The Stacks project

62.6 Relative cycles

Here is the definition we will work with; see Section 62.15 for a comparison with the definitions in [SV].

Definition 62.6.1. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. A relative $r$-cycle on $X/S$ is a family $\alpha $ of $r$-cycles on fibres of $X/S$ such that for every morphism $g : S' \to S$ where $S'$ is the spectrum of a discrete valuation ring we have

\[ sp_{X'/S'}(\alpha _\eta ) = \alpha _0 \]

where $sp_{X'/S'}$ is as in Section 62.4 and $\alpha _\eta $ (resp. $\alpha _0$) is the value of the base change $g^*\alpha $ of $\alpha $ at the generic (resp. closed) point of $S'$. The group of all relative $r$-cycles on $X/S$ is denoted $z(X/S, r)$.

Lemma 62.6.2. Let $\alpha $ be a relative $r$-cycle on $X/S$ as in Definition 62.6.1. Then any restriction, base change, flat pullback, or proper pushforward of $\alpha $ is a relative $r$-cycle.

Proof. For flat pullback use Lemma 62.4.4. Restriction is a special case of flat pullback. To see it holds for base change use that base change is transitive. For proper pushforward use Lemma 62.4.5. $\square$

Lemma 62.6.3. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Let $\{ g_ i : S_ i \to S\} $ be a h covering (More on Flatness, Definition 38.34.2). Then $\alpha $ is a relative $r$-cycle if and only if each base change $g_ i^*\alpha $ is a relative $r$-cycle.

Proof. If $\alpha $ is a relative $r$-cycle, then each base change $g_ i^*\alpha $ is a relative $r$-cycle by Lemma 62.6.2. Assume each $g_ i^*\alpha $ is a relative $r$-cycle. Let $g : S' \to S$ be a morphism where $S'$ is the spectrum of a discrete valuation ring. After replacing $S$ by $S'$, $X$ by $X' = X \times _ S S'$, and $\alpha $ by $\alpha ' = g^*\alpha $ and using that the base change of a h covering is a h covering (More on Flatness, Lemma 38.34.9) we reduce to the problem studied in the next paragraph.

Assume $S$ is the spectrum of a discrete valuation ring with closed point $0$ and generic point $\eta $. We have to show that $sp_{X/S}(\alpha _\eta ) = \alpha _0$. Since a h covering is a V covering (by definition), there is an $i$ and a specialization $s' \leadsto s$ of points of $S_ i$ with $g_ i(s') = \eta $ and $g_ i(s) = 0$, see Topologies, Lemma 34.10.13. By Properties, Lemma 28.5.10 we can find a morphism $h : S' \to S_ i$ from the spectrum $S'$ of a discrete valuation ring which maps the generic point $\eta '$ to $s'$ and maps the closed point $0'$ to $s$. Denote $\alpha ' = h^*g_ i^*\alpha $. By assumption we have $sp_{X'/S'}(\alpha '_{\eta '}) = \alpha '_{0'}$. Since $g = g_ i \circ h : S' \to S$ is the morphism of schemes induced by an extension of discrete valuation rings we conclude that $sp_{X/S}$ and $sp_{X'/S'}$ are compatible with base change maps on the fibres, see Lemma 62.4.3. We conclude that $sp_{X/S}(\alpha _\eta ) = \alpha _0$ because the base change map $Z_ r(X_0) \to Z_ r(X'_{0'})$ is injective as discussed in Section 62.3. $\square$

Lemma 62.6.4. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Let $\{ f_ i : X_ i \to X\} $ be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension $e$. Then $\alpha $ is a relative $r$-cycle if and only if each flat pullback $f_ i^*\alpha $ is a relative $r$-cycle.

Proof. If $\alpha $ is a relative $r$-cycle, then each pull back $f_ i^*\alpha $ is a relative $r$-cycle by Lemma 62.6.2. Assume each $f_ i^*\alpha $ is a relative $r$-cycle. Let $g : S' \to S$ be a morphism where $S'$ is the spectrum of a discrete valuation ring. After replacing $S$ by $S'$, $X$ by $X' = X \times _ S S'$, and $\alpha $ by $\alpha ' = g^*\alpha $ we reduce to the problem studied in the next paragraph.

Assume $S$ is the spectrum of a discrete valuation ring with closed point $0$ and generic point $\eta $. We have to show that $sp_{X/S}(\alpha _\eta ) = \alpha _0$. Denote $f_{i, 0} : X_{i, 0} \to X_0$ the base change of $f_ i$ to the closed point of $S$. Similarly for $f_{i, \eta }$. Observe that

\[ f_{i, 0}^*sp_{X/S}(\alpha _\eta ) = sp_{X_ i/S}(f_{i, \eta }^*\alpha _\eta ) = f_{i, 0}^*\alpha _0 \]

Namely, the first equality holds by Lemma 62.4.4 and the second by assumption. Since the family of maps $f_{i, 0}^* : Z_ r(X_0) \to Z_ r(X_{i, 0})$ is jointly injective (due to the fact that $f_{i, 0}$ is jointly surjective), we conclude what we want. $\square$

Lemma 62.6.5. Let $S$ be a locally Noetherian scheme. Let $i : X \to Y$ be a closed immersion of schemes locally of finite type over $S$. Let $r \geq 0$. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Then $\alpha $ is a relative $r$-cycle on $X/S$ if and only if $i_*\alpha $ is a relative $r$-cycle on $Y/S$.

Proof. Since base change commutes with $i_*$ (Lemma 62.5.1) it suffices to prove the following: if $S$ is the spectrum of a discrete valuation ring with generic point $\eta $ and closed point $0$, then $sp_{X/S}(\alpha _\eta ) = \alpha _0$ if and only if $sp_{Y/S}(i_{\eta , *}\alpha _\eta ) = i_{0, *}\alpha _0$. This is true because $i_{0, *} : Z_ r(X_0) \to Z_ r(Y_0)$ is injective and because $i_{0, *}sp_{X/S}(\alpha _\eta ) = sp_{Y/S}(i_{\eta , *}\alpha _\eta )$ by Lemma 62.4.5. $\square$

The following lemma will be strengthened in Lemma 62.6.12.

Lemma 62.6.6. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ locally of finite type. Let $r \geq 0$. Let $\alpha $ and $\beta $ be relative $r$-cycles on $X/S$. The following are equivalent

  1. $\alpha = \beta $, and

  2. $\alpha _\eta = \beta _\eta $ for any generic point $\eta \in S$ of an irreducible component of $S$.

Proof. The implication (1) $\Rightarrow $ (2) is immediate. Assume (2). For every $s \in S$ we can find an $\eta $ as in (2) which specializes to $s$. By Properties, Lemma 28.5.10 we can find a morphism $g : S' \to S$ from the spectrum $S'$ of a discrete valuation ring which maps the generic point $\eta '$ to $\eta $ and maps the closed point $0$ to $s$. Then $\alpha _ s$ and $\beta _ s$ are elements of $Z_ r(X_ s)$ which base change to the same element of $Z_ r(X_{0'})$, namely $sp_{X_{S'}/S'}(\alpha _{\eta '})$ where $\alpha _{\eta '}$ is the base change of $\alpha _\eta $. Since the base change map $Z_ r(X_ s) \to Z_ r(X_{0'})$ is injective as discussed in Section 62.3 we conclude $\alpha _ s = \beta _ s$. $\square$

Lemma 62.6.7. In the situation of Example 62.5.2 assume $S$ is locally Noetherian and $\mathcal{F}$ is flat over $S$ in dimensions $\geq r$ (More on Flatness, Definition 38.20.10). Then $[\mathcal{F}/X/S]_ r$ is a relative $r$-cycle on $X/S$.

Proof. By More on Flatness, Lemma 38.20.9 the hypothesis on $\mathcal{F}$ is preserved by any base change. Also, formation of $[\mathcal{F}/X/S]_ r$ is compatible with any base change by Lemma 62.5.3. Since the condition of being compatible with specializations is checked after base change to the spectrum of a discrete valuation ring, this reduces us to the case where $S$ is the spectrum of a valuation ring. In this case the set $U = \{ x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\} $ is open in $X$ by More on Flatness, Lemma 38.13.11. Since the complement of $U$ in $X$ has fibres of dimension $< r$ over $S$ by assumption, we see that restriction along the inclusion $U \subset X$ induces an isomorphism on the groups of $r$-cycles on fibres after any base change, compatible with specialization maps and with formation of the relative cycle associated to $\mathcal{F}$. Thus it suffices to show compatibility with specializations for $[\mathcal{F}|_ U / U /S]_ r$. Since $\mathcal{F}|_ U$ is flat over $S$, this follows from Lemma 62.4.1 and the definitions. $\square$

Lemma 62.6.8. In the situation of Example 62.5.4 assume $S$ is locally Noetherian and $Z$ is flat over $S$ in dimensions $\geq r$. Then $[Z/X/S]_ r$ is a relative $r$-cycle on $X/S$.

Proof. The assumption means that $\mathcal{O}_ Z$ is flat over $S$ in dimensions $\geq r$. Thus applying Lemma 62.6.7 with $\mathcal{F} = (Z \to X)_*\mathcal{O}_ Z$ we conclude. $\square$

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism which is of finite type. Let $r \geq 0$. Denote $Hilb(X/S, r)$ the set of closed subschemes $Z \subset X$ such that $Z \to S$ is flat and of relative dimension $\leq r$. By Lemma 62.6.8 for each $Z \in Hilb(X/S, r)$ we have an element $[Z/X/S]_ r \in z(X/S, r)$. Thus we obtain a group homomorphism

62.6.8.1
\begin{equation} \label{relative-cycles-equation-cycle-classes} \text{free abelian group on }Hilb(X/S, r) \longrightarrow z(X/S, r) \end{equation}

sending $\sum n_ i[Z_ i]$ to $\sum n_ i[Z_ i/X/S]_ r$. A key feature of relative $r$-cycles is that they are locally (on $X$ and $S$ in suitable topologies) in the image of this map.

Lemma 62.6.9. Let $f : X \to S$ be a finite type morphism of schemes with $S$ Noetherian. Let $r \geq 0$. Let $\alpha $ be a relative $r$-cycle on $X/S$. Then there is a proper, completely decomposed (More on Morphisms, Definition 37.78.1) morphism $g : S' \to S$ such that $g^*\alpha $ is in the image of (62.6.8.1).

Proof. By Noetherian induction, we may assume the result holds for the pullback of $\alpha $ by any closed immersion $g : S' \to S$ which is not an isomorphism.

Let $S_1 \subset S$ be an irreducible component (viewed as an integral closed subscheme). Let $S_2 \subset S$ be the closure of the complement of $S'$ (viewed as a reduced closed subscheme). If $S_2 \not= \emptyset $, then the result holds for the pullback of $\alpha $ by $S_1 \to S$ and $S_2 \to S$. If $g_1 : S'_1 \to S_1$ and $g_2 : S'_2 \to S_2$ are the corresponding completely decomposed proper morphisms, then $S' = S'_1 \amalg S'_2 \to S$ is a completely decomposed proper morphism and we see the result holds for $S$1 . Thus we may assume $S' \to S$ is bijective and we reduce to the case described in the next paragraph.

Assume $S$ is integral. Let $\eta \in S$ be the generic point and let $K = \kappa (\eta )$ be the function field of $S$. Then $\alpha _\eta $ is an $r$-cycle on $X_ K$. Write $\alpha _\eta = \sum n_ i[Y_ i]$. Taking the closure of $Y_ i$ we obtain integral closed subschemes $Z_ i \subset X$ whose base change to $\eta $ is $Y_ i$. By generic flatness (for example Morphisms, Proposition 29.27.1), we see that $Z_ i$ is flat over a nonempty open $U$ of $S$ for each $i$. Applying More on Flatness, Lemma 38.31.1 we can find a $U$-admissible blowing up $g : S' \to S$ such that the strict transform $Z'_ i \subset X_{S'}$ of $Z_ i$ is flat over $S'$. Then $\beta = \sum n_ i[Z'_ i/X_{S'}/S']_ r$ is in the image of (62.6.8.1) and $\beta = g^*\alpha $ by Lemma 62.6.6.

However, this does not finish the proof as $S' \to S$ may not be completely decomposed. This is easily fixed: denoting $T \subset S$ the complement of $U$ (viewed as a closed subscheme), by Noetherian induction we can find a completely decomposed proper morphism $T' \to T$ such that $(T' \to S)^*\alpha $ is in the image of (62.6.8.1). Then $S' \amalg T' \to S$ does the job. $\square$

Lemma 62.6.10. Let $f : X \to S$ be a finite type morphism of schemes with $S$ the spectrum of a discrete valuation ring. Let $r \geq 0$. Then (62.6.8.1) is surjective.

Proof. This of course follows from Lemma 62.6.9 but we can also see it directly as follows. Say $\alpha $ is a relative $r$-cycle on $X/S$. Write $\alpha _\eta = \sum n_ i[Z_ i]$ (the sum is finite). Denote $\overline{Z}_ i \subset X$ the closure of $Z_ i$ as in Section 62.4. Then $\alpha = \sum n_ i[\overline{Z}_ i/X/S]$. $\square$

Lemma 62.6.11. Let $f : X \to S$ be a morphism of schemes. Let $r \geq 0$. Assume $S$ locally Noetherian and $f$ smooth of relative dimension $r$. Let $\alpha \in z(X/S, r)$. Then the support of $\alpha $ is open and closed in $X$ (see proof for a more precise result).

Proof. Let $x \in X$ with image $s \in S$. Since $f$ is smooth, there is a unique irreducible component $Z(x)$ of $X_ s$ which contains $x$. Then $\dim (Z(x)) = r$. Let $n_ x$ be the coefficient of $Z(x)$ in the cycle $\alpha _ s$. We will show the function $x \mapsto n_ x$ is locally constant on $X$.

Let $g : S' \to S$ be a morphism of locally Noetherian schemes. Let $X'$ be the base change of $X$ and let $\alpha ' = g^*\alpha $ be the base change of $\alpha $. Let $x' \in X'$ map to $s' \in S'$, $x \in X$, and $s \in S$. We claim $n_{x'} = n_ x$. Namely, since $Z(x)$ is smooth over $\kappa (s)$ we see that $Z(x) \times _{\mathop{\mathrm{Spec}}(\kappa (s))} \mathop{\mathrm{Spec}}(\kappa (s'))$ is reduced. Since $Z(x')$ is an irreducible component of this scheme, we see that the coefficient $n_{x'}$ of $Z(x')$ in $\alpha '_{s'}$ is the same as the coefficient $n_ x$ of $Z(x)$ in $\alpha _ s$ by the definition of base change in Section 62.3 thereby proving the claim.

Since $X$ is locally Noetherian, to show that $x \mapsto n_ x$ is locally constant, it suffices to show: if $x' \leadsto x$ is a specialization in $X$, then $n_{x'} = n_ x$. Choose a morphism $S' \to X$ where $S'$ is the spectrum of a discrete valuation ring mapping the generic point $\eta $ to $x'$ and the closed point $0$ to $x$. See Properties, Lemma 28.5.10. Then the base change $X' \to S'$ of $f$ by $S' \to S$ has a section $\sigma : S' \to X'$ such that $\sigma (\eta ) \leadsto \sigma (0)$ is a specialization of points of $X'$ mapping to $x' \leadsto x$ in $X$. Thus we reduce to the claim in the next paragraph.

Let $S$ be the spectrum of a discrete valuation ring with generic point $\eta $ and closed point $0$ and we have a section $\sigma : S \to X$. Claim: $n_{\sigma (\eta )} = n_{\sigma (0)}$. By the discussion in More on Morphisms, Section 37.29 and especially More on Morphisms, Lemma 37.29.6 after replacing $X$ by an open subscheme, we may assume the fibres of $X \to S$ are connected. Since these fibres are smooth, they are irreducible. Then we see that $\alpha _\eta = n[X_\eta ]$ with $n = n_{\sigma (\eta )}$ and the relation $sp_{X/S}(\alpha _\eta ) = \alpha _0$ implies $\alpha _0 = n[X_0]$, i.e., $n_{\sigma (0)} = n$ as desired. $\square$

Lemma 62.6.12. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ and $\alpha , \beta \in z(X/S, r)$. The set $E = \{ s \in S : \alpha _ s = \beta _ s\} $ is closed in $S$.

Proof. The question is local on $S$, thus we may assume $S$ is affine. Let $X = \bigcup U_ i$ be an affine open covering. Let $E_ i = \{ s \in S : \alpha _ s|_{U_{i, s}} = \beta _ s|_{U_{i, s}}\} $. Then $E = \bigcap E_ i$. Hence it suffices to prove the lemma for $U_ i \to S$ and the restriction of $\alpha $ and $\beta $ to $U_ i$. This reduces us to the case discussed in the next paragraph.

Assume $X$ and $S$ are quasi-compact. Set $\gamma = \alpha - \beta $. Then $E = \{ s \in S : \gamma _ s = 0\} $. By Lemma 62.6.8 there exists a jointly surjective finite family of proper morphisms $\{ g_ i : S_ i \to S\} $ such that $g_ i^*\gamma $ is in the image of (62.6.8.1). Observe that $E_ i = g_ i^{-1}(E)$ is the set of point $t \in S_ i$ such that $(g_ i^*\gamma )_ t = 0$. If $E_ i$ is closed for all $i$, then $E = \bigcup g_ i(E_ i)$ is closed as well. This reduces us to the case discussed in the next paragraph.

Assume $X$ and $S$ are quasi-compact and $\gamma = \sum n_ i[Z_ i/X/S]_ r$ for a finite number of closed subschemes $Z_ i \subset X$ flat and of relative dimension $\leq r$ over $S$. Set $X' = \bigcup Z_ i$ (scheme theoretic union). Then $i : X' \to X$ is a closed immersion and $X'$ has relative dimension $\leq r$ over $S$. Also $\gamma = i_*\gamma '$ where $\gamma ' = \sum n_ i[Z_ i/X'/S]_ r$. Since clearly $E = E' = \{ s \in S : \gamma '_ s = 0\} $ we reduce to the case discussed in the next paragraph.

Assume $X$ has relative dimension $\leq r$ over $S$. Let $s \in S$, $s \not\in E$. We will show that there exists an open neighbourhood $V \subset S$ of $s$ such that $E \cap V$ is empty. The assumption $s \not\in E$ means there exists an integral closed subscheme $Z \subset X_ s$ of dimension $r$ such that the coefficient $n$ of $[Z]$ in $\gamma _ s$ is nonzero. Let $x \in Z$ be the generic point. Since $\dim (Z) = r$ we see that $x$ is a generic point of an irreducible component (namely $Z$) of $X_ s$. Thus after replacing $X$ by an open neighbourhood of $x$, we may assume that $Z$ is the only irreducible component of $X_ s$. In particular, we have $\gamma _ s = n[Z]$.

At this point we apply More on Morphisms, Lemma 37.47.1 and we obtain a diagram

\[ \xymatrix{ X \ar[dd] & X' \ar[l]^ g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^ h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] } \]

with all the properties listed there. Let $\gamma ' = g^*\gamma $ be the flat pullback. Note that $E \subset E' = \{ s \in S: \gamma '_ s = 0\} $ and that $s \not\in E'$ because the coefficient of $Z'$ in $\gamma '_ s$ is nonzero, where $Z' \subset X'_ s$ is the closure of $x'$. Similarly, set $\gamma '' = \pi _*\gamma '$. Then we have $E' \subset E'' = \{ s \in S: \gamma ''_ s = 0\} $ and $s \not\in E''$ because the coefficient of $Z''$ in $\gamma ''_ s$ is nonzero, where $Z'' \subset Y_ s$ is the closure of $y$. By Lemma 62.6.11 and openness of $Y \to S$ we see that an open neighbourhood of $s$ is disjoint from $E''$ and the proof is complete. $\square$

Lemma 62.6.13. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ be the limit of a directed inverse system of Noetherian schemes with affine transition morphisms. Let $0 \in I$ and let $X_0 \to S_0$ be a finite type morphism of schemes. For $i \geq 0$ set $X_ i = S_ i \times _{S_0} X_0$ and set $X = S \times _{S_0} X_0$. If $S$ is Noetherian too, then

\[ z(X/S, r) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} z(X_ i/S_ i, r) \]

where the transition maps are given by base change of relative $r$-cycles.

Proof. Suppose that $i \geq 0$ and $\alpha _ i, \beta _ i \in z(X_ i/S_ i, r)$ map to the same element of $z(X/S, r)$. Then $S \to S_ i$ maps into the closed subset $E \subset S_ i$ of Lemma 62.6.12. Hence for some $j \geq i$ the morphism $S_ j \to S_ i$ maps into $E$, see Limits, Lemma 32.4.10. It follows that the base change of $\alpha _ i$ and $\beta _ i$ to $S_ j$ agree. Thus the map is injective.

Let $\alpha \in z(X/S, r)$. Applying Lemma 62.6.9 a completely decomposed proper morphism $g : S' \to S$ such that $g^*\alpha $ is in the image of (62.6.8.1). Set $X' = S' \times _ S X$. We write $g^*\alpha = \sum n_ a [Z_ a/X'/S']_ r$ for some $Z_ a \subset X'$ closed subscheme flat and of relative dimension $\leq r$ over $S'$.

Now we bring the machinery of Limits, Section 32.10 ff to bear. We can find an $i \geq 0$ such that there exist

  1. a completely decomposed proper morphism $g_ i : S'_ i \to S_ i$ whose base change to $S$ is $g : S' \to S$,

  2. setting $X'_ i = S'_ i \times _{S_ i} X_ i$ closed subschemes $Z_{ai} \subset X'_ i$ flat and of relative dimension $\leq r$ over $S'_ i$ whose base change to $S'$ is $Z_ a$.

To do this one uses Limits, Lemmas 32.10.1, 32.8.5, 32.8.7, 32.13.1, and 32.18.1 and More on Morphisms, Lemma 37.78.5. Consider $\alpha '_ i = \sum n_ a [Z_{ai}/X'_ i/S'_ i]_ r \in z(X'_ i/S'_ i, r)$. The image of $\alpha '_ i$ in $z(X'/S', r)$ agrees with the base change $g^*\alpha $ by construction.

Set $S''_ i = S'_ i \times _{S_ i} S'_ i$ and $X''_ i = S''_ i \times _{S_ i} X_ i$ and set $S'' = S' \times _ S S'$ and $X'' = S'' \times _ S X$. We denote $\text{pr}_1, \text{pr}_2 : S'' \to S'$ and $\text{pr}_1, \text{pr}_2 : S''_ i \to S'_ i$ the projections. The two base changes $\text{pr}_1^*\alpha '_ i$ and $\text{pr}_1^*\alpha '_ i$ map to the same element of $z(X''/S'', r)$ because $\text{pr}_1^*g^*\alpha = \text{pr}_1^*g^*\alpha $. Hence after increasing $i$ we may assume that $\text{pr}_1^*\alpha '_ i = \text{pr}_1^*\alpha '_ i$ by the first paragraph of the proof. By Lemma 62.5.9 we obtain a unique family $\alpha _ i$ of $r$-cycles on fibres of $X_ i/S_ i$ with $g_ i^*\alpha _ i = \alpha '_ i$ (this uses that $S'_ i \to S_ i$ is completely decomposed). By Lemma 62.6.3 we see that $\alpha _ i \in z(X_ i/S_ i, r)$. The uniqueness in Lemma 62.5.9 implies that the image of $\alpha _ i$ in $z(X/S, r)$ is $\alpha $ and the proof is complete. $\square$

Lemma 62.6.14. Let $S$ be a locally Noetherian scheme. Let $i : X \to X'$ be a thickening of schemes locally of finite type over $S$. Let $r \geq 0$. Then $i_* : z(X/S, r) \to z(X'/S, r)$ is a bijection.

Proof. Since $i_ s : X_ s \to X'_ s$ is a thickening it is clear that $i_*$ induces a bijection between families of $r$-cycles on the fibres of $X/S$ and families of $r$-cycles on the fibres of $X'/S$. Also, given a family $\alpha $ of $r$-cycles on the fibres of $X/S$ $\alpha \in z(X/S, r) \Leftrightarrow i_*\alpha \in z(X'/S, r)$ by Lemma 62.6.5. The lemma follows. $\square$

Lemma 62.6.15. Let $S$ be a locally Noetherian scheme. Let $X$ be a scheme locally of finite type over $S$. Let $r \geq 0$. Let $U \subset X$ be an open such that $X \setminus U$ has relative dimension $< r$ over $S$, i.e., $\dim (X_ s \setminus U_ s) < r$ for all $s \in S$. Then restriction defines a bijection $z(X/S, r) \to z(U/S, r)$.

Proof. Since $Z_ r(X_ s) \to Z_ r(U_ s)$ is a bijection by the dimension assumption, we see that restriction induces a bijection between families of $r$-cycles on the fibres of $X/S$ and families of $r$-cycles on the fibres of $U/S$. These restriction maps $Z_ r(X_ s) \to Z_ r(U_ s)$ are compatible with base change and with specializations, see Lemma 62.5.1 and 62.4.4. The lemma follows easily from this; details omitted. $\square$

Lemma 62.6.16. Let $g : S' \to S$ be a universal homeomorphism of locally Noetherian schemes which induces isomorphisms of residue fields. Let $f : X \to S$ be locally of finite type. Set $X' = S' \times _ S X$. Let $r \geq 0$. Then base change by $g$ determines a bijection $z(X/S, r) \to z(X'/S', r)$.

Proof. By Lemma 62.5.10 we have a bijection between the group of families of $r$-cycles on fibres of $X/S$ and the group of families of $r$-cycles on fibres of $X'/S'$. Say $\alpha $ is a families of $r$-cycles on fibres of $X/S$ and $\alpha ' = g^*\alpha $ is the base change. If $R$ is a discrete valuation ring, then any morphism $h : \mathop{\mathrm{Spec}}(R) \to S$ factors as $g \circ h'$ for some unique morphism $h' : \mathop{\mathrm{Spec}}(R) \to S'$. Namely, the morphism $S' \times _ S \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(R)$ is a universal homomorphism inducing bijections on residue fields, and hence has a section (for example because $R$ is a seminormal ring, see Morphisms, Section 29.47). Thus the condition that $\alpha $ is compatible with specializations (i.e., is a relative $r$-cycle) is equivalent to the condition that $\alpha '$ is compatible with specializations. $\square$

[1] Namely, any closed subscheme of $S'_1 \times _ S X$ flat and of relative dimension $\leq r$ over $S'_1$ may be viewed as a closed subscheme of $S' \times _ S X$ flat and of relative dimension $\leq r$ over $S'$.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0H4Z. Beware of the difference between the letter 'O' and the digit '0'.