $\mathcal {P} \rightarrow (-\infty, \mathcal {P}] \cap \mathcal {P}' \times \mathcal {P}$, with the projection to the first factor in $\mathcal {P}'$. It is evidently a retraction of $\mathcal {P}$ onto $\mathcal {P}'$, so we only need to show that $Nr$, viewed as a map from $N\mathcal {P}$ to itself, is homotopic to the identity. We define a homotopy

\[ N\mathcal{P} \times \Delta^1 \rightarrow N\mathcal{P} \]

on simplices as follows: given a $k$-simplex $(p_0, \ldots, p_k) \times \alpha$ where $\alpha :[k]\rightarrow [1]$ is non-decreasing, send it to $(r^{1-\alpha (0)}(p_0), r^{1-\alpha (1)}(p_1), \ldots )$ where $r^0=\operatorname {Id}$ and $r^1=r$. The identity $r(p_i) \leq p_i$ ensures this is a chain because $\alpha$ is non-decreasing, and evidently it is a homotopy from $r$ to the identity.

For the second case, we define $r$ using the minimum, then obtain a homotopy from $r$ to $\operatorname {Id}_{N\mathcal {P}}$ similarly by replacing $r^{1-\alpha (i)}$ with $r^{\alpha (i)}$ above.

As an application, we find the following.

4.1 Technical criteria for cohomological descent

The following summarizes the application of proper base change that allows us to spread out cohomological descent from (geometric) points when it holds; in the schematic case we must put some restrictions in place to obtain a suitable proper base change theorem.

Proof. We first treat case (i): to establish cohomological descent for $K$, it suffices to show that $C:=\tilde {C}(N\mathcal {P}/X, K)$ is quasi-isomorphic to zero. This is equivalent to checking that each cohomology sheaf $\mathcal {H}^i(C)$ of this complex is zero, which can be checked by showing the stalks $\mathcal {H}^i(C)_x$ are zero, and this is equivalent to showing $C_x$ is quasi-isomorphic to zero. To conclude, we invoke proper base change [Sta19, Tag 09V6] to identify $C_x$ with

\[ \tilde{C}(N\mathcal{P}_x/\{x\}, K_x), \]

which is quasi-isomorphic to zero by the assumption that cohomological descent holds on fibers.

The argument in case (ii) is identical after replacing points with geometric points and applying proper base change in the form of [Sta19, Tag 0DDE]. For case (iii), it is not true that a general pro-étale sheaf is zero if it is zero after evaluation at all geometric points, but this is true for a constructible pro-étale sheaf (where the statement reduces to the étale case). Thus, the same argument also goes through in case (iii), applying proper base change in the form of [Reference Bhatt and ScholzeBS15, Lemma 6.7.5 and Proposition 6.8.14].

To show a pospace or poscheme over a point satisfies cohomological descent, we will typically appeal to contractibility of the nerve; in § 3.2 above we already established some useful tools that can be used to deduce this contractibility. For schemes we will often need one more reduction before we can apply these contractibility criteria; the result is encoded in the following lemma. We say $f: \mathcal {P}_1 /X \rightarrow \mathcal {P}_2/X$ is a universal homeomorphism of poschemes over $X$ if it is a map of poschemes over $X$ that is universal homeomorphism as a map of schemes and induces a universal homeomorphism $\mathord {\le }_{\mathcal {P}_1} \rightarrow \mathord {\le }_{\mathcal {P}_2}$.

4.2 Weak centers and the proof of Theorem 4.0.1

The last lemma used in the proof is an alternative characterization of weak centers: recall from § 3.1.6 that, for $\mathcal {P}/X$ a poscheme, $c \in \mathcal {P}(X)$ is a weak center/minimum/maximum if the defining decomposition of $\mathcal {P}$ holds topologically but not necessarily scheme-theoretically.

4.3 Construction of filtration and computation of graded components

Proof Proof of Theorem 4.0.2

Let $\mathcal {P}/X$ be a finitely ranked pospace or poscheme (with no assumptions of cohomological descent or contractibility!). We consider a split simplicial filtration $F_i N\mathcal {P}=\mathrm {rk}^{-1}(-\infty, i]$ of $N\mathcal {P}$. By the formalism of § 2.3, we obtain

\[ \operatorname{Fil}_{F_\bullet N\mathcal{P}}: D^+(N\mathcal{P}) \rightarrow DF^+( N\mathcal{P}) \]

and a canonical isomorphism

\[ \operatorname{Id}_{D^+(N\mathcal{P})} = \operatorname{Forget} \circ \operatorname{Fil}_{F_\bullet N\mathcal{P}}. \]

We then consider the functor $\operatorname {Fil}_{\mathcal {P},\mathrm {rk}}: D^+(X) \rightarrow DF^+(X)$ given by composition as

\[ \operatorname{Fil}_{\mathcal{P},\mathrm{rk}}:= R\epsilon^f_* \circ \operatorname{Fil}_{F_\bullet N(\mathcal{P})} \circ \epsilon^*, \]

where $\epsilon :N\mathcal {P}\rightarrow X$ is the augmentation. We have a canonical natural transformation

(4.3.0.1)\begin{equation} \operatorname{Id}_{D^+(X)} \rightarrow \operatorname{Forget} \circ \operatorname{Fil}_{\mathcal{P},\mathrm{rk}} \end{equation}

given as the composition of

\[ \operatorname{Id}_{D^+(X)} \xrightarrow{u_{\mathcal{P}}} R\epsilon_* \circ \epsilon^* \cong R\epsilon_* \circ \operatorname{Forget} \circ \operatorname{Fil}_{F_\bullet N\mathcal{P}} \circ \epsilon^* \cong \operatorname{Forget} \circ R\epsilon^f_* \circ \operatorname{Fil}_{F_\bullet N\mathcal{P}} \circ \epsilon^*. \]

In particular, (4.3.0.1) restricts to an isomorphism of functors $D \rightarrow D^+(X)$ on any subcategory $D$ on which $\mathcal {P}/X$ is of cohomological descent. We thus obtain the desired statement, up to the computation of the graded pieces (which no longer requires the finiteness hypothesis for the rank). This computation is given in Lemma 4.3.3 below.

Before giving the general computation of graded pieces in Lemma 4.3.3 to complete the above proof, we consider a useful example that computes the graded pieces for a simplicial cocone.

Example 4.3.1 If $\epsilon :A \rightarrow X$, then $\epsilon _{\mathrm {Cocone}(A)}: \mathrm {Cocone}(A) \rightarrow X$ is contractible, and thus satisfies cohomological descent. If we take the filtration as in Example 2.3.3, then

(4.3.1.1)\begin{equation} R{\epsilon_{\mathrm{Cocone}(A)}}_* \operatorname{Gr}^p_{F_\bullet\mathrm{Cocone}(A)/X} K \cong \begin{cases} R{\epsilon_{A}}_*\epsilon_{A}^*K & \text{if } p=-1 \\ \tilde{C}(A/X, K)[-1] & \text{if } p=0. \end{cases} \end{equation}

Indeed, because $F_1 \mathrm {Cocone}(A)=\mathrm {Cocone}(A)$, $F^1 \epsilon _{\mathrm {Cocone}(A)}^*K=0$, thus

\[ F^{0}\epsilon_{\mathrm{Cocone}(A)}^* K =\operatorname{Gr}^0\epsilon_{\mathrm{Cocone}(A)}^*K. \]

In particular, we have an exact triangle

\[ \operatorname{Gr}^{0}\epsilon_{\mathrm{Cocone}(A)}^* K \rightarrow F^{-1} \epsilon_{\mathrm{Cocone}(A)}^*K \rightarrow \operatorname{Gr}^{-1} \epsilon_{\mathrm{Cocone}(A)}^*K \rightarrow \cdots. \]

Since $F_{-1} \mathrm {Cocone}(A)=0$, the middle term is $\epsilon _{\mathrm {Cocone}(A)}^*K$. Expanding the definition, we find the right term is $\epsilon _{A}^*K$, where we have identified $A=F_0 \mathrm {Cocone}(A)$. Since $\mathrm {Cocone}(A)$ is contractible, applying $R{\epsilon _{\mathrm {Cocone}(A)}}_*$ gives an identification of the exact triangle

\begin{align*} R{\epsilon_{\mathrm{Cocone}(A)}}_*\operatorname{Gr}^{0}\epsilon_{\mathrm{Cocone}(A)}^* K &\rightarrow R{\epsilon_{\mathrm{Cocone}(A)}}_* F^{-1} \epsilon_{\mathrm{Cocone}(A)}^*K\\ &\rightarrow R{\epsilon_{\mathrm{Cocone}(A)}}_*\operatorname{Gr}^{-1} \epsilon_{\mathrm{Cocone}(A)}^*K \rightarrow \cdots. \end{align*}

with

\[ R{\epsilon_{\mathrm{Cocone}(A)}}_* (\operatorname{Gr}^0\epsilon_{\mathrm{Cocone}(A)}^* K) \rightarrow K \rightarrow R{\epsilon_{A}}_*\epsilon_{A}^*K \rightarrow \cdots \]

and (4.3.1.1) follows from the definition of $\tilde {C}(A/X, K)$ (Definition 2.2.1).

The argument for computing the graded pieces then is a reduction to this example via excision. Consider for each $p$ the natural map (cf. Example 3.1.10(iv))

\begin{align*} r_p: \mathrm{Cocone}(N(-\infty, \mathcal{P}_p)/\mathcal{P}_p) \cong N(-\infty, \mathcal{P}_p] \rightarrow N\mathcal{P}. \end{align*}

Proof. The point is that $(r_p)_m$ maps $F_0 \mathrm {Cocone}(N(-\infty, \mathcal {P}_p) )$ into $F_{p-1} N\mathcal {P}$ and on $m$-simplices restricts to an isomorphism

\[ \mathrm{Cocone}(N(-\infty, \mathcal{P}_p)/\mathcal{P}_p)_m \backslash F_0 \mathrm{Cocone}(N(-\infty, \mathcal{P}_p)/\mathcal{P}_p)_m \xrightarrow{\sim} (F_p N\mathcal{P})_m \backslash (F_{p-1} N\mathcal{P})_m. \]

Suppose we represent $K$ by a complex of injectives. Then, $K_m$ is a complex of injectives for each $m$, and similarly for $F^{p-1}K_m$. Then $(r_p^*F^{p-1}K)_m = (F^0 r_p^*K)_m=(\operatorname {Gr}^0 r_p^*K)_m$ is a complex of injectives. Thus, pushforward of this complex computes $R{r_p}_*$, but that is just identified with the pushpull from $F_p N\mathcal {P}$ and by Lemma 2.3.4 we conclude this is $\operatorname {Gr}^{p-1} K$.

Lemma 4.3.3 Let $X$ be a topological space (respectively, scheme) and let $\mathcal {P}/X$ be a finitely ranked pospace (respectively, poscheme). For $K \in D^+(X)$, there is a functorial (in $K$) identification

\[ \operatorname{Gr}_{\mathcal{P}, \mathrm{rk}}^p K = R\pi_* \tilde{C}(N(-\infty, \mathcal{P}_{p+1})/\mathcal{P}_{p+1}, \pi^*K)[-1]. \]

Proof. Combining Lemma 4.3.2 with Example 4.3.1, we obtain

(4.3.3.1) \begin{align} \operatorname{Gr}_{\mathcal{P}, \mathrm{rk}}^p K & = R\epsilon_* \operatorname{Gr}^p \epsilon^*K\nonumber\\ & = R{(\epsilon \circ r_{p+1})}_* \operatorname{Gr}^0 (\epsilon \circ r_{p+1})^*K \nonumber\\ & = R{(\pi \circ \epsilon_{\mathrm{Cocone}(N(-\infty, \mathcal{P}_{p+1}))})}_* \operatorname{Gr}^0(\pi \circ \epsilon_{\mathrm{Cocone}(N(-\infty, \mathcal{P}_{p+1}))})^* K \nonumber\\ &= R\pi_* \tilde{C}(N(-\infty, \mathcal{P}_{p+1})/\mathcal{P}_{p+1}, \pi^*K)[-1]. \end{align}

4.4 Spectral sequences

Proof Proof of Theorem 4.0.3

Write $\tilde {K}=\operatorname {Fil}_{\mathcal {P},\mathrm {rk}}(K|_Z)$. Then, since we assume $\mathcal {P}/Z$ is of cohomological descent for $K|_Z$, we have $\operatorname {Forget}(\tilde {K})=K|_Z$. We then apply [Sta19, Tag 015W] to obtain the spectral sequence (i), whose terms are described by (4.3.3.1); it remains only to observe that $\pi$ is proper by assumption so that when $\star =!$, $R(f \circ \pi )_!=Rf_! \circ R\pi _*$. For part (ii) we extend to a filtration of $j_!j^*K$ by

\[ i_*i^*K[-1]\rightarrow j_!j^*K\rightarrow K \quad \textrm{for } j: U \hookrightarrow X \textrm{ and } i:Z\hookrightarrow X \]

before applying the spectral sequence (and make a similar observation when $\star =!$).

4.4.1 The first spectral sequence of a stratified space

Recall the setup from § 1.2.7: $X = \bigcup _{\alpha } S_\alpha$ for disjoint locally closed sets $S_\alpha$ and $Z_{\alpha } := \overline {S_\alpha } = \bigcup _{\beta \le \alpha } S_\beta$. We consider the pospace or poscheme given by $\mathcal {P} = \bigsqcup _{\alpha } Z_\alpha \rightarrow X$ with $z_\alpha \geq z_\beta$ if $z_\alpha = z_\beta$ as elements of $X$ and $\alpha \ge \beta$ and with a ranking given by a ranking on the indexing set.

We have, $\mathcal {P}_{p+1} = \bigsqcup _{\mathrm {rk}(\alpha )=p+1} Z_\alpha$, and we will also write $j:U_{p+1} \rightarrow \mathcal {P}_{p+1}$ and $i: \partial \mathcal {P}_{p+1} \rightarrow \mathcal {P}_{p+1}$ where

\[ U_{p+1}:=\bigsqcup_{\mathrm{rk}(\alpha)=p+1} S_\alpha, \quad \partial \mathcal{P}_{p+1}:=\mathcal{P}_{p+1}\backslash U_{p+1}=\bigsqcup_{\mathrm{rk}(\alpha)=p+1}Z_\alpha \backslash S_\alpha. \]

We then claim that $\tilde {C}(N(-\infty, \mathcal {P}_{p+1})/\mathcal {P}_{p+1}, K)=j_!j^*K[1]$. Indeed, $(-\infty, \mathcal {P}_{p+1})/\mathcal {P}_{p+1}$ is proper with image $\partial \mathcal {P}_{p+1}$, and the fiber over any geometric point in $\partial \mathcal {P}_{p+1}$ has a minimum. Thus, writing $\epsilon : N(-\infty, \mathcal {P}_{p+1}) \rightarrow \mathcal {P}_{p+1}$, $R\epsilon _* \epsilon ^* K= i_*i^*K$ and, thus, the cone is $j_!j^*K[1]$, as desired. In particular, if $X$ is a topological space or a variety over an algebraically closed field, then taking $f$ to be the structure map to a point and $\star =!$, we obtain the standard spectral sequence for a stratified space

\[ E_1^{p,q}=\bigoplus_{\mathrm{rk}(\alpha)=p+1}H^{p+q}_{\rm c}(S_\alpha, K) \Rightarrow H^{p+q}_{\rm c} (X, K). \]

4.4.2 The Banerjee spectral sequence of a symmetric semisimplicial filtration

Here we recover the spectral sequence of Banerjee [Reference BanerjeeBan19, Theorem 1]. In that setting, we are given ‘face maps’ $f_i : M^p \times X_{n + e} \to M^{p-1} \times X_n$ for $0 \le i < p$, fixed $e$ and $n, p > 0$, satisfying certain conditions that define a symmetric semisimplicial filtration of $\{X_n\}$ by powers of $M$. Take

\[ \mathcal{P} = \bigsqcup_{p > 0} \mathbf{S}^p M \times X_{n - ep} \]

over $Z_n := f_0(M \times X_{n-p}) \subset X_n$, where $\mathbf {S}^p M = M^p/\mathfrak {S}_p$ and $\mathord {\le }_{\mathcal {P}}$ is given by all possible compositions of the face maps (this is well-defined on $\mathbf {S}^\bullet M$ by the assumptions on the face maps [Reference BanerjeeBan19, Definition 2.10]). Given $x \in Z_n$, the ‘equalizer’ and ‘embedding’ assumptions imply that there is a maximal $p$ and a unique $(S, x) \in \mathbf {S}^p(M) \times X_{n - ep}$ that maps to $x$. To us, this means that the pospace $\mathcal {P}$ admits a fiberwise maximum. Then Theorem C gives the desired

\[ E_1^{p,q} = H_{\rm c}^q(M^p \times X_{n - ep}; \mathbb{Q}) \otimes_{\mathfrak{S}_p} \mathrm{sgn} \Rightarrow H_{\rm c}^{p + q}(X_n - Z_n; \mathbb{Q}). \]

The sign representation $\mathrm {sgn}$ appears here for the same reason as in Corollary 7.2.6.

5.1 Decategorifications

Suppose $X$ is a noetherian scheme; fix $L/\mathbb {Q}_\ell$ an algebraic extension for $\ell$ invertible on $X$. Then we can consider the abelian category $\operatorname {Cons}(X,L)$ of constructible $L$-sheaves on (the pro-étale site of) $X$ and its Grothendieck ring $K_0(\operatorname {Cons}(X,L))$. We consider the constructible derived category $D_{\operatorname {Cons}}(X,L)$, the subcategory of the bounded derived category $D^b(X,L)$ consisting of complexes with constructible cohomology sheaves. There is an Euler characteristic $\mathrm {Ob}(D_{\operatorname {Cons}}(X,L)) \rightarrow K_0(\operatorname {Cons}(X,L))$,

\[ K \mapsto [K]=\sum_k (-1)^k [H^i(K)]. \]

By [Reference Bhatt and ScholzeBS15, Remark 6.8.15], the constructible derived category is preserved by proper pushforward. Thus, we can decategorify the work of the previous section: for $\pi : \mathcal {P} \rightarrow X$ a proper finitely ranked poscheme of cohomological descent for $K$, $\operatorname {Fil}_{\mathcal {P},\mathrm {rk}}(K)$ induces

(5.1.0.1)\begin{equation} [K] = \sum_p [\operatorname{Gr}^p_{\mathcal{P},\mathrm{rk}}(K)] = - \sum_p [R\pi_* \tilde{C}(-\infty, \mathcal{P}_{p+1}, K)] \end{equation}

where the shift in (4.3.3.1) manifests as a minus sign. In fact, there is an inclusion–exclusion formula independent of any rank function: under the same hypotheses, we have

(5.1.0.2)\begin{equation} [K] = \sum_{k\geq 0} (-1)^k [R{\epsilon^\circ_k}_*{\epsilon^\circ_k}^*K)], \end{equation}

where $\epsilon _k^\circ$ denotes the restriction of $\epsilon _k$ to the non-degenerate $k$-simplices $N\mathcal {P}_k^\circ$ (i.e. the scheme of strict $(k+1)$ chains). Indeed, since $\mathcal {P}$ admits a rank function, Lemma 3.1.11 shows that $N\mathcal {P}$ is split, i.e. that each degeneracy map is an isomorphism onto connected components. Then, we apply the spectral sequence (2.2.0.1) and use that each column on the $E_1$ page is quasi-isomorphic to the normalized complex given by the kernel of the total degeneracy map (see § 2.1.6), which by the above consideration is exactly the restriction to the space of non-degenerate simplices. The choice of a rank function gives a way to break up each simplex space into connected components, thus breaks up each term of (5.1.0.2); rearranging and reassembling, one can recover (5.1.0.1); indeed, we saw exactly this rearrangement phenomenon already in our toy model for classical inclusion–exclusion in the introduction (§ 1.1).

It turns out we can also give a combinatorial decategorification of the inclusion–exclusion formula that lifts (5.1.0.1) and (5.1.0.2): we work in the modified Grothendieck ring of varieties $K_0(\mathrm {Var}/X)$; recall that in characteristic zero this is the standard Grothendieck ring defined by cut and paste relations, but in non-zero characteristic one must also mod out by radicial surjective maps more general than constructible decompositions (e.g. purely inseparable field extensions). We refer to [Reference Bilu and HoweBH21] for a detailed discussion and other perspectives. Here let us just highlight that there is a natural compactly supported cohomology homomorphism

\[ K_0(\mathrm{Var}/X) \rightarrow K_0(D_{\operatorname{Cons}}(X,L)), [f:Y \rightarrow X] \mapsto [Rf_! L], \]

so that it makes sense to ask for a combinatorial lift of (5.1.0.2) to $K_0(\mathrm {Var}/X)$. This lift is exactly what is provided by Theorem D, which says that if $\mathcal {P}/X$ is a bounded poscheme with weak geometric centers, then

\[ [X/X]=\chi(N\mathcal{P}/X)=\sum_{k\geq 0} (-1)^k [N\mathcal{P}_k^\circ/X] \textrm{ in } K_0(\mathrm{Var}/X). \]

5.2 Euler characteristic contractibility criterion and proof of Theorem D

To prove Theorem D, we will spread out our geometric weak centers to reduce to the following elementary Euler characteristic version of Lemma 3.2.4.

Lemma 5.2.1 Suppose $X$ is a Noetherian scheme and $\mathcal {P}/X$ is a finite type bounded poscheme with a weak center/maximum/minimum. Then

\[ \chi(N\mathcal{P}/X) = [X/X] \textrm{ in } K_0(\mathrm{Var}/X). \]

Proof. Since we are working in the Grothendieck ring, we can apply Lemma 4.2.1 to assume there is a genuine center/maximum/minimum. Then, in the case of a maximum or minimum, by passing to a constructible decomposition we can assume it is isolated so that it is a center.

Thus, we may assume we have a center $c$. We write $A_k \subset N\mathcal {P}^\circ _k$ for the clopen subscheme of strict chains passing through $c$, and $B_k$ for its complement, the strict chains that do not pass through $c$. Then for $k \geq 1$, $B_k \cong A_{k+1}$ by insertion of $c$ in the unique possible spot. Thus, the sum

\[ \sum_{k \geq 0} (-1)^k[N\mathcal{P}^\circ_k] = \sum_k (-1)^k ([A_k] + [B_k]) \]

telescopes and is equal to $A_0=[X]$.

Proof Proof of Theorem D

By Noetherian induction it suffices to show that if $X$ is irreducible and reduced, then the identity holds after restriction to a non-empty open $U \subset X$. We write $\eta$ for the generic point of $X$ and fix an algebraic closure $\overline {K(\eta )}$ of $K(\eta )$ and a weak center $c \in \mathcal {P}(\overline {K(\eta )})$. We first observe that $c$ is defined over a finite purely inseparable extension $L/K(\eta )$: first, since $\mathcal {P}/X$ is of finite type, $c$ can be defined over some finite normal extension $M/K(\eta )$. Writing $G=\mathrm {Aut}(M/K(\eta ))$, we have $L=M^G/K(\eta )$ is purely inseparable, thus it suffices to show $c$ is fixed by $G$. Thus, suppose $\sigma \in G$. Then, by the definition of a weak center, since $c$ is a field-valued point, either $c \leq \sigma (c)$ or $c \geq \sigma (c)$; by replacing $\sigma$ with $\sigma ^{-1}$, we can assume $c \leq \sigma (c)$. Then, since $\sigma$ preserves the order relation (because it is defined over $K(\eta )$), we find $\sigma (c) \leq \sigma ^2(c)$, $\sigma ^2(c) \leq \sigma ^3(c)$, etc., so that for $k>1$ a multiple of the order of $G$,

\[ c \leq \sigma(c) \leq \dotsm \leq \sigma^k(c)=c. \]

Thus, $c \leq \sigma (c) \leq c$ so $\sigma (c)=c$, as desired.

Now, we can spread out $c:\mathrm {Spec}\, L \rightarrow \mathrm {Spec}\, K(\eta )$ to a finite radicial surjective map $\tilde {U}\rightarrow U$ over a non-empty open $U \subset X$. By shrinking $U$ further, we can assume this spreading out is a weak center of $\mathcal {P} \times _X \tilde {U}$. Lemma 5.2.1 then gives

\[ \sum_{k\geq 0} (-1)^k[N(\mathcal{P} \times_X \tilde{U} )^\circ_k/ \tilde{U}] = [\tilde{U}/\tilde{U}] \textrm{ in } K_0(\mathrm{Var}/\tilde{U}). \]

By composing with the map $\tilde {U} \rightarrow U$, we find

\begin{align*}\sum_{k \geq 0}(-1)^k[N\mathcal{P}^{\circ}_k|_U][\tilde{U}/U] & = \sum_{k\geq 0} (-1)^k[N\mathcal{P}^\circ_k \times_X \tilde{U} /U] \\ &= \sum_{k\geq 0} (-1)^k[N(\mathcal{P} \times_X \tilde{U} )^\circ_k/U] = [\tilde{U}/U] \textrm{ in } K_0(\mathrm{Var}/U). \end{align*}

Recalling that in the modified Grothendieck ring $[\tilde {U}/U]=[U/U]=1$, we conclude.

Because it is will be useful in later sections, we also give an analog of Lemma 3.2.1 using a similar argument. Both this and the previous criterion should be special cases of a more general poscheme Euler characteristic version of Quillen's Theorem A [Reference QuillenQui73].

Theorem 5.2.2 Let $X$ be a Noetherian scheme, let $\mathcal {P}/X$ be a poscheme that is bounded and of finite type. Let $\mathcal {P}'\subset \mathcal {P}$ be a sub-poscheme and suppose that, for every geometric point $t: \mathrm {Spec}\, \kappa \rightarrow X$, one of the following holds:

1. (i) $(-\infty, \mathcal {P}_t] \cap \mathcal {P}_t' \times \mathcal {P}_t /\mathcal {P}_t$ has a weak maximum; or

2. (ii) $[\mathcal {P}_t, \infty ) \cap \mathcal {P}_t \times \mathcal {P}_t' / \mathcal {P}_t$ has a weak minimum.

Then

(5.2.2.1)\begin{equation} \chi(N\mathcal{P} /X )=\chi(N\mathcal{P}' /X ) \textrm{ in } K_0(\mathrm{Var}/X). \end{equation}

Proof. By Noetherian induction, it suffices to assume $X$ is irreducible and to show that there exists a non-empty open $U \subset X$ where (5.2.2.1) holds. We write $\eta$ for the generic point and $\overline {\eta }: \mathrm {Spec}\, \overline {K(\eta )} \rightarrow X$ for a geometric point above $\eta$. We assume case (i) holds at $\overline {\eta }$, case (ii) being similar. We write

\[ m_{\overline{\eta}}: \mathcal{P}_{\overline{\eta}} \rightarrow (-\infty, \mathcal{P}_{\overline{\eta}}] \cap (\mathcal{P}' \times_X \mathcal{P})_{\overline{\eta}} \]

for the weak maximum. First note that $m_{\overline {\eta }}$ can be defined over a finite subextension $M/K(\eta )$, which we may take to be normal. Then, if we let $G=\mathrm {Aut}(M/K(\eta ))$, we must have that $m_{\overline {\eta }}$ is fixed by the action of $G$ because it is a maximum and the order relation is defined over $K(\eta )$. It follows that $m_{\overline {\eta }}$ can be defined over $L=M^G$, a finite purely inseparable extension of $K(\eta )$: that is, writing $\eta _L: \mathrm {Spec}\, L \rightarrow X$, we obtain $m_{\overline {\eta }}$ as the base change to $\overline {K(\eta )}$ of

\[ m_{\eta_L}: \mathcal{P}_{\eta_L} \rightarrow (-\infty, \mathcal{P}_{\eta_L}]\cap (\mathcal{P}'\times_X\mathcal{P})_{\eta_L}. \]

It is a weak maximum still because this can be checked on geometric points (for a section to be a weak maximum it is necessary and sufficient that it be a maximum on any set of geometric points). Now, we can spread out the purely inseparable map $\mathrm {Spec}\, L \rightarrow \mathrm {Spec}\, K(\eta )$ to a radicial surjective $\tilde {U} \rightarrow U$ where $U$ is open in $X$ such that $m_{\eta _L}$ spreads out to

\[ m_{\tilde{U}}: \mathcal{P}_{\tilde{U}} \rightarrow (-\infty, \mathcal{P}_{\tilde{U}}]\cap (\mathcal{P}'\times_X\mathcal{P})_{\tilde{U}}. \]

Now $(\infty, m_{\tilde {U}}]^c$ is open, so its image in $\tilde {U}$ is constructible by Chevalley's theorem [Sta19, Tag 054K]. Since this image does not contain $\eta _L$, we deduce that its complement, the locus where $m_{\tilde {U}}$ is a weak maximum, contains a non-empty open set. Thus, replacing $\tilde {U}$ with this non-empty open set, we can assume, furthermore, that $m_{\tilde {U}}$ is a weak maximum.

Now, for any geometric point $p_0 < p_1 < \dotsm < p_k$ of $(N\mathcal {P}_{\tilde {U}})^\circ _k$, either the chain stays entirely in $\mathcal {P}'_{\tilde {U}}$ or first leaves at an index $i$. Of those that leave, we can break them up into the subset $A_k$ such that for this first $i$, $p_{i-1}=m_{\tilde {U}}(p_i)$, and the subset $B_k$ where this is not satisfied; this gives a constructible decomposition

(5.2.2.2)\begin{equation} (N{\mathcal{P}_{\tilde{U}}})^\circ_k = (N{\mathcal{P}'_{\tilde{U}}})^\circ_k \sqcup A_k \sqcup B_k. \end{equation}

Moreover, we claim that $[A_{k+1}/\tilde {U}]=[B_k/\tilde {U}]$. Indeed, we can decompose $B_k$ as $\bigsqcup _{i=0}^k B_{k,i}$ where $B_{k,i}$ is the constructible set that first leaves at $i$. Then we have a map $B_{k,i} \rightarrow A_{k+1}$ such that

\[ p_0 < \dotsm < p_k \mapsto p_0 < \dotsm < p_{i-1} < m_{\tilde{U}}(p_i) < p_i \dotsm < p_k \]

and the induced map $\bigsqcup _{i=0}^k B_{k,i} \rightarrow A_k$ is a bijection on geometric points, so gives the desired equality in the Grothendieck ring. Because we have $A_0=\emptyset$, the $A_\bullet$ and $B_{\bullet }$ terms cancel when we use (5.2.2.2) to compute the Euler characteristic, giving the result.

6.1 Configuration and symmetric posets

For $Z$ a non-empty set, the configuration poset $\mathbf {C}^\bullet (Z)$ is the lattice of finite non-empty subsets of $Z$. The symmetric semigroup $\mathbf {S}^\bullet (Z)$ is the semigroup of finite non-empty multisets of $Z$ under disjoint union, i.e. the free commutative semigroup on $Z$. It is contained in the symmetric monoid $\mathbf {S}^{\bullet,+}(Z)$, the free commutative monoid on $Z$, where we allow also the empty set (which gives an identity element for the monoid operation). We may view $\mathbf {S}^{\bullet,+}(Z)$ (and, thus, also the subset $\mathbf {S}^\bullet (Z)$) as a poset, where $a \leq c$ if and only if there is a (necessarily unique) $b$ in $\mathbf {S}^{\bullet,+}(Z)$ such that $ab = c$. In other words, if we think of this poset as a category, then each morphism is labeled by an element of $\mathbf {S}^{\bullet,+}(Z)$ (i.e. the underlying graph is the directed Cayley graph of the monoid with the identity vertex removed). In particular, we can think of elements of the nerve as labeled by multisets in two different ways. The poset interpretation gives that a $k$-simplex in $N(\mathbf {S}^{\bullet,+}(Z))$ is a chain of multisets $I_0 \leq \dotsm \leq I_k$, while the monoid interpretation gives that a $k$-simplex is an ordered list of finite multisets in $Z$, $(J_0, J_1, \ldots, J_k)$, with the bijection given by $I_s = J_0 + \dotsm +J_s$, $J_s = I_s - I_{s-1}$ (here set $I_{-1} = \emptyset$), and where the subtraction exists by definition of the order relation and is unique because the monoid is cancelative. The simplicial subset $N\mathbf {S}^{\bullet } \subseteq N\mathbf {S}^{\bullet,+}$ consists of the simplices where $I_0=J_0 \neq \emptyset$, and $N\mathbf {C}^{\bullet,+} \subset N\mathbf {S}^{\bullet,+}$ consists of the simplices where each $I_s$ is a set (i.e. each element has multiplicity zero or one) or where the $J_s$ are pairwise disjoint.

There is a natural rank function $\mathbf {S}^{\bullet,+}(Z) \rightarrow \mathbb {Z}_{\geq 0}$, given by cardinality, and the rank $k$ component can be described as

\[ \mathbf{S}^{k}(Z) = Z^k / \mathfrak{S}_k. \]

In the monoid interpretation of the nerve, we can write

\[ N\mathbf{S}^{\bullet,+}(Z)_p = \bigsqcup_{ \underline{k}=(k_0, \ldots, k_p) \in \mathbb{Z}_{\ge 0}^p} \mathbf{S}^{\underline{k}}(Z), \]

where $\mathbf {S}^{\underline {k}}(Z) = \prod _{i=0}^{p} \mathbf {S}^{k_i}(Z)$. The nerve of $\mathbf {S}^{\bullet }(Z)$ consists of those simplices where $k_0 \neq 0$. In this description, the face map

\[ \delta_i: N\mathbf{S}^{\bullet,+}(Z)_p \rightarrow N\mathbf{S}^{\bullet,+}(Z)_{p-1} \]

is described as follows: for $0 \leq i < p$, it merges the $i$th and $(i+1)$th set, while for $i=p$ it forgets the $p$th set. The nerve of the configuration poset $\mathbf {C}^{\bullet }(Z)$ consists of the simplex spaces

\[ \mathbf{C}^{\underline{k}}(Z) = \big( Z^{\sum \underline{k}} \backslash \Delta \big) / \mathfrak{S}_{\underline{k}}, \]

where $\sum \underline {k} = k_0 + \cdots + k_p$ and $\mathfrak {S}_{\underline {k}} = \prod _{i = 0}^p \mathfrak {S}_{k_i}$ denotes the subgroup of $\mathfrak {S}_{\sum \underline {k}}$ preserving subsequent blocks of size $k_i$.

An element of $\mathbf {S}^\bullet (Z)$ can be written as a formal sum $\sum _{z \in Z} n_z z$ where $n_z \in \mathbb {Z}_{\geq 0}$, $n_z = 0$ for all but finitely many $z \in Z$ and $n_z>0$ for at least one $z$; the rank is given by $\sum _{z \in Z} n_z$. There is a natural support map

\[ \mathrm{Support}: \mathbf{S}^\bullet(Z) \rightarrow \mathbf{C}^\bullet(Z), \sum n_x z \mapsto \{ z \in Z | n_z>0 \}. \]

We equip both $N\mathbf {S}^\bullet (Z)$ and $N\mathbf {C}^{\bullet }(Z)$ with the rank filtration. We then obtain immediately the following result.

6.2 The poscheme of effective zero-cycles

In the following we assume that $Z/X$ satisfies the condition (AF) that any finite set in a single fiber is contained in a quasi-affine open (see [Reference RydhRyd08, Paper III, Appendix A.1]); this holds, in particular, if $Z/X$ is quasi-projective. The symmetric powers $\mathbf {S}^k_X(Z)$ defined by the quotient (6.0.0.2) then exist as schemes, and we consider the symmetric power monoid

\[ \mathbf{S}^{\bullet,+}_X(Z) = \bigsqcup_{k=0}^\infty \mathbf{S}^k_X(Z), \]

where the monoid multiplication

\[ \mathbf{S}^{k_1}_X(Z) \times \mathbf{S}^{k_2}_X(Z) \rightarrow \mathbf{S}^{k_1+k_2}_X(Z) \]

is induced by the quotient property from the natural map

\[ Z^{\times_X k_1} \times_X Z^{\times_X k_2} \rightarrow Z^{\times_X k_1+k_2} \rightarrow \mathbf{S}^{k_1+k_2}_X(Z) \]

and the identification

\[ \big( Z^{\times_X k_1} \times_X Z^{\times_X k_2} \big) / (\mathfrak{S}_{k_1} \times \mathfrak{S}_{k_2}) = \mathbf{S}^{k_1}_X(Z) \times \mathbf{S}^{k_2}_X(Z). \]

In characteristic zero, this provides a good notion of moduli of effective zero-cycles, compatible with arbitrary base change, and the monoid map can be used to define a poscheme structure compatible with the obvious poset structure on geometric points (see below). In positive and mixed characteristic there are some well-known perversities of symmetric powers (see, e.g., [Reference LundkvistLun08]) and, in particular, it is not clear that the monoid structure defines a poscheme structure for a general $Z/X$.

We can address this by using divided powers schemes: in [Reference RydhRyd08, Paper III], there is defined, for any scheme $Z/X$ and $r \geq 0$, a divided powers schemes $\Gamma ^r(Z/X)$, which we write here as $\Gamma ^r_X(Z)$. Because we have assumed that $Z/X$ satisfies the condition (AF), each $\Gamma ^r_X(Z)$ is represented by a scheme. In [Reference RydhRyd08, Paper III] it is shown that:

1. (i) the formation of $\Gamma ^r_X(Z)$ is stable under arbitrary change of base $X' \rightarrow X$;

2. (ii) if $X=\mathrm {Spec}\, A$, $Z=\mathrm {Spec}\, B$, then $\Gamma ^r_X(Z)=\mathrm {Spec}\,\Gamma ^r_A(B)$, where $\Gamma ^r_A(B)$ is the $r$th divided power of $B$ as an $A$-module;

3. (iii) there is a commutative monoid structure on

   \[ \Gamma^{\bullet,+}_X(Z)=\bigsqcup_{k=0}^\infty \Gamma^k_X(Z) \]
   compatible with the degree, i.e. restricting to maps
   \[ \Gamma^{r}_X(Z) \times_X \Gamma^{s}_X(Z) \rightarrow \Gamma^{r+s}_X(Z); \]

4. (iv) there is a canonical universal homeomorphism that induces isomorphisms on residue fields

   \[ \mathrm{SG}:\mathbf{S}^{\bullet,+}_X(Z) \rightarrow \Gamma^{\bullet,+}_X(Z); \]
   the map $\mathrm {SG}$ is compatible with the monoid structures, and if $Z/X$ is flat (e.g. if $X=\mathrm {Spec}\, \kappa$ for $\kappa$ a field) or if $X/\mathbb {Q}$, then it is an isomorphism;

5. (v) there is a dense open non-degenerate locus $\Gamma ^r_X(Z)^{\mathrm {nd}}$ such that $\mathrm {SG}$ restricts to an isomorphism $\mathbf {C}^r_X(Z) \rightarrow \Gamma ^r_X(Z)^{\mathrm {nd}}$;

6. (vi) if $Z/X$ is projective, then, for any sufficiently high power $\mathcal {L}^n$ of a relatively ample bundle $\mathcal {L}$, $\Gamma ^r_X(Z)$ is naturally identified with the Chow scheme of effective zero-cycles of degree $r$ on $Z$ for $\mathcal {L}^n$ (or, more accurately, its reduced subscheme is identified with the Chow variety, and this identification equips the latter with a natural scheme structure).

By part (iv), $\Gamma ^\bullet _X(Z)$ agrees with $\mathbf {S}^\bullet _X(Z)$ for $X\!$ of characteristic zero or when $X\!$ is a geometric point, but $\Gamma ^\bullet _X(Z)$ is better behaved for $X$ of positive or mixed characteristic. Over a geometric point, we will sometimes write the more familiar $\mathbf {S}^\bullet$ instead of $\Gamma ^\bullet$, with the implicit understanding that these are canonically isomorphic in this case.

Proposition 6.2.2 below says that the monoid scheme $\Gamma ^{\bullet,+}_X(Z)$ is cancelative in a scheme-theoretic sense. As a consequence, we will see that the monoid multiplication induces a natural poscheme structure. In the proof, we use some standard properties of divided powers modules recalled in the following lemma.

Proposition 6.2.2 For any map of schemes $Z\rightarrow X$ satisfying (AF), the map

\begin{align*} \Gamma^{r}_X(Z) \times \Gamma^{s}_X(Z) & \rightarrow \Gamma^{r}_X(Z) \times \Gamma^{r+s}_X(Z) \\ (x,y) & \mapsto (x, x+y) \end{align*}

is a closed immersion.