2.1. The Levine–Weibel Chow group

Let k be any field and X a reduced quasiprojective scheme of dimension $d \ge 2$ over k. Recall from [Reference Binda and Krishna3, Definition 3.5] that a Cartier curve on X is a purely $1$ -dimensional reduced closed subscheme $C \hookrightarrow X$ , none of whose irreducible components is contained in $X_{\textrm {sing}}$ , and whose defining sheaf of ideals is a local complete intersection in ${\mathcal O}_X$ at every point of $C \cap X_{\textrm {sing}}$ . Let $k(C)$ denote the ring of total quotients for a Cartier curve C on X, and let $\{C_1, \dotsc , C_r\}$ be the set of irreducible components of C. For $f \in {\mathcal O}^{\times }_{C, C \cap X_{\textrm {sing}}} \subset k(C)^{\times }$ , let $f_i \in k(C_i)^{\times }$ be the image of f under the projection ${\mathcal O}^{\times }_{C, C \cap X_{\textrm {sing}}} \hookrightarrow k(C)^{\times } \twoheadrightarrow k(C_i)^{\times }$ . We let ${\textrm {div}}(f) = \stackrel {r}{\underset {i =1}\sum } {\textrm {div}}(f_i) \in {\mathcal Z}_0(X^o)$ be the cycle associated to f. We let ${\mathcal R}^{LW}_0(X)$ denote the subgroup of ${\mathcal Z}_0(X^o)$ generated by ${\textrm {div}}(f)$ , where $C \subset X$ is a Cartier curve and $f \in {\mathcal O}^{\times }_{C, C \cap X_{\textrm {sing}}}$ . The Levine–Weibel Chow group of X is the quotient ${{\mathcal Z}_0(X^o)}/{{\mathcal R}^{LW}_0(X)}$ and is denoted by ${\textrm {CH}}^{LW}_0(X)$ .

It is immediate from the definition that ${\textrm {CH}}^{LW}_0(X)$ coincides with ${\textrm {CH}}^F_0(X)$ if X is regular. We also remark that the foregoing definition of ${\textrm {CH}}^{LW}_0(X)$ is slightly different from that of [Reference Levine and Weibel50]. However, this difference disappears if k is infinite, as a consequence of [Reference Levine49, Lemma 1.4]. If X is integral and projective over k, there is a degree map $\deg \colon {\textrm {CH}}^{LW}_0(X) \to {\mathbb Z}$ and we let ${\textrm {CH}}^{LW}_0(X)^0$ be the kernel of this map.

Let ${\mathcal R}^{{\textrm {lci}}}_0(X)$ be the subgroup of ${\mathcal Z}_0(X^o)$ generated by $\nu _*({\textrm {div}}(f))$ for $f \in {\mathcal O}^{\times }_{C, \nu ^{-1}\left (X_{\textrm {sing}}\right )}$ , where $\nu \colon C \to X$ is a finite morphism from a reduced curve of pure dimension $1$ over k such that the image of none of the irreducible components of C is contained in $X_{\textrm {sing}}$ and $\nu $ is a local complete intersection (lci) morphism at every point $x \in C$ such that $\nu (x) \in X_{\textrm {sing}}$ . Such a curve C is called a good curve relative to $X_{\textrm {sing}}$ . The lci Chow group of $0$ -cycles for X is the quotient ${\textrm {CH}}^{{\textrm {lci}}}_0(X) = {{\mathcal Z}_0(X^o)}/{{\mathcal R}^{{\textrm {lci}}}_0(X)}$ . This modification of the Levine–Weibel Chow group was introduced in [Reference Binda and Krishna3].

Clearly, the identity map of ${\mathcal Z}_0(X^o)$ induces a surjection ${\textrm {CH}}^{LW}_0(X) \twoheadrightarrow {\textrm {CH}}^{{\textrm {lci}}}_0(X)$ . If k is infinite, we can say the following:

Proof. Let $\nu \colon C \to X$ be a good curve relative to $X_{\textrm {sing}}$ , and let $Z=\nu ^{-1}\left (X_{\textrm {sing}}\right )$ . It suffices to show that $\nu _*({\textrm {div}}(f)) \in {\mathcal R}^{LW}_0(X)$ for any $f \in {\mathcal O}^{\times }_{C, Z}$ . Since $\nu $ is finite, we can find a factorization

(2.1)

where i is a closed immersion and g is the canonical projection. Setting $X' = {\mathbb P}^n_X$ , we see that $X'$ is an $R_1$ -scheme and $X^{\prime }_{\textrm {sing}} = g^{-1}\left (X_{\textrm {sing}}\right )$ . In particular, $i^{-1}\left (X^{\prime }_{\textrm {sing}}\right ) \subseteq Z$ . Since $\nu $ is an lci morphism along $X_{\textrm {sing}}$ and g is smooth, it follows that the closed immersion i is regular along $X^{\prime }_{\textrm {sing}}$ . In other words, C is embedded as a Cartier curve on $X'$ . One deduces that ${\textrm {div}}(f) \in {\mathcal R}^{LW}_0(X')$ .

Since $X'$ is $R_1$ and k is infinite, it follows from [Reference Biswas and Srinivas7, Lemma 2.1] (see also [Reference Levine49, Lemma 1.4]) that there are closed reduced curves $C^{\prime }_i \subset X'$ and rational functions $f_i \in k\left (C^{\prime }_i\right )^{\times }$ such that $C^{\prime }_i \subset X^{\prime {o}}$ and ${\textrm {div}}(f) = \sum _i {\textrm {div}}(f_i) \in {\mathcal Z}_0(X^{\prime {o}})$ . If $g\left (C^{\prime }_i\right )$ is a point, then clearly $g_*({\textrm {div}}(f_i)) = 0$ , as the map $C^{\prime }_i \to X$ then factors through a regular closed point and $g_*({\textrm {div}}(f_i))$ is already zero in the Chow group of the closed point. Otherwise, we let $C_i = g\left (C^{\prime }_i\right )$ and assume that the map $C^{\prime }_i \to C_i$ is finite. Then each $C_i \subset X$ is clearly a Cartier curve, as it does not meet $X_{\textrm {sing}}$ . Let $N_i: k\left (C^{\prime }_i\right )^{\times } \to k(C_i)^{\times }$ denote the norm map. We then have

$$ \begin{align*} \nu_*({\textrm{div}}(f)) = g_*({\textrm{div}}(f)) = \sum_i g_*({\textrm{div}}(f_i)) = \sum_i {\textrm{div}}(N_i(f_i)), \end{align*} $$

and it is clear that ${\textrm {div}}(N_i(f_i)) \in {\mathcal R}^{LW}_0(X)$ for every i.

2.2. The moving lemma

One of the key ingredients for proving Theorem 1.2 is a moving lemma for the Levine–Weibel Chow group over finite fields, which we shall prove in this subsection.

Let k be any field and $X \in {\operatorname {\mathbf {Sch}}}_k$ an integral $R_1$ -scheme. Let $A \subset X$ be a closed subset of X of codimension $\geq 2$ such that $X_{\textrm {sing}} \subseteq A$ . We let ${\mathcal R}^{LW}_0(X, A)$ be the subgroup of ${\mathcal Z}_0(X \setminus A)$ generated by ${\textrm {div}}(f)$ , where f is a nonzero rational function on an integral curve $C \subset X$ such that $C \cap A = \emptyset $ . We define ${\textrm {CH}}^{LW}_0(X, A) := {{\mathcal Z}_0(X \setminus A)}/{{\mathcal R}^{LW}_0(X, A)}$ . We define ${\textrm {CH}}^{{\textrm {lci}}}_0(X,A)$ in an analogous way. The inclusion ${\mathcal Z}_0(X \setminus A) \hookrightarrow {\mathcal Z}_0(X^o)$ preserves the subgroups of rational equivalences. Hence, we get canonical maps

$$ \begin{align*} {\textrm{CH}}^{LW}_0(X,A) \to {\textrm{CH}}^{LW}_0(X), \qquad {\textrm{CH}}^{{\textrm{lci}}}_0(X,A) \to {\textrm{CH}}^{{\textrm{lci}}}_0(X). \end{align*} $$

We shall need the following application of the Bertini theorems of Altman and Kleiman [Reference Altman and Kleiman1] and Wutz [Reference Wutz77] (which is a small modification of the Bertini theorem of Poonen [Reference Poonen59]):

The moving lemma we want to prove is the following:

Lemma 2.4. Let k be any field. Then the canonical map

(2.2) $$ \begin{align} {\mathrm{CH}}^{LW}_0(X,A) \to {\mathrm{CH}}^{LW}_0(X) \end{align} $$

is an isomorphism. The same holds also for the lci Chow group.

Proof. We can assume that $\dim (X) \ge 2$ because the lemma is trivial otherwise. Before we begin the proof, we note that one always has the commutative diagram

(2.3)

Suppose now that k is infinite. Since X is $R_1$ , the top horizontal arrow is known to be an isomorphism (see [Reference Biswas and Srinivas7, Lemma 2.1] and [Reference Levine49, Lemma 1.4]). The right (resp., left) vertical arrow is an isomorphism by Lemma 2.1 (resp., Lemma 2.2). It follows that the bottom horizontal arrow is an isomorphism.

We now assume that k is finite and prove the injectivity of formula (2.2). Let $\alpha \in {\textrm {CH}}^{{\textrm {lci}}}_0(X,A)$ be a cycle which dies in ${\textrm {CH}}^{{\textrm {lci}}}_0(X)$ . Let $\ell _1 \neq \ell _2$ be two distinct prime numbers and let $k_i$ be the pro- $\ell _i$ algebraic extension of k for $i =1,2$ . For $i = 1,2$ , [Reference Binda and Krishna3, Proposition 6.1] says that there is a commutative diagram

(2.4)

The bottom horizontal arrow is an isomorphism because $k_i$ is infinite. It follows that $\alpha $ dies in ${\textrm {CH}}^{{\textrm {lci}}}_0\left (X_{k_i}, A_{k_i}\right )$ . It follows from [Reference Binda and Krishna3, Proposition 6.1] (by a straightforward modification, explained in [Reference Gupta and Krishna27, Proposition 8.5]) that there is a finite algebraic extension $k \hookrightarrow k^{\prime }_i$ inside $k_i$ such that $\alpha $ dies in ${\textrm {CH}}^{{\textrm {lci}}}_0\left (X_{k^{\prime }_i}, A_{k^{\prime }_i}\right )$ . We apply [Reference Binda and Krishna3, Proposition 6.1] once again to conclude that $\ell ^n_i \alpha = 0$ for $i =1, 2$ and some $n \gg 1$ . Since $\ell ^n_1$ and $\ell ^n_2$ are relatively prime, it follows that $\alpha = 0$ . We have thus shown that the map ${\textrm {CH}}^{{\textrm {lci}}}_0(X,A) \to {\textrm {CH}}^{{\textrm {lci}}}_0(X)$ is injective.

To show the same for the Levine–Weibel Chow group, we use diagram (2.3) again. The left vertical arrow is an isomorphism by Lemma 2.2. We just showed that the bottom horizontal arrow is injective. It follows that the top horizontal arrow is also injective.

It remains to show that formula (2.2) is surjective when k is finite. For this, we fix a closed point $x \in X^o$ . We need to show that there is a $0$ -cycle $\alpha \in {\mathcal Z}_0(X^o)$ supported on $X^o \setminus A$ such that the cycle class $[x]$ coincides with $\alpha $ in ${\textrm {CH}}^{LW}_0(X)$ .

To that end, we let $C \subset X$ be a curve as in Lemma 2.3 with $W = \{x\}$ and $B = X_{\textrm {sing}}$ . We let ${\textrm {CH}}^{F}_0(C, C \cap A)$ be the cokernel of the map ${\mathcal O}^{\times }_{C, C\cap A} \xrightarrow {{\textrm {div}}} {\mathcal Z}_0(C \setminus A)$ . Using the isomorphism ${\textrm {CH}}^{F}_0(C) \cong H^1_{\textrm {zar}}\left (C, {\mathcal O}^{\times }_C\right )$ and taking the colimit over the exact sequences for the cohomology of ${\mathcal O}^{\times }_C$ with support in finite closed subsets $S \subset C \setminus A$ , one easily checks that the canonical map ${\textrm {CH}}^{F}_0(C, C \cap A) \to {\textrm {CH}}^{F}_0(C)$ is an isomorphism. In particular, the $0$ -cycle $[x]$ on C coincides with a $0$ -cycle $\alpha ' \in {\textrm {CH}}^{F}_0(C)$ supported on $C \setminus A$ .

Since $C \cap X_{\textrm {sing}} = \emptyset $ , we have a push-forward map $\iota _* \colon {\textrm {CH}}^F_0(C) \to {\textrm {CH}}^{LW}_0(X)$ , where $\iota \colon C \hookrightarrow X$ is the inclusion. Letting $\alpha = \iota _*(\alpha ')$ , we achieve our claim. This proves the surjectivity of formula (2.2) for the Levine–Weibel Chow group. Since the map ${\textrm {CH}}^{LW}_0(X) \to {\textrm {CH}}^{{\textrm {lci}}}_0(X)$ is surjective by definition, it follows that the map ${\textrm {CH}}^{{\textrm {lci}}}_0(X,A) \to {\textrm {CH}}^{{\textrm {lci}}}_0(X)$ is also surjective. This concludes the proof.

We now draw some consequences of the moving lemma. The first is an extension of Lemma 2.1 for integral $R_1$ -schemes over finite fields.

The next two applications show that the Levine–Weibel Chow groups of $R_1$ -schemes admit pullback and push-forward maps. Note that neither of these maps was previously known to exist.

Corollary 2.6. Let k be any field and $X \in {\operatorname {\mathbf {Sch}}}_k$ an integral $R_1$ -scheme. Let $f \colon X' \to X$ be a morphism in ${\operatorname {\mathbf {Sch}}}_k$ such that $X^{\prime }_{\mathrm {sing}} \subseteq f^{-1}\left (X_{\mathrm {sing}}\right )$ and the resulting map $f^{-1}\left (X_{\mathrm {reg}}\right ) \to X_{\mathrm {reg}}$ is finite and surjective. Then $f^* \colon {\mathcal Z}_0(X^o) \to {\mathcal Z}_0(X^{\prime {o}})$ induces a pullback homomorphism

$$ \begin{align*} f^* \colon {\mathrm{CH}}^{LW}_0(X) \to {\mathrm{CH}}^{LW}_0(X'). \end{align*} $$

This is an isomorphism if f is the normalization morphism.

Corollary 2.7. Let $f \colon X' \to X$ be a proper morphism of integral quasiprojective schemes over a field k. Assume that $f^{-1}\left (X_{\mathrm {sing}}\right )$ has codimension $\ge 2$ in $X'$ and contains $X^{\prime }_{\mathrm {sing}}$ . Then the push-forward between the cycle groups $f_* \colon {\mathcal Z}_0\left (X' \setminus f^{-1}\left (X_{{\mathrm {sing}}}\right )\right ) \to {\mathcal Z}_0(X^o)$ descends to a homomorphism

$$ \begin{align*} f_* \colon {\mathrm{CH}}^{LW}_0(X') \to {\mathrm{CH}}^{LW}_0(X). \end{align*} $$

2.3. The cycle class map

Let k be any field and $X \in {\operatorname {\mathbf {Sch}}}_k$ of pure dimension d. Let ${\mathcal K}^M_{i, X}$ be the Zariski (or Nisnevich) sheaf of Milnor K-theory on X [Reference Kato33, §0]. Let ${\mathcal K}_{i,X}$ denote the Zariski (or Nisnevich) sheaf of Quillen K-theory on X. The product structures on the Milnor and Quillen K-theory yield a natural map of sheaves ${\mathcal K}^M_{i, X} \to {\mathcal K}_{i,X}$ . In formula (1.1), we defined the cycle class homomorphism (this could be trivial, for example, if $X^o = \emptyset $ )

(2.5) $$ \begin{align} {\operatorname{\textrm{cyc}}}_{X} \colon {\mathcal Z}_0(X^o) \to H^d_{\textrm{zar}}\left(X, {\mathcal K}^M_{d, X}\right). \end{align} $$

Using the Thomason–Trobaugh spectral sequence for Quillen K-theory [Reference Thomason and Trobaugh75, Theorem 10.3], there are canonical maps $H^d_{\textrm {zar}}\left (X, {\mathcal K}_{d, X}\right ) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}_{d, X}\right ) \to K_0(X)$ . These maps fit into a commutative diagram [Reference Gupta and Krishna26, Lemma 3.2]

(2.6)

such that the composite map ${\mathcal Z}_0(X^o) \to K_0(X)$ is the cycle class map [Reference Levine and Weibel50, Proposition 2.1] which takes a closed point $x \in X^o$ to the class $[k(x)] \in K_0(X)$ . Note also that any closed point $x \in X^o$ has a class $[k(x)] \in K_0(X, D)$ , where the latter is the $0$ th homotopy group of the $S^1$ -spectrum $K(X, D)$ defined as the homotopy fiber of the pullback map of spectra $K(X) \to K(D)$ for any closed subscheme D supported on $X_{\textrm {sing}}$ . Moreover, there is a factorization ${\mathcal Z}_0(X^o) \to K_0(X, D) \to K(X)$ .

As an analogue of Bloch’s formula, one asks whether the cycle class homomorphism ${\mathcal Z}_0(X^o) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d, X}\right )$ factors through ${\textrm {CH}}^{LW}_0(X)$ and whether the resulting map is an isomorphism when X is reduced. We shall prove some new results on this question in this paper. We consider a special case in what follows.

2.4. The case of isolated singularities

Let us now assume that X is reduced and equidimensional, and has only isolated singularities. Let S denote the finite set of singular points. We consider the sequence of Zariski sheaves of Milnor K-groups [Reference Kato33, §0]:

(2.7) $$ \begin{align} {\mathcal K}^M_{m, X} \xrightarrow{\epsilon} \left(\begin{array}{@{}c@{}}{\underset{x \in X^{(0)}}\coprod}\ (i_x)_* K^M_{m}(k(x)) \\ \oplus \\ {\underset{P \in S}\coprod}\ (i_P)_* K^M_{m}\left({\mathcal O}_{X,P}\right) \end{array}\right) \xrightarrow{e_0} \left(\begin{array}{@{}c@{}} {\underset{x \in X^{(1)}}\coprod} \ (i_x)_* K^M_{m-1}(k(x)) \\ \oplus \\ {\underset{P \in S}\coprod} {\underset{P \in \overline{\{x\}}, x \in X^{(0)}}\coprod}\ (i_P)_* K^M_{m}(k(x)) \end{array}\right) \xrightarrow{e_1} \dotsb \\ \nonumber \dotsb \xrightarrow{e_{d-1}} \left(\begin{array}{@{}c@{}} {\underset{x \in X^{(d)}}\coprod} \ (i_x)_* K^M_{m-d}(k(x)) \\ \oplus \\ {\underset{P \in S}\coprod} {\underset{P \in \overline{\{x\}}, x \in X^{(d-1)}}\coprod}\ (i_P)_* K^M_{m-d+1}(k(x)) \end{array}\right) \xrightarrow{e_d} \left(\begin{array}{@{}c@{}} 0 \\ \oplus \\ {\underset{P \in S}\coprod} \ (i_P)_* K^M_{m-d}(k(P)) \end{array}\right) \to 0. \end{align} $$

Here the map $\epsilon $ is induced by the inclusion into both terms and the other maps are given by the matrices

$$ \begin{align*} e_0 = \left(\begin{array}{cc} \partial_1 & 0 \\ -\Delta & \epsilon \end{array}\right), e_1 = \left(\begin{array}{cc} \partial_1 & 0 \\ \Delta & \partial_2 \end{array}\right), \dotsb , e_d = \left(\begin{array}{cc} 0 & 0 \\ \pm \Delta & \partial_2 \end{array}\right), \end{align*} $$

with $\partial _1$ and $\partial _2$ being the differentials of the Gersten–Quillen complex for Milnor K-theory sheaves as described in [Reference Kato33] and $\Delta $ s being the diagonal maps.

Proposition 2.9. Let $X \in {\operatorname {\mathbf {Sch}}}_k$ be integral of dimension $d \ge 1$ with only isolated singularities, and let $\tau $ denote Zariski or Nisnevich topology. Then there are canonical maps

$$ \begin{align*} {\mathrm{CH}}^{LW}_0(X) \xrightarrow{{\operatorname{\mathrm{cyc}}}_X} H^d_\tau\left(X, {\mathcal K}^M_{d,X}\right) \to H^d_\tau\left(X, {\mathcal K}_{d,X}\right) \to {\mathrm{CH}}^F_0(X), \end{align*} $$

in which the middle arrow is an isomorphism. Furthermore, all arrows in the middle square of diagram (2.6) are isomorphisms.

Proof. The case $d = 1$ is well known [Reference Levine and Weibel50, Proposition 1.4]. We can thus assume that $d \ge 2$ . Let S denote the singular locus of X and let $X^{\left (j\right )}_S$ denote the set of points $x \in X$ of codimension j such that $S \cap \overline {\{x\}} = \emptyset $ . We first observe that the map of sheaves ${\mathcal K}^M_{d, X} \twoheadrightarrow \epsilon \left ({\mathcal K}^M_{d, X}\right )$ is generically an isomorphism, and the same holds for the Quillen K-theory sheaves. It follows [Reference Hartshorne29, Exercise II.1.19, Lemma III.2.10] that the map $H^d_\tau \left (X, {\mathcal K}^M_{d,X}\right ) \to H^d_\tau \left (X, \epsilon \left ({\mathcal K}^M_{d,X}\right )\right )$ is an isomorphism, and ditto for the Quillen K-theory sheaves. It follows from Lemma 2.8 that both $H^d_\tau \left (X, {\mathcal K}^M_{d,X}\right )$ and $H^d_\tau \left (X, {\mathcal K}_{d,X}\right )$ are given by the middle homology of the complex ${\mathcal C}_X$ :

$$ \begin{align*} \left(\begin{array}{c} {\underset{x \in X^{(d-1)}}\coprod} \ K_{1}(k(x)) \\ \oplus \\ {\underset{P \in S}\coprod} {\underset{P \in \overline{\{x\}}}\coprod}\ K_{2}(k(x)) \end{array}\right) \xrightarrow{e_{d-1}} \left(\begin{array}{c} {\underset{x \in X^{(d)}}\coprod} \ K_{0}(k(x)) \\ \oplus \\ {\underset{P \in S}\coprod} {\underset{P \in \overline{\{x\}}}\coprod}\ K_{1}(k(x)) \end{array}\right) \xrightarrow{e_d} \left(\begin{array}{c} 0 \\ \oplus \\ {\underset{P \in S}\coprod} \ K_{0}(k(P)) \end{array}\right). \end{align*} $$

On the other hand, letting ${\mathcal C}^0_X$ and ${\mathcal C}^{F,0}_X$ denote, respectively, the complexes

$$ \begin{align*} {\underset{x \in X^{(d-1)}_S}\coprod} \ K_{1}(k(x)) \xrightarrow{{\textrm{div}}} {\underset{x \in X^{(d)}_S}\coprod} \ K_{0}(k(x)) \to 0, \\ {\underset{x \in X^{(d-1)}}\coprod} \ K_{1}(k(x)) \xrightarrow{{\textrm{div}}} {\underset{x \in X^{(d)}}\coprod} \ K_{0}(k(x)) \to 0, \end{align*} $$

we see that there are canonical maps of chain complexes ${\mathcal C}^0_X \hookrightarrow {\mathcal C}_X \twoheadrightarrow {\mathcal C}^{F,0}_X$ . This yields canonical maps $H_1\left ({\mathcal C}^0_X\right ) \to H^d_\tau \left (X, {\mathcal K}^M_{d,X}\right ) \xrightarrow {\cong } H^d_\tau \left (X, {\mathcal K}_{d,X}\right ) \to H_1\left ({\mathcal C}^{F,0}_X\right )$ . It follows, however, from Lemma 2.4 that $H_1\left ({\mathcal C}^0_X\right ) \cong {\textrm {CH}}^{LW}_0(X)$ . It is also clear that $H_1\left ({\mathcal C}^{F,0}_X\right ) \cong {\textrm {CH}}^F_0(X)$ . The second assertion is clear. This concludes the proof.

3.1. The reciprocity maps

Let k be a finite field and $X \in {\operatorname {\mathbf {Sch}}}_k$ an integral $R_1$ -scheme of dimension $d \ge 1$ . Given a closed point $x \in X^o$ , we have the push-forward map $(\iota _x)_* \colon \pi _1({\textrm {Spec}}(k(x))) \to \pi _1(X^o)$ , where $\iota \colon {\textrm {Spec}}(k(x)) \hookrightarrow X^o$ is the inclusion. We let $\phi _X([x]) = (\iota _x)_*(F_x)$ , where $F_x$ is the Frobenius automorphism of $\overline {k(x)}$ , which is the topological generator of ${\textrm {Gal}}\left ({\overline {k(x)}}/{k(x)}\right ) \cong \pi _1({\textrm {Spec}}(k(x))) \cong \widehat {{\mathbb Z}}$ . Extending it linearly, we get the reciprocity map $\phi _X \colon {\mathcal Z}_0(X^o) \to \pi ^{\textrm {ab}}_1(X^o)$ . By [Reference Kato and Saito35, Theorem 2.5], the cycle class map ${\operatorname {\textrm {cyc}}}_X \colon {\mathcal Z}_0(X^o) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )$ is surjective.

Proof. We let ${\mathcal K}^M_{i, \left (X,Y\right )} = {\textrm {Ker}}\left ({\mathcal K}^M_{i,X} \twoheadrightarrow {\mathcal K}^M_{i,Y}\right )$ for any closed subscheme $Y \subset X$ . We now look at the diagram

(3.1)

where Y ranges over all closed subschemes of X contained in $X_{\textrm {sing}}$ and m ranges over all nonzero integers. The top horizontal arrow is the map induced on the cohomology by the inclusion of Nisnevich sheaves ${\mathcal K}^M_{d, \left (X,Y\right )} \hookrightarrow {\mathcal K}^M_{d,X}$ . All such maps are isomorphisms, because $\dim (Y) \le d-2$ . This explains why the top horizontal arrow is an isomorphism. The cycle class map ${\operatorname {\textrm {cyc}}}_X$ clearly factors through ${\mathcal Z}_0(X^o) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,\left (X, Y\right )}\right )$ for any $Y \subset X_{\textrm {sing}}$ , because $H^d_{\{x\}}\left (X, {\mathcal K}^M_{d,\left (X, Y\right )}\right ) \xrightarrow {\cong } H^d_{\{x\}}\left (X, {\mathcal K}^M_{d, X}\right )$ for any closed point $x \in X^o$ . The limit of all these maps is ${\operatorname {\textrm {cyc}}}^{\prime }_X$ . Hence, ${\operatorname {\textrm {cyc}}}_X$ is the composition of ${\operatorname {\textrm {cyc}}}^{\prime }_X$ with the top horizontal arrow.

The middle vertical arrow is induced by the canonical surjections $H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,\left (X, Y\right )}\right ) \twoheadrightarrow {H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,\left (X, Y\right )}\right )}/m$ . The bottom horizontal arrow on the right is given by [Reference Kato and Saito35, Theorem 9.1(3)], and is an isomorphism. The arrows $\eta _X$ and $\psi _X$ are defined so that the triangles on the two sides of the middle vertical arrow commute. It follows from [Reference Kato and Saito35, Proposition 3.8(2)] that $\psi _X \circ {\operatorname {\textrm {cyc}}}^{\prime }_X = \phi _X$ . We deduce from this that there is a unique homomorphism $\rho _X \colon H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right ) \to \pi ^{\textrm {ab}}_1(X^o)$ , which factors $\phi _X$ via ${\operatorname {\textrm {cyc}}}_X$ .

Lemma 3.2. For any closed subset $A \subset X$ of codimension $\ge 2$ and containing $X_{\mathrm {sing}}$ , the reciprocity map $\phi _X$ descends to a homomorphism

$$ \begin{align*} \phi_X \colon {\mathrm{CH}}^{LW}_0(X, A) \to \pi^{\mathrm{ab}}_1(X \setminus A). \end{align*} $$

Proof. If $d =1$ , then X is a connected smooth projective curve over k (note that k is perfect) with $A = \emptyset $ , and one knows that ${\textrm {CH}}^{LW}_0(X) \cong {\textrm {CH}}^F_0(X)$ . The lemma therefore follows from the classical class field theory in this case.

To prove that $\phi _X$ kills ${\mathcal R}^{LW}_0(X,A)$ in general, we need to show that $\phi _X({\textrm {div}}(f)) = 0$ if f is a nonzero rational function on an integral projective curve $C \subset X \setminus A$ . So we choose any such curve C and $f \in k(C)^{\times }$ . Let $\nu \colon C_n \to C \hookrightarrow X$ be the composite map, where $\nu $ is the normalization morphism. We then have a commutative diagram [Reference Raskind62, Lemma 5.1(1)]

(3.2)

Since ${\textrm {div}}((f)_C) = \nu _*\left ({\textrm {div}}\left ((f)_{C_n}\right )\right )$ , this diagram reduces the problem to showing that $\phi _{C_n}\left ({\textrm {div}}\left ((f)_{C_n}\right )\right ) = 0$ . But this has already been shown.

In view of Lemma 2.4 (with $A=X_{{\textrm {sing}}})$ , Lemma 3.2 implies the following:

Corollary 3.3. The reciprocity map $\phi _X$ descends to a homomorphism

$$ \begin{align*} \phi_X \colon {\mathrm{CH}}^{LW}_0(X) \to \pi^{\mathrm{ab}}_1(X^o). \end{align*} $$

It is clear that there is a commutative diagram (with exact rows)

(3.3)

3.2. Reciprocity map for surfaces

We assume now that X is an integral projective $R_1$ -scheme of dimension $2$ over a field k. By Proposition 2.9, there are canonical maps

(3.4) $$ \begin{align} {\textrm{CH}}^{LW}_0(X) \xrightarrow{{\operatorname{\textrm{cyc}}}_X} H^2_\tau\left(X, {\mathcal K}^M_{2,X}\right) \to K_0(X), \end{align} $$

where $\tau = {{\textrm {zar}}}/{{\textrm {nis}}}$ and the composite arrow is the cycle class map to K-theory. We let $F^2K_0(X)$ denote the image of this composite arrow. For any closed subscheme $D \subset X$ supported on $X_{\textrm {sing}}$ , we let $F^2K_0(X, D)$ denote the image of the cycle class map ${\mathcal Z}_0(X^o) \to K_0(X,D)$ (see §2.3). The main result about ${\textrm {CH}}^{LW}_0(X)$ is the following:

Proof. By [Reference Krishna and Srinivas44, Lemma 2.2], there is an exact sequence

$$ \begin{align*} SK_1(D) \to F^2K_0(X,D) \to F^2K_0(X) \to 0, \end{align*} $$

where $SK_1(D)$ is the kernel of the edge map $K_1(D) \to H^0_{\textrm {zar}}\left (D, {\mathcal O}^\times _D\right )$ in the Thomason–Trobaugh spectral sequence [Reference Thomason and Trobaugh75, Theorem 10.3]. But this edge map is an isomorphism, because D is a $0$ -dimensional k-scheme.

By Lemma 2.4 and [Reference Pedrini and Weibel58, Theorem 2.2], there is an exact sequence

$$ \begin{align*} SK_1\left(X_{\textrm{sing}}\right) \to {\textrm{CH}}^{LW}_0(X) \to F^2K_0(X) \to 0. \end{align*} $$

Item (1) now follows because $SK_1\left (X_{\textrm {sing}}\right ) = 0$ . To prove (2), it is enough to work with the Nisnevich topology by Proposition 2.9. We now have the maps ${\textrm {CH}}^{LW}_0(X) \xrightarrow {{\operatorname {\textrm {cyc}}}_X} H^2_{\textrm {nis}}\left (X, {\mathcal K}^M_{2,X}\right ) \twoheadrightarrow F^2K_0(X)$ . The map ${\operatorname {\textrm {cyc}}}_X$ is surjective by [Reference Kato and Saito35, Theorem 2.5] (see [Reference Gupta and Krishna27, §3.5] for an explanation as to why it suffices to know that $X^o$ is regular instead of being nice). The map ${\operatorname {\textrm {cyc}}}_X$ in (2) is now an isomorphism by (1). The other map in (2) is an isomorphism for dimension reasons.

3.3. Zero-cycles on a surface and its desingularizations

Let k be any field. We recall the definition of the Chow group of $0$ -cycles with modulus from [Reference Binda and Saito6]. Let X be a quasiprojective scheme over k and $D \subset X$ an effective Cartier divisor. Given a finite map $\nu \colon C \to X$ from a regular integral curve whose image is not contained in D, we let $E = \nu ^*(D)$ . We say that a rational function f on C has modulus E if $f \in {\textrm {Ker}}\left ({\mathcal O}^{\times }_{C,E} \twoheadrightarrow {\mathcal O}^{\times }_E\right )$ . We let $G(C,E)$ denote the multiplicative subgroup of such rational functions in $k(C)^{\times }$ . Then ${\textrm {CH}}_0(X\mid D)$ is the quotient ${\mathcal Z}_0(X \setminus D)$ by the subgroup ${\mathcal R}_0(X\mid D)$ generated by $\nu _*({\textrm {div}}(f))$ for all possible choices of $\nu \colon C \to X$ and $f \in G(C,E)$ as above.

Let $D \subset X$ be as before. We have seen in §2.3 that there is a cycle class homomorphism $\widetilde {{\operatorname {\textrm {cyc}}}}_{X\mid D} \colon {\mathcal Z}_0(X^o \setminus D) \to K_0(X,D)$ . If k is perfect and $X \in {\mathbf {Sm}}_k$ , then it was shown in [Reference Binda and Krishna3, Theorem 12.4] that ${\operatorname {\textrm {cyc}}}_{X\mid D}$ descends to a group homomorphism

(3.5) $$ \begin{align} {\operatorname{\textrm{cyc}}}_{X|D} \colon {\textrm{CH}}_0(X\mid D) \to K_0(X,D). \end{align} $$

If $\dim (X) \le 2$ , this map exists without any condition on k by [Reference Krishna41, Theorem 1.2]. We let $F^dK_0(X,D)$ be the image of this map if X is of pure dimension d.

Let us now assume that k is any field and X is an integral projective $R_1$ -scheme of dimension $2$ over k. Let $f \colon X_n \to X$ be the normalization morphism and $f' \colon \widetilde {X} \to X_n$ a resolution of singularities of $X_n$ (assuming it exists). Let $E \subset \widetilde {X}$ be the reduced exceptional divisor. Then $\pi = f \circ f' \colon \widetilde {X} \to X$ is a resolution of singularities of X with reduced exceptional divisor E. We let $E' = \pi ^{-1}\left (X_{\textrm {sing}}\right )_{\textrm {red}}$ so that $E'$ is a union of E and a finite set of closed points. Note that such a resolution of singularities exists (for example, by [Reference Lipman51]). We write $S = X_{\textrm {sing}}$ and $S' = (X_n)_{\textrm {sing}}$ with reduced structures.

Proposition 3.5. Let k be a field and X an integral and projective k-scheme of dimension $2$ which is $R_1$ . Let $\widetilde {X} \xrightarrow {f'} X_n \xrightarrow {f} X$ be the desingularization and normalization morphisms as before. Let $m \ge 1$ be any integer. With the foregoing notations, we have a commutative diagram

(3.6)

Moreover, all arrows are isomorphisms for $m \gg 1$ .

3.4. Reciprocity theorem for surfaces

Assume now that k is a finite field and X an integral projective $R_1$ -scheme of dimension $2$ over k. It follows from Proposition 2.9 and Corollary 3.3 that the cycle class and the reciprocity homomorphisms give rise to the degree-preserving maps

(3.7) $$ \begin{align} {\textrm{CH}}^{LW}_0(X) \xrightarrow{{\operatorname{\textrm{cyc}}}_X} H^2_{\textrm{nis}}\left(X, {\mathcal K}^M_{2,X}\right) \xrightarrow{\rho_X} \pi^{\textrm{ab}}_1(X^o). \end{align} $$

Our main result on the class field theory of X is the following:

Theorem 3.6. The cycle class and the reciprocity homomorphisms induce isomorphisms of finite groups

(3.8) $$ \begin{align} {\mathrm{CH}}^{LW}_0(X)^0 \xrightarrow{\cong} H^2_{\mathrm{nis}}\left(X, {\mathcal K}^M_{2,X}\right)^0 \xrightarrow{\cong} \pi^{\mathrm{ab}}_1(X^o)^0. \end{align} $$

Proof. By Proposition 3.4, we only have to show that the composition of the two maps in formula (3.8) is an isomorphism of finite groups. We now choose a resolution of singularities $\pi \colon \widetilde {X} \to X$ (which exists by [Reference Lipman51]) with the reduced exceptional divisor $E \subset \widetilde {X}$ . For every integer $m \ge 1$ , we then have a commutative diagram

(3.9)

where $\pi ^{\textrm {ab}}_1\left (\widetilde {X}, mE\right )$ is the abelianized étale fundamental group with modulus (see [Reference Binda, Krishna and Saito5, §8.3] for the definition) and $\pi ^{\textrm {ab}}_1\left (\widetilde {X}, mE\right )^0$ is the kernel of the map $\pi ^{\textrm {ab}}_1\left (\widetilde {X}, mE\right ) \to {\textrm {Gal}}\left ({\overline {k}}/k\right )$ .

The left vertical arrow in diagram (3.9) is an isomorphism for all $m \gg 0$ by Proposition 3.5. Combining this with [Reference Kerz and Saito38, Theorem III] (if ${\textrm {char}}(k) \neq 2$ ) and [Reference Binda, Krishna and Saito5, Lemma 8.4, Theorem 8.5] (in general), it follows that all arrows in the diagram are isomorphisms for all $m \gg 0$ . Moreover, these groups are finite by [Reference Binda, Krishna and Saito5, Corollary 8.3].

4.1. Reflexive sheaves

Let $(X, {\mathcal O}_X)$ be a locally ringed space. If $(X, {\mathcal O}_X)$ is an integral locally ringed space (that is, the stalks of ${\mathcal O}_X$ are integral domains), we let ${\mathcal K}_X$ denote the sheaf of field of fractions of ${\mathcal O}_X$ . Recall that for a sheaf of ${\mathcal O}_X$ -modules ${\mathcal E}$ on X, the dual ${\mathcal E}^\vee $ is the sheaf of ${\mathcal O}_X$ -modules ${\mathcal {H}{om}}_{{\mathcal O}_X}({\mathcal E}, {\mathcal O}_X)$ . There is a natural evaluation map ${\textrm {ev}}_{{\mathcal E}} \colon {\mathcal E} \to {\mathcal E}^{\vee \vee }$ whose kernel is the subsheaf of torsion submodules of ${\mathcal E}$ , where the latter is defined as the kernel of the canonical map ${\mathcal E} \to {\mathcal E} \otimes _{{\mathcal O}_X} {\mathcal K}_X$ . We denote either of these kernels by ${\mathcal E}_{\textrm {tor}}$ . If ${\mathcal E}$ is torsion-free, one calls ${\mathcal E}^{\vee \vee }$ the reflexive hull of ${\mathcal E}$ . One says that ${\mathcal E}$ is reflexive if the evaluation map ${\mathcal E} \to {\mathcal E}^{\vee \vee }$ is an isomorphism. The following lemma is elementary:

Let $(X, {\mathcal O}_X)$ be a locally ringed space with Noetherian stalks and let ${\mathcal E}$ be a sheaf of ${\mathcal O}_X$ -modules. Recall that ${\mathcal E}$ is said to satisfy the $S_i$ property for some $i \ge 0$ if for every $x \in X$ , the ${\mathcal O}_{X,x}$ -module ${\mathcal E}_x$ satisfies Serre’s $S_i$ condition – that is, ${\textrm {depth}}({\mathcal E}_x) \ge \min \{i, \dim ({\mathcal E}_x)\}$ for every point $x \in X$ . One says that X is a locally $S_i$ -ringed space (or an $S_i$ -scheme, if X is a Noetherian scheme) if it has Noetherian stalks and ${\mathcal O}_X$ satisfies the $S_i$ property. The following is easy to verify using [Reference de Jong11, Lemma 0AV6]:

Let k be a field. Suppose that X is an integral k-scheme and there is a locally closed immersion $X \hookrightarrow {\mathbb P}^N_k$ . We let ${\mathcal O}^r_X(m)$ be the restriction of the sheaf $\left ({\mathcal O}_{{\mathbb P}^N_k}(m)\right )^{\oplus r}$ onto X. Let ${\mathcal E}$ be a reflexive coherent sheaf on X. We can then prove the following:

Lemma 4.3. There are integers $q, q' \in {\mathbb Z}$ , $r, r' \ge 1$ , and a coherent sheaf ${\mathcal E}'$ together with exact sequences

(4.1) $$ \begin{align} & 0 \to {\mathcal E} \to {\mathcal O}^r_X(q) \to {\mathcal E}' \to 0, \\ \nonumber & 0 \to {\mathcal E}' \to {\mathcal O}^{r'}_X(q') \to {\mathcal H} \to 0. \end{align} $$

In particular, each of ${\mathcal E}$ and ${\mathcal E}'$ is torsion-free (or zero). If ${\mathcal E}$ is locally free, so is ${\mathcal E}'$ .

Proof. Since ${\mathcal E}^\vee $ is coherent, there is a surjection ${\mathcal O}^r_X(-q) \twoheadrightarrow {\mathcal E}^\vee $ for some $q, r \gg 0$ . We let ${\mathcal F}$ be the kernel of this surjection. Since ${\mathcal F}$ is necessarily coherent, we also have a surjection ${\mathcal O}^{r'}_X(-q') \twoheadrightarrow {\mathcal F}$ for some $q', r' \gg 0$ . Letting ${\mathcal F}'$ be its kernel, we get short exact sequences of coherent sheaves

$$ \begin{align*} & 0 \to {\mathcal F} \to {\mathcal O}^r_X(-q) \to {\mathcal E}^\vee \to 0, \\ & 0 \to {\mathcal F}' \to {\mathcal O}^{r'}_X(-q') \to {\mathcal F} \to 0. \end{align*} $$

By dualizing, we get exact sequences

(4.2) $$ \begin{align} & 0 \to {\mathcal E} \to {\mathcal O}^r_X(q) \to {\mathcal F}^\vee \to {\mathcal{E}{xt}}^1_{{\mathcal O}_X}({\mathcal E}^\vee, {\mathcal O}_X) \to 0, \\ \nonumber & 0 \to {\mathcal F}^\vee \to {\mathcal O}^{r'}_X(q') \to {\mathcal F}^{\prime\vee} \to {\mathcal{E}{xt}}^1_{{\mathcal O}_X}({\mathcal F}, {\mathcal O}_X) \to 0. \end{align} $$

Letting ${\mathcal E}'$ be the cokernel of ${\mathcal E} \to {\mathcal O}^r_X(q)$ , we get the two exact sequences (4.1). If ${\mathcal E}$ is locally free, then so is ${\mathcal E}^\vee $ . In this case, ${\mathcal {E}{xt}}^1_{{\mathcal O}_X}({\mathcal E}^\vee , {\mathcal O}_X) = 0$ and ${\mathcal F}$ must also be locally free. This implies that ${\mathcal E}'$ is locally free.

4.2. The Hartogs lemma

We need to prove a version of the Hartogs lemma for formal schemes. Before we do this, we recall this result for the ordinary schemes.

Proof. Since E is reflexive, we can find an exact sequence of A-modules

(4.3) $$ \begin{align} 0 \to E \to A^r \to A^{r'} \end{align} $$

for some $r, r' \ge 1$ . This gives rise to a commutative diagram of exact sequences

(4.4)

where $j \colon U \hookrightarrow X$ is the inclusion. A diagram chase shows that we can assume that $E = A$ . But this case follows from Lemma 4.5.

The following lemma is well known:

Proof. The proof of this lemma is identical to that of [Reference de Jong11, Lemma 031T(2)] once we observe that in a Noetherian integral domain which is an $S_2$ -ring, a prime ideal has height $1$ if and only if its depth is $1$ . We add a proof of the lemma for completeness.

Suppose $a, b \in A$ are two nonzero elements such that $a \in bA_{{\mathfrak p}}$ for every prime ideal ${\mathfrak p} \subset A$ of depth $1$ . We need to show that $a \in (b)$ .

Let ${\textrm {Ass}}(b)$ denote the set of associated primes of $(b)$ . Since the integral domain A is an $S_2$ -ring and $b \neq 0$ , an easy consequence of [Reference Grothendieck and Dieudonné25, Propositions 16.4.6(ii), 16.4.10(i)] is that ${A}/{(b)}$ is an $S_1$ -ring (see, for example, [Reference Ghosh and Krishna20, Lemma 2.1]). In particular, all associated primes of ${A}/{(b)}$ have height $0$ . The latter statement is equivalent to saying that all associated primes of $(b)$ are minimal, and hence have height $1$ by Krull’s principal ideal theorem [Reference Matsumura52, Theorem 13.5]. Since $1 \le {\textrm {depth}}\left (A_{{\mathfrak p}}\right ) \le \dim \left (A_{{\mathfrak p}}\right ) \le 1$ , we conclude that ${\textrm {depth}}\left (A_{{\mathfrak p}}\right ) = 1$ for every ${\mathfrak p} \in {\textrm {Ass}}(b)$ . This implies by our hypothesis that $a \in bA_{{\mathfrak p}}$ for every ${\mathfrak p} \in {\textrm {Ass}}(b)$ . We are now done, because the canonical map

$$ \begin{align*} {A}/{(b)} \to {\underset{{\mathfrak p} \in {\textrm{Ass}}(b)}\prod} \ {A_{{\mathfrak p}}}/{bA_{{\mathfrak p}}} \end{align*} $$

is injective (see, for example, [Reference de Jong11, Lemma 0311]).

4.3. The formal Hartogs lemma

Let X be a Noetherian scheme and $Y \subset X$ a closed subscheme. Let $\widehat {X}$ denote the formal completion of X along Y (see [Reference Grothendieck21, Chapitre 0, §9] or [Reference Hartshorne29, Chapter II, §10]). Let ${\mathcal I}_Y$ denote the sheaf of ideals on X defining Y. Let $Y_m$ denote the closed subscheme of X defined by the ideal sheaf ${\mathcal I}^m_Y$ . For any open subscheme $U \subset X$ such that $V:= U \cap Y$ is dense in Y, we have a commutative diagram of Noetherian formal schemes

(4.5)

where $V_m := U\cap Y_m$ . In the right square, all arrows are flat morphisms (of locally ringed spaces), the vertical arrows are open immersions (see [Reference de Jong11, §01HD] for the definition of open immersion of locally ringed spaces), and the horizontal arrows are the completion morphisms. In the left square, the vertical arrows are open immersions and the horizontal arrows are closed immersions. The two squares are Cartesian. The compositions of the two horizontal arrows are the given closed immersions $Y_m \hookrightarrow X$ and $V_m \hookrightarrow U$ of schemes.

For any quasicoherent sheaf ${\mathcal F}$ on X, we let $\widehat {{\mathcal F}}$ denote the pullback of ${\mathcal F}$ under $c_X$ . Note that the canonical map $\widehat {{\mathcal F}} \to {\underset {m}\varprojlim } \ {\mathcal F} \otimes _{{\mathcal O}_X} {\mathcal O}_{Y_m}$ is an isomorphism if ${\mathcal F}$ is coherent. We shall use this isomorphism later. For any morphism $f \colon X' \to X$ and quasicoherent sheaf ${\mathcal F}$ on X, we let ${\mathcal F}_{X'} = f^*({\mathcal F})$ . We shall write ${\mathcal F}_{Y_m}$ simply as ${\mathcal F}_m$ for $m \ge 1$ . Our goal in this subsection is to prove a formal version of the Hartogs lemma.

Let A be an excellent normal integral domain and $J \subset A$ the radical ideal such that $V(J) = X_{\textrm {sing}}$ , where $X = {\textrm {Spec}}(A)$ . Let ${\mathfrak p} \subset A$ be a complete intersection prime ideal such that ${\textrm {ht}}({\mathfrak p} + J) \ge {\textrm {ht}}({\mathfrak p}) + {\textrm {ht}}(J)$ and $A/{{\mathfrak p}}$ is normal. Let $U \subset X$ be an open subscheme containing $V({\mathfrak p}) \cap X^o$ whose complement has codimension $\ge 2$ in X. Let $\widehat {A}$ be the ${\mathfrak p}$ -adic completion of A and $\widehat {X} = {\textrm {Spf}}\left (\widehat {A}\right )$ , the formal spectrum of $\widehat {A}$ . Let $\widehat {U}$ be the formal completion of U along $V({\mathfrak p}) \cap U$ . We let $Y = {\textrm {Spec}}(A/{{\mathfrak p}})$ and $A_m:=A/{{\mathfrak p}^m}$ . Since A is excellent, so is U. It follows therefore from [Reference Grothendieck22, Corollaire 6.5.4, Scholie 7.8.3(v)] that $\widehat {X}$ and $\widehat {U}$ are both normal integral formal schemes. Under these conditions, we have the following:

Proof. Since E is reflexive, we can find an exact sequence of A-modules

(4.6) $$ \begin{align} 0 \to E \to A^r \to A^{r'} \end{align} $$

for some $r, r' \ge 1$ . Since all arrows in the right square of diagram (4.5) are flat, we have an exact sequence of coherent sheaves

(4.7) $$ \begin{align} 0 \to \widehat{{\mathcal E}} \to {\mathcal O}^r_{\widehat{X}} \to {\mathcal O}^{r'}_{\widehat{X}} \end{align} $$

on $\widehat {X}$ . This gives rise to a commutative diagram of exact sequences

(4.8)

where $j \colon U \hookrightarrow X$ is the inclusion. A diagram chase shows that we can assume that $E = A$ .

We now let $V = Y \cap U$ . Let $j \colon U \hookrightarrow X$ and $\widehat {j} \colon \widehat {U} \hookrightarrow \widehat {X}$ denote the inclusion maps. We then have a commutative diagram

(4.9)

The top horizontal arrow is clearly an isomorphism, and the Milnor exact sequence for the cohomology of inverse limit sheaves (see, for example, [Reference Jannsen31, Proposition 1.6]) implies that the bottom horizontal arrow is also an isomorphism. It suffices therefore to show that the right vertical arrow is an isomorphism. We shall show by induction the stronger assertion that the restriction map $A_m \to H^0\left (V_m, {\mathcal O}_{V_m}\right )$ is an isomorphism for all $m \ge 1$ .

Let us first assume that $m =1$ . It follows from our assumption that Y is a Noetherian normal integral scheme in which the codimension of $Y \setminus V$ is $\ge 2$ . Therefore, the map $A_1 \to H^0(V, {\mathcal O}_{V}) = H^0(V_1, {\mathcal O}_{V_1})$ is an isomorphism by Lemma 4.4. We now prove the general case by induction on m. We consider the exact sequence of $A_{m+1}$ -modules

(4.10) $$ \begin{align} 0 \to {{\mathfrak p}^m}/{{\mathfrak p}^{m+1}} \to A_{m+1} \to A_m \to 0. \end{align} $$

Since ${\mathfrak p} \subset A$ is a complete intersection, it follows that ${{\mathfrak p}^m}/{{\mathfrak p}^{m+1}} \cong S^m\left ({{\mathfrak p}}/{{\mathfrak p}^{2}}\right )$ is a free $A/{{\mathfrak p}}$ -module. That is, ${{\mathfrak p}^m}/{{\mathfrak p}^{m+1}} \cong A^{q}_1$ for some $q \ge 1$ . We thus get an exact sequence of $A_{m+1}$ -modules

(4.11) $$ \begin{align} 0 \to A^q_1 \to A_{m+1} \to A_m \to 0. \end{align} $$

This gives rise to the exact sequence of coherent ${\mathcal O}_X$ -modules

(4.12) $$ \begin{align} 0 \to {\mathcal O}^q_{Y} \to {\mathcal O}_{Y_{m+1}} \to {\mathcal O}_{Y_m} \to 0. \end{align} $$

Considering the cohomology groups and comparing them on X and U, we get a commutative diagram of exact sequences

(4.13)

The left and right vertical arrows are isomorphisms by induction on m. It follows from the five lemma that the middle vertical arrow must also be an isomorphism. This concludes the proof of the claim and the lemma.

Let X be an excellent normal integral scheme and $Y \subset X$ a local complete intersection closed subscheme which is integral and normal. Let $U \subset X$ be an open subscheme containing $Y \cap X^o$ such that $X \setminus U$ has codimension $\ge 2$ in X. Under these hypotheses, we have the following ‘formal Hartogs lemma’:

4.4. The ${\textrm {Lef}}(X,Y)$ condition

Let X be a Noetherian scheme and $Y \subset X$ a closed (resp., open) subscheme. We shall then say that $(X,Y)$ is a closed (resp., open) pair. Recall from [Reference Grothendieck23, §2, Exposé X] that a closed pair $(X,Y)$ is said to satisfy the Lefschetz condition, and one says that ${\textrm {Lef}}(X,Y)$ holds if, for any open subscheme $U \subset X$ containing Y and any coherent locally free sheaf ${\mathcal E}$ on U, the restriction map $H^0(U, {\mathcal E}) \to H^0\left (\widehat {X}, \widehat {{\mathcal E}}\right )$ is an isomorphism. We shall not recall the effective Lefschetz condition, because we do not need it.

We shall work with the following setup throughout this subsection. We fix a field k and let $X \hookrightarrow {\mathbb P}^N_k$ be an integral normal projective scheme over k of dimension $d \ge 3$ . Suppose we are given a closed immersion $\iota \colon Y \hookrightarrow X$ of integral normal schemes such that $X^o \cap Y \subset Y^o$ and $2 \le \dim (Y) \le d-1$ . We further assume that $Y \subset X$ is cut out by e general hypersurfaces in ${\mathbb P}^N_k$ such that Y is a complete intersection in X, where $e = {\textrm {codim}}(Y,X)$ . Let $U \subset X$ be an open subscheme containing $X^o \cap Y$ such that $X \setminus U$ has codimension $\ge 2$ in X. Let $j \colon U \hookrightarrow X$ be the inclusion map. We shall let $\widehat {X}$ (resp., $\widehat {U}$ ) denote the formal completion of X (resp., U) along Y (resp., $U \cap Y$ ). We shall continue to use the notations of diagram (4.5).

Proof. We let ${\mathcal E}' = j_* {\mathcal E}$ . It follows from Corollary 4.8 that ${\mathcal E}'$ is a reflexive coherent sheaf on X. We now consider the commutative diagram

(4.14)

The top horizontal arrow is an isomorphism by Lemma 4.12. Corollary 4.7 implies that the left vertical arrow is an isomorphism. Since X is excellent and normal, Corollary 4.11 implies that the right vertical arrow is an isomorphism. A diagram chase now finishes the proof.

It follows from Lemma 4.14 that $(X^o, Y^o)$ is a closed pair. We can now prove the following:

Proof. Let $U \subset X^o$ be an open subscheme containing $Y^o$ . In view of Lemma 4.13, we only need to show that $X \setminus U$ has codimension $\ge 2$ in X. Suppose to the contrary that there is an irreducible closed subscheme $Z \subset X \setminus U$ of dimension $d-1$ . We must then have that the scheme theoretic intersections $Z \cap Y$ and $Z \cap \left (Y \cap X_{\textrm {sing}}\right )$ have same support, and therefore $\dim (Z \cap Y) \le \dim \left (Y\cap X_{{\textrm {sing}}}\right )$ . We also get

$$ \begin{align*} \dim(Z\cap Y) \ge \dim(Z) - e = d -1- (d-\dim(Y))=\dim(Y)-1> \dim\left(Y \cap X_{{\textrm{sing}}}\right), \end{align*} $$

and since $Y \subset X$ is cut out by e general hypersurfaces in ${\mathbb P}^N_k$ , the intersection of Y and $X_{\textrm {sing}}$ is proper and X is catenary. This is clearly a contradiction.

5.1. The Enriques–Severi–Zariski theorem

We shall need a version of the Enriques–Severi–Zariski theorem for some singular projective schemes and their regular loci. Suppose that $H \subset {\mathbb P}^N_k$ is a hypersurface of degree $m \ge 1$ such that $Y=X\cap H$ is good. Let $C_{Y/X}$ denote the conormal sheaf on Y associated to the regular embedding $\iota \colon Y \hookrightarrow X$ so that $C_{Y/X} = {{\mathcal I}_Y}/{{\mathcal I}^2_Y} \cong {\mathcal O}_Y(-m)$ , where ${\mathcal I}_Y$ is the sheaf of ideals on X defining Y. For any coherent sheaf ${\mathcal F}$ on Y and integer $n \ge 1$ , we define ${\mathcal F}^{[n]} := {\mathcal F} \otimes _{{\mathcal O}_Y} S^n\left (C_{Y/X}\right ) \cong {\mathcal F}(-nm)$ . For any coherent sheaf ${\mathcal F}$ on $Y^o$ and integer $n \ge 1$ , we define ${\mathcal F}^{[n]} := {\mathcal F} \otimes _{{\mathcal O}_{Y^o}} S^n\left (C_{Y^o/X^o}\right ) \cong {\mathcal F}(-nm)$ .

Proof. Let $\omega ^{\bullet }_{X/k}$ denote the dualizing complex for X. Under the assumption of the lemma, it follows from [Reference Lee and Nakayama47, Lemma 4.27] that $\omega ^{\bullet }_{X/k}\in D^{[-d, -b]}(X)$ . Moreover, we have ${\mathcal H}^{-d}\left (\omega ^{\bullet }_{X/k}\right ) \cong j_*\left (\omega _{{X^o}/k}\right )$ , where $\omega _{{X^o}/k}$ is the canonical sheaf of $X^o$ . Since X is an $(R_1 + S_b)$ -scheme for some $b \ge 2$ , it is normal (see, for example, [Reference Matsumura52, Theorem 23.8]). We can therefore apply the Grothendieck–Serre duality for normal projective schemes to get

$$ \begin{align*} H^i_{\textrm{zar}}(X, {\mathcal E}(-j)) \cong {\mathbb H}^{-i}_{\textrm{zar}}\left(X, {\mathcal E}^\vee\otimes^L_{{\mathcal O}_X} \omega^{\bullet}_{X/k}(j)\right). \end{align*} $$

The desired assertion now follows easily from the Serre vanishing theorem.

Proof. It follows from Corollary 4.8 that $\tilde {j}_* {\mathcal E}$ is a coherent reflexive sheaf on Y. We denote this extension by ${\mathcal E}$ itself. Using the exact sequences of Lemma 4.3 and tensoring them with $S^n\left (C_{Y/X}\right )$ (which is invertible on Y) and subsequently taking the cohomology, we get exact sequences

(5.1) $$ \begin{align} 0 \to H^0\left(Z, {\mathcal E}^{[n]}\right) \to H^0\left(Z, {{\mathcal O}^r_Z(q)}^{[n]}\right) \to H^0\left(Z, {\mathcal E}^{\prime[n]}\right) \to H^1\left(Z, {\mathcal E}^{[n]}\right) \to H^1\left(Z, {{\mathcal O}^r_Z(q)}^{[n]}\right) \end{align} $$

and

(5.2) $$ \begin{align} 0 \to H^0\left(Z, {\mathcal E}^{\prime[n]}\right) \to H^0\left(Z, {{\mathcal O}^{r'}_Z(q')}^{[n]}\right), \end{align} $$

where $Z \in \{Y, Y^o\}$ . Corollary 4.7 and Lemma 5.1 together tell us that $H^0\left (Z, {{\mathcal O}^r_Z(q)}^{[n]}\right ) = 0 = H^0\left (Z, {{\mathcal O}^{r'}_Z(q')}^{[n]}\right )$ for all $n \gg 0$ . We conclude that $H^0\left (Z, {\mathcal E}^{[n]}\right ) = 0$ and

(5.3) $$ \begin{align} H^1\left(Z, {\mathcal E}^{[n]}\right) \hookrightarrow H^1\left(Z, {{\mathcal O}^r_Z(q)}^{[n]}\right) \end{align} $$

for all $n \gg 0$ .

5.2. The Gysin map and Poincaré duality

We fix a prime-to-p integer n and let $\Lambda = {{\mathbb Z}}/n$ be the constant sheaf of rings on the étale site of ${\operatorname {\mathbf {Sch}}}_k$ . For any integer $m \in {\mathbb Z}$ , we let $\Lambda (m)$ denote the usual Tate twist of the sheaf of nth roots of unity $\mu _n$ on the étale site of ${\operatorname {\mathbf {Sch}}}_k$ . For any étale sheaf of $\Lambda $ -modules ${\mathcal F}$ on ${\operatorname {\mathbf {Sch}}}_k$ , we let ${\mathcal F}(m) = {\mathcal F} \otimes _\Lambda \Lambda (m)$ . Let $D^{+}(X, \Lambda )$ denote the derived category of the bounded-below complexes of the sheaves of $\Lambda $ -modules on the small étale site of X.

Let $\iota \colon Y \hookrightarrow X$ be as in §5.1. Recall from Gabber’s construction [Reference Fujiwara16] (see also [Reference Navarro56, Definition 2.1]) that the regular closed embedding $\iota \colon Y \hookrightarrow X$ induces the Gysin homomorphism $\iota _*: H^i_{{\acute {e}t}}(Y, \Lambda (m)) \to H^{i+2}_{{\acute {e}t}}(X, \Lambda (m+1))$ for any pair of integers $i \ge 0$ and $m \in {\mathbb Z}$ . This homomorphism is defined as follows. It follows from [Reference Deligne14, §2] that the line bundle ${\mathcal O}_X(Y)$ (which we shall write for short as ${\mathcal O}(Y)$ ) on X has a canonical class $[{\mathcal O}(Y)] \in H^1_{Y, {\acute {e}t}}(X, {\mathbb G}_m)$ , and its image via the boundary map $H^1_{Y, {\acute {e}t}}(X, {\mathbb G}_m) \to H^2_{Y, {\acute {e}t}}(X, \Lambda )$ is Deligne’s localized Chern class $c_1(Y)$ . Here, $H^*_{Y, {\acute {e}t}}(X, -)$ denotes the étale cohomology with support in Y.

At the level of the derived category $D^{+}(Y, \Lambda )$ , this Chern class is given in terms of the map $c_1(Y) \colon \Lambda _Y \to \iota ^{!}\Lambda _X(1)[2]$ . Using this Chern class, Gabber’s Gysin homomorphism $\iota _* \colon H^*_{{\acute {e}t}}(Y, \Lambda (m)) \to H^{*+2}_{{\acute {e}t}}(X, \Lambda (m+1))$ is the one induced on the cohomology by the composite map $\iota _* \colon \iota _*(\Lambda _Y) \to \iota _* \iota ^{!}(\Lambda _X(1)[2]) \to \Lambda _X(1)[2]$ in $D^{+}(X, \Lambda )$ , where the second arrow is the counit-of-adjunction map.

The local complete intersection closed immersions $Y \hookrightarrow X$ and $Y_s \hookrightarrow X_{\textrm {sing}}$ induce a diagram of distinguished triangles in $D^{+}(X, \Lambda )$ given by

(5.4)

The Cartesian square

(5.5)

and the functoriality of Deligne’s localized Chern class imply (see [Reference Navarro56, Corollary 2.12] or [Reference Fujiwara16, Proposition 1.1.3]) that the right-side square in diagram (5.4) is commutative. It follows that there is a Gysin morphism $c^{j_!}_1(Y^o) \colon \iota _* \circ \tilde {j}_{!}\left (\Lambda _{Y^o}\right ) \to j_{!}\left (\Lambda _{X^o}\right )(1)[2]$ in $D^{+}(X, \Lambda )$ such that the resulting left square in diagram (5.4) is commutative.

Applying the cohomology functor on $D^{+}(X, \Lambda )$ , we get a commutative diagram of long exact sequences

(5.6)

where $H^*_{c, {\acute {e}t}}(-, \Lambda (m))$ denotes the étale cohomology with compact support [Reference Milne54, Chapter VI, §3] and the vertical arrows are the Gysin homomorphisms.

Assume now that k is either finite or algebraically closed. Recall in this case (see [Reference Jannsen, Saito and Zhao32, Introduction] and [Reference Milne54, Chapter VI, §11], as well as [Reference Deglise12, Corollary 4.2.3]) that there is a perfect pairing

(5.7) $$ \begin{align} H^i_{c, {\acute{e}t}}(X^o, \Lambda(m)) \times H^{2d+e-i}_{{\acute{e}t}}(X^o, \Lambda(d-m)) \to H^{2d+e}_{c, {\acute{e}t}}(X^o, \Lambda(d)) \cong \Lambda \end{align} $$

for $m \in {\mathbb Z}$ , where $e = 1$ if k is finite and $e = 0$ if k is algebraically closed. This pairing exists even if $X^o$ is not smooth, but may not be perfect in the latter case. It is well known and follows from its construction (see [Reference Milne54, proof of Theorem VI.11.1] or directly use [Reference Deglise12, §3.3.13 and Remark 4.2.5]) that formula (5.7) is compatible with the closed immersion $\iota ^o \colon Y^o \hookrightarrow X^o$ – that is, there is a commutative diagram

(5.8)

5.3. A Lefschetz theorem for étale cohomology of $X^o$

Let $\iota \colon Y \hookrightarrow X$ be as in §5.1. We shall now prove a Lefschetz theorem for the étale cohomology of $X^o$ . This is of independent interest in the study of singular varieties. In this paper, we shall use it in the proofs of the main results.

Proof. Using diagram (5.8), the proposition is reduced to showing that the Gysin homomorphism

(5.9) $$ \begin{align} \iota^o_* \colon H^i_{c, {\acute{e}t}}(Y^o, \Lambda(d-1)) \to H^{i+2}_{c, {\acute{e}t}}(X^o, \Lambda(d)) \end{align} $$

is an isomorphism for $i \ge 2d+e-3$ and surjective for $i = 2d+e-4$ .

Using the long exact sequences of diagram (5.6) and the known bounds on the étale cohomological dimensions of $X_{{\textrm {sing}}}$ and $Y_s$ , the assertion that formula (5.9) is an isomorphism is equivalent to asserting that the Gysin homomorphism

(5.10) $$ \begin{align} \iota_* \colon H^i_{{\acute{e}t}}(Y, \Lambda(d-1)) \to H^{i+2}_{{\acute{e}t}}(X, \Lambda(d)) \end{align} $$

is an isomorphism for $i \ge 2d+e-3$ and surjective for $i = 2d+e-4$ . For $e = 0$ , this follows from [Reference Skorobogatov71, Theorem 2.1]. We shall prove the case $e = 1$ using a similar strategy as follows.

We let $U = X \setminus Y$ . The Leray spectral sequence for the inclusion $j' \colon U \hookrightarrow X$ yields a strongly convergent spectral sequence

(5.11) $$ \begin{align} E^{a,b}_2 = H^a_{{\acute{e}t}}\left(X, R^bj^{\prime}_*(\Lambda_U(d))\right) \Rightarrow H^{a+b}_{{\acute{e}t}}(U, \Lambda(d)). \end{align} $$

We know that the canonical map $\Lambda _X(d) \xrightarrow {\cong } j^{\prime }_*(\Lambda _U(d))$ is an isomorphism. Furthermore, the canonical map $R^bj^{\prime }_*(\Lambda _U(d)) \to \iota _* \iota ^{*}R^bj^{\prime }_*(\Lambda _U(d))$ is an isomorphism for $b> 0$ . We let ${\mathcal F}_b = \iota ^{*}R^bj^{\prime }_*(\Lambda _U(d))$ . The sheaf exact sequence

$$ \begin{align*} 0 \to \iota_* \iota^{!}(\Lambda_X(d)) \to \Lambda_X(d) \to j^{\prime}_*(\Lambda_U(d)) \to 0 \end{align*} $$

shows that the resulting boundary map $\partial ' \colon {\mathcal F}_1 \to \iota ^{!}(\Lambda _X(d)[2])$ is an isomorphism. Moreover, it follows from the construction of the spectral sequence (5.11) that the map on the cohomology groups $H^a_{{\acute {e}t}}(Y, {\mathcal F}_1) \xrightarrow {\partial '} H^{a+2}_{{\acute {e}t}}(X, \Lambda _X(d))$ , induced by the composite map $\iota _*({\mathcal F}_1) \to \iota _* \iota ^{!}(\Lambda _X(d)[2]) \to \Lambda _X(d)[2]$ , is the map

$$ \begin{align*} E^{a,1}_2 = H^a_{{\acute{e}t}}(Y, {\mathcal F}_1) \xrightarrow{\partial} H^{a+2}_{{\acute{e}t}}(X, \Lambda_X(d)) = E^{a+2, 0}_2. \end{align*} $$

It follows that the composition

$$ \begin{align*} {\mathbb H}^a_{{\acute{e}t}}\left(X, \iota_* \iota^{!}(\Lambda_X(d)[2])\right) \xrightarrow{\partial^{\prime-1}} H^a_{{\acute{e}t}}(Y, {\mathcal F}_1) \xrightarrow{\partial} H^{a+2}_{{\acute{e}t}}(X, \Lambda_X(d)) \end{align*} $$

is induced by the canonical adjunction map $\iota _* \iota ^{!}(\Lambda _X(d)[2]) \to \Lambda _X(d)[2]$ . Precomposing the latter with the Gysin map (see §5.2) $\iota _* \Lambda _Y(d-1) \xrightarrow {c_1(Y)} \iota _* \iota ^{!}(\Lambda _X(d)[2])$ , we see that the composition

(5.12) $$ \begin{align} H^a_{{\acute{e}t}}(Y, \Lambda_Y(d-1)) \xrightarrow{\partial^{\prime-1} \circ c_1(Y)} H^a_{{\acute{e}t}}(Y, {\mathcal F}_1) \xrightarrow{\partial} H^{a+2}_{{\acute{e}t}}(X, \Lambda_X(d)) \end{align} $$

coincides with the Gysin homomorphism $\iota _*$ in diagram (5.6).

We next study the map $H^a_{{\acute {e}t}}(Y, {\mathcal F}_1) \xrightarrow {\partial } H^{a+2}_{{\acute {e}t}}(X, \Lambda _X(d))$ . Using the purity theorem for the closed pair $(X^o, Y^o)$ of smooth schemes [Reference Milne54, Theorem VI.5.1], it follows that

(5.13) $$ \begin{align} {\mathcal F}_b\rvert_{Y^o} \cong \begin{cases} \Lambda_{Y^o}(d) & \text{if }b =1, \\ 0 &\text{if }b> 1. \end{cases} \end{align} $$

On the other hand, the affine Lefschetz theorem [Reference Milne54, Theorem VI.7.3(d)] implies that the sheaf $R^bj^{\prime }_*(\Lambda _U(d))$ is supported on a closed subscheme of Y whose dimension is bounded by $d - b$ . We conclude that

(5.14) $$ \begin{align} H^a_{{\acute{e}t}}\left(X, R^bj^{\prime}_*(\Lambda_U(d))\right) = 0 \text{ if}\ b \ge 2\ \text{and}\ a> \min\{2q+1, 2d-2b+1\}. \end{align} $$

Using formulas (5.11), (5.13) and (5.14), we get exact sequences

(5.15) $$ \begin{align} & H^{i-1}_{{\acute{e}t}}(U, \Lambda_U(d)) \to H^{i-2}_{{\acute{e}t}}(Y, {\mathcal F}_1) \xrightarrow{\partial} H^{i}_{{\acute{e}t}}(X, \Lambda_X(d)) \to H^{i}_{{\acute{e}t}}(U, \Lambda_U(d)), \\ \nonumber & H^{2d-3}_{{\acute{e}t}}(Y, {\mathcal F}_1) \xrightarrow{\partial} H^{2d-1}_{{\acute{e}t}}(X, \Lambda_X(d)) \to H^{2d-1}_{{\acute{e}t}}(U, \Lambda_U(d)) \end{align} $$

for $i \ge 2d$ . Since $d \ge 3$ and the étale cohomological dimension of k is $1$ , another application of the affine Lefschetz theorem reduces these exact sequences to an isomorphism $H^{i-2}_{{\acute {e}t}}(Y, {\mathcal F}_1) \xrightarrow {\cong } H^{i}_{{\acute {e}t}}(X, \Lambda _X(d))$ for $i \ge 2d$ and a surjection $H^{2d-3}_{{\acute {e}t}}(Y, {\mathcal F}_1) \twoheadrightarrow H^{2d-1}_{{\acute {e}t}}(X, \Lambda _X(d))$ .

Finally, we consider the long exact sequences

The left vertical isomorphism on the top is a consequence of formula (5.13). Using the bound on the cohomological dimension of $Y_s$ , this diagram shows that the right vertical arrow is an isomorphism. Since the composite right vertical arrow is the Gysin homomorphism as observed before, we conclude that the map $\iota _*$ in formula (5.10) is an isomorphism for $i \ge 2d-2$ and surjection for $i = 2d-3$ . This finishes the proof.

5.4. A Lefschetz theorem for étale fundamental groups

Recall that the Lefschetz theorem for the étale fundamental groups of smooth projective schemes over a field was proven by Grothendieck [Reference Grothendieck23, Exposé XII, Corollaire 3.5]. However, this is a very challenging problem for smooth nonprojective schemes, due to the presence of ramification when we extend étale covers to compactifications. We shall prove the following version of the Lefschetz theorem for the étale fundamental groups of smooth but nonprojective schemes:

Proof. Since $\pi ^{\textrm {ab}}_1(X^o) \xrightarrow {\cong } {\underset {n \in {\mathbb Z}}\varprojlim } {\pi ^{\textrm {ab}}_1(X^o)}/n$ and the same holds for $Y^o$ , it suffices to show that the map ${\pi ^{\textrm {ab}}_1(Y^o)}/n \to {\pi ^{\textrm {ab}}_1(X^o)}/n$ is an isomorphism for every integer $n \in {\mathbb Z}$ . Using the Pontryagin duality (see, for example, the middle of [Reference Raskind62, proof of Lemma 1.9, p. 99]), $\left ({\pi ^{\textrm {ab}}_1(Z)}/n\right )^\vee \cong H^1_{{\acute {e}t}}(Z, {{\mathbb Z}}/n)$ , for $Z \in \{X^o, Y^o\}$ , we are reduced to showing that the pullback map

(5.16) $$ \begin{align} (\iota^o)^* \colon H^1_{\acute{e}t}(X^o, {{\mathbb Z}}/n) \to H^1_{\acute{e}t}(Y^o, {{\mathbb Z}}/n) \end{align} $$

is an isomorphism for all $n \in {\mathbb Z}$ . In view of Proposition 5.3, we can assume $n = p^r$ , where ${\textrm {char}}(k) = p> 0$ . We shall prove this by induction on $r \ge 1$ .

We have an exact sequence of constant étale sheaves on ${\operatorname {\mathbf {Sch}}}_k$ :

$$ \begin{align*} 0 \to {{\mathbb Z}}/{p} \to {{\mathbb Z}}/{p^r} \to {{\mathbb Z}}/{p^{r-1}} \to 0 \end{align*} $$

for every $r \ge 2$ , where $ {{\mathbb Z}}/{p} \to {{\mathbb Z}}/{p^r}$ is multiplication by $p^{r-1}$ . This yields a long exact sequence of étale cohomology groups

$$ \begin{align*} &H^0_{\acute{e}t}(X^o, {{\mathbb Z}}/{p^r}) \to H^0_{\acute{e}t}\left(X^o, {{\mathbb Z}}/{p^{r-1}}\right) \to H^1_{\acute{e}t}(X^o, {{\mathbb Z}}/{p}) \to H^1_{\acute{e}t}(X^o, {{\mathbb Z}}/{p^r}) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\to H^1_{\acute{e}t}\left(X^o, {{\mathbb Z}}/{p^{r-1}}\right) \to H^2_{\acute{e}t}(X^o, {{\mathbb Z}}/{p}). \end{align*} $$

Since $X^o$ is integral, the first arrow from the left in this exact sequence is surjective. The same applies to $Y^o$ as well. We thus get a commutative diagram of exact sequences

(5.17)

where the vertical arrows are the pullback maps. Using a diagram chase and an induction on r, we are reduced to showing that the pullback map

(5.18) $$ \begin{align} (\iota^o)^* \colon H^i_{\acute{e}t}(X^o, {{\mathbb Z}}/p) \to H^i_{\acute{e}t}(Y^o, {{\mathbb Z}}/p) \end{align} $$

is an isomorphism for $i = 1$ and injective for $i = 2$ .

We let $U = X \setminus Y$ so that $U^o = X^o \setminus Y^o$ . We let $u \colon U \hookrightarrow X$ and $u^o \colon U^o \hookrightarrow X^o$ denote the inclusion maps. Then formula (5.18) is equivalent to the assertion that $H^i_{{\acute {e}t}}\left (X^o, u^o_{!} \left ({{{\mathbb Z}}/p}\rvert _{U^o}\right )\right )= 0$ for $i = 1, 2$ . Using the cohomology exact sequence associated to the relative Artin–Schreier sheaf exact sequence (where F is the Frobenius of ${\mathcal O}_{X^o}$ )

$$ \begin{align*} 0 \to u^o_{!} \left({{{\mathbb Z}}/p}\rvert_{U^o}\right) \to {\mathcal O}_{X^o}(-Y^o) \xrightarrow{1 - F} {\mathcal O}_{X^o}(-Y^o) \to 0, \end{align*} $$

we get the long exact sequence

$$ \begin{align*} \dotsb \to H^{i-1}_{\acute{e}t}\left(X^o, {\mathcal O}_{X^o}(-Y^o)\right) \to H^i_{{\acute{e}t}}\left(X^o, u^o_{!} \left({{{\mathbb Z}}/p}\rvert_{U^o}\right)\right) &\to H^i_{\acute{e}t}\left(X^o, {\mathcal O}_{X^o}(-Y^o)\right) \\ &\xrightarrow{1 - F} H^i_{\acute{e}t}\left(X^o, {\mathcal O}_{X^o}(-Y^o)\right) \to \dotsc \end{align*} $$

for $i \ge 1$ . Using this, it suffices to show that $H^i_{\acute {e}t}\left (X^o, {\mathcal O}_{X^o}(-Y^o)\right ) = 0$ for $i \le 2$ . Equivalently, it suffices to show that $H^i_{\textrm {zar}}(X^o, {\mathcal O}_{X^o}(-Y^o)) = 0$ for $i \le 2$ .

We now consider the exact sequence of Zariski cohomology groups

$$ \begin{align*} \dotsb \to H^{i-1}\left(X^o, {\mathcal O}_{X^o}(-Y^o)\right) \to H^{i}_{X_{\textrm{sing}}}(X, {\mathcal O}_{X}(-Y)) &\to H^{i}(X, {\mathcal O}_{X}(-Y)) \\ &\qquad\to H^{i}\left(X^o, {\mathcal O}_{X^o}(-Y^o)\right) \to \dotsb. \end{align*} $$

Since X is an $(R_3+ S_4)$ -scheme, we have ${\underset {x \in X_{\textrm {sing}}}{\textrm {inf}}} \{{\textrm {depth}} ({\mathcal O}_{X}(-Y)_x)\} \ge 4$ . We conclude from [Reference Hartshorne28, Theorem 3.8] and a spectral sequence argument that $H^{i}_{X_{\textrm {sing}}}(X, {\mathcal O}_{X}(-Y)) = 0$ for $i \le 3$ . The long exact sequence then tells us that the map $H^{i}(X, {\mathcal O}_{X}(-Y)) \to H^{i}\left (X^o, {\mathcal O}_{X^o}(-Y^o)\right )$ of Zariski cohomology groups is an isomorphism for $i \le 2$ . The theorem therefore is finally reduced to showing that $H^{i}(X, {\mathcal O}_{X}(-Y)) = 0$ for $i \le 2$ . But this follows from Lemma 5.1, since the degree of the hypersurface H is very large.

6.1. Proof of Theorem 1.2

Let k be a finite field and $X \in {\operatorname {\mathbf {Sch}}}_k$ an integral projective $R_1$ -scheme of dimension $d \ge 1$ over k. It was shown in Corollary 3.3 that the Frobenius substitution associated to the regular closed points gives rise to a reciprocity homomorphism $\phi _X \colon {\textrm {CH}}^{LW}_0(X) \to \pi ^{\textrm {ab}}_1(X^o)$ . Also, the left vertical arrow of diagram (3.3) gives us the restriction map $\phi _{X}^{0}$ . What remains to show is that $\phi ^0_X \colon {\textrm {CH}}^{LW}_0(X)^0 \to \pi ^{\textrm {ab}}_1(X^o)^0$ is surjective, and it is an isomorphism of finite groups under either condition (1) or (2) of the theorem. We shall prove all of these by induction on d. Since the case $d \le 2$ already follows from Theorem 3.6, we shall assume that $d \ge 3$ .

We let $\pi \colon X_n \to X$ be the normalization morphism. Using Corollaries 2.6 and 2.7, the Zariski–Nagata purity theorem [Reference Grothendieck24, Exposé X, Théorème 3.1] for $\pi _1((X_n)^o)$ , and [Reference Raskind62, Lemma 5.1(1)], we can assume that X is normal.

Part 1. We first consider the case when X has isolated singularities. Let $\alpha \in {\textrm {CH}}^{LW}_0(X)$ be a $0$ -cycle such that $\phi _X(\alpha ) = 0$ . We now fix an embedding $X \hookrightarrow {\mathbb P}^N_k$ and apply [Reference Wutz77, Theorem 3.1] to find a hypersurface $H \subset {\mathbb P}^N_k$ containing ${\textrm {Supp}}(\alpha )$ such that the scheme-theoretic intersection $Y = X \cap H$ is smooth and does not meet $X_{\textrm {sing}}$ . In particular, $Y \subset X^o$ . We can then find a cycle $\alpha ' \in {\textrm {CH}}^{F}_0(Y)$ such that $\alpha = \iota _*(\alpha ')$ , where $\iota \colon Y \hookrightarrow X$ is the inclusion. Since X is an $S_2$ -scheme of dimension $\ge 3$ and $Y \subset X^o$ , it follows from [Reference Grothendieck23, Exposé XII, Corollaire 3.5] that Y is connected (hence integral) and the map $\pi _1(Y) \to {\underset {W}\varprojlim } \pi _1(W)$ is an isomorphism, where the limit is taken over all open neighborhoods of Y in X.

Since $Y \subset X^o$ , there is a factorization

(6.1) $$ \begin{align} \pi_1(Y) \to {\underset{U}\varprojlim} \pi_1(U) \to {\underset{W}\varprojlim} \pi_1(W), \end{align} $$

where the first limit is over all open neighborhoods of Y contained in $X^o$ . Note that the second arrow is an isomorphism, and therefore so is the first arrow. On the other hand, we showed in the proof of Lemma 4.15 that for any open $U \subset X^o$ containing Y, the codimension of $X^o \setminus U$ is $\ge 2$ . We conclude from the Zariski–Nagata purity theorem that the first limit in formula (6.1) is $\pi _1(X^o)$ . It follows that the map $\pi _1(Y) \to \pi _1(X^o)$ is an isomorphism. We therefore conclude that there is a commutative diagram

(6.2)

such that the right vertical arrow is an isomorphism and $\alpha ' \in {\textrm {CH}}^F_0(Y)$ .

It follows that $\iota _* \circ \phi _Y(\alpha ') = 0$ . Since $\phi _Y$ is injective by induction, we get $\alpha ' = 0$ . In particular, we get $\alpha = \iota _*(\alpha ') = 0$ . This shows that $\phi _X$ is injective. The surjectivity of $\phi ^0_X$ and finiteness of ${\textrm {CH}}^{LW}_0(X)^0$ and $\pi ^{\textrm {ab}}_1(X^o)^0$ also follow from diagram (6.2) and induction on d. We have thus finished the proof of Theorem 1.2 when X has isolated singularities.

Part 2. We now assume that X is an $(R_3 + S_4)$ -scheme. In particular, it is normal. Moreover, it is regular if $d \le 3$ , in which case the theorem is due to Kato and Saito [Reference Kato and Saito34]. If $d = 4$ , then X has isolated singularities, in which case we have already proven the theorem in part 1. We can therefore assume that $d \ge 5$ . As before, we let $\alpha \in {\textrm {CH}}^{LW}_0(X)$ be a $0$ -cycle such that $\phi _X(\alpha ) = 0$ .

We can now apply [Reference Ghosh and Krishna20, Theorem 6.3], which says that for every integer $m \gg 1$ there exists a hypersurface $H \subset {\mathbb P}^N_k$ of degree m containing ${\textrm {Supp}}(\alpha )$ such that the hypersurface section $Y = X \cap H$ satisfies the following:

1. (1) $Y \cap X^o = Y^o$ is regular.

2. (2) Y is an $(R_3 + S_4)$ -scheme.

3. (3) Y contains no irreducible component of $X_{\textrm {sing}}$ .

Since X is normal and integral, it follows from [Reference Grothendieck23, Exposé XII, Corollaire 3.5] that Y is connected. Since (2) implies that Y is also normal, it follows that it must be integral. In particular, it is good (see the beginning of §5). Moreover, there is a $0$ -cycle $\alpha ' \in {\textrm {CH}}^{LW}_0(Y)$ such that $\alpha = \iota _*(\alpha ')$ if we let $\iota \colon Y \hookrightarrow X$ be the inclusion.

Using Corollary 2.7 and [Reference Raskind62, Lemma 5.1(1)], we get a commutative diagram

(6.3)

The right vertical arrow is an isomorphism by Theorem 5.4, since $m \gg 0$ . It follows that $\phi _Y(\alpha ') = 0$ . We conclude by induction that $\alpha ' = 0$ . The surjectivity of $\phi _X$ and finiteness of $\pi ^{\textrm {ab}}_1(X^o)^0$ also follow from the diagram, because $\phi _Y$ is surjective and $\pi ^{\textrm {ab}}_1(Y^o)^0$ is finite by induction. This finishes the proof of the theorem, except that we still need to show that $\phi ^0_X$ is surjective without condition (1) or (2) of the theorem.

As $d \ge 3$ and X is normal (equivalently, $(R_1 + S_2)$ ), we can find a hypersurface section $Y = X \cap H$ which satisfies conditions (1) and (3) in part 2, and it is an $(R_1 + S_2)$ -scheme. We already observed that Y is then an integral normal scheme. Now, the map $\iota _*: \pi ^{\textrm {ab}}_1(Y^o) \to \pi ^{\textrm {ab}}_1(X^o)$ is surjective by Proposition 4.15 and [Reference Grothendieck23, Exposé X, Corollaire 2.6]. Since the map $\phi _Y$ in diagram (6.3) is surjective on degree $0$ subgroups by induction, we conclude that $\phi ^0_X$ is surjective.

6.2. Proof of Theorem 1.4

Let X be as in Theorem 1.4. We need some preparation before we prove the theorem.

By [Reference Raskind62, Theorem 6.2], the norm maps $N_x \colon K_0(k(x)) \to K_0(k)$ for $x \in X^o_{(0)}$ induce a natural homomorphism $\deg \colon H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right ) \to {\mathbb Z}$ . Since these norms are multiplication by the degrees of the field extensions ${k(x)}/k$ , we see that the composition ${\mathcal Z}_0(X^o) \xrightarrow {{\operatorname {\textrm {cyc}}}_X} H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right ) \xrightarrow {\deg } {\mathbb Z}$ is the degree homomorphism. We let $H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )^0$ be the kernel of this map. We then get a commutative diagram

(6.4)

where the right vertical arrow is the canonical inclusion into the profinite completion and $\pi ^{\textrm {ab}}_1(X^o) \to \widehat {{\mathbb Z}}$ is the push-forward map induced by the structure map of $X^o$ .

Since the last term of the top row of diagram (6.4) is torsion free, we have an exact sequence of inverse limits

(6.5) $$ \begin{align} 0\to {\underset{m}\varprojlim} H^d_{\textrm{nis}}\left(X, {\mathcal K}^M_{d,X}\right)^0/m \to {\underset{m}\varprojlim} {H^d_{\textrm{nis}}\left(X, {\mathcal K}^M_{d,X}\right)}/m \to \widehat{{\mathbb Z}}. \end{align} $$

Proof. If $\pi \colon X_n \to X$ denotes the normalization of X, then the maps $\pi ^* \colon H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right ) \to H^d_{\textrm {nis}}\left (X_n, {\mathcal K}^M_{d,X_n}\right )$ and $\pi ^* \colon \pi ^{\textrm {ab}}_1(X^o) \to \pi ^{\textrm {ab}}_1(X^o_n)$ are isomorphisms. The first isomorphism holds for dimension reasons and by the exactness of $\pi _*$ on Nisnevich sheaves. The second isomorphism holds by the Zariski–Nagata purity theorem [Reference Grothendieck24, Exposé X, Théorème 3.1]. We can therefore assume that X is normal. By diagram (6.4), it suffices to show that $\rho ^0_X$ is injective.

We assume first that X is geometrically connected. Since $H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )^0$ is a finite group (hence profinite complete) by [Reference Raskind62, Theorem 6.2(1)] (take $I = {\mathcal O}_X$ and $T = X$ ) and $\pi ^{\textrm {ab}}_1(X^o)$ is profinite complete, diagram (6.4) and formula (6.5) give rise to a commutative diagram of exact sequences

(6.6)

The middle vertical arrow is an isomorphism by [Reference Kato and Saito35, Theorem 9.1(3)]. It follows that the left vertical arrow must also be an isomorphism.

Suppose now that X is not geometrically connected. Since $\widehat {\rho }_X$ is always an isomorphism by [Reference Kato and Saito35, Theorem 9.1(3)], it suffices to show, using formula (6.5), that $H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )^0$ is profinite complete. To show the latter, it suffices to prove the stronger claim that $H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )^0$ is torsion of bounded exponent. Since the image of $\rho ^0_X$ is finite by Theorem 1.2, the claim is equivalent to proving that the kernel of $\rho ^0_X$ is torsion of bounded exponent. Since $\widehat {\rho }_X$ is an isomorphism, it is enough to show that the kernel of the completion map $\alpha _X \colon H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right ) \to {\underset {m}\varprojlim } {H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )}/m$ is torsion of bounded exponent.

Now, we know that there exists a finite field extension ${k'}/k$ such that $X_{k'}$ is a disjoint union of geometrically connected integral normal schemes. We pick any irreducible component Y of $X_{k'}$ and let $\pi \colon Y \to X$ be the projection map. Then $\pi $ is a finite étale morphism of normal schemes whose degree divides $[k':k]$ . Since we have shown that $\alpha _Y$ is injective, it suffices to show that the kernel of the map $\pi ^* \colon H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )\to H^d_{\textrm {nis}}\left (Y, {\mathcal K}^M_{d,Y}\right )$ is torsion of bounded exponent. But this follows directly from [Reference Kato and Saito35, Lemma 4.5(1)].

We now prove Theorem 1.4. The claim that the cycle class map ${\operatorname {\textrm {cyc}}}_X \colon {\mathcal Z}_0(X^o) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )$ factors through ${\textrm {CH}}^{LW}_0(X)$ is a direct consequence of Lemmas 3.1 and 6.1, and Corollary 3.3. The resulting map ${\operatorname {\textrm {cyc}}}_X \colon {\textrm {CH}}^{LW}_0(X) \to H^d_{\textrm {nis}}\left (X, {\mathcal K}^M_{d,X}\right )$ is surjective by [Reference Kato and Saito35, Theorem 2.5]. It is injective because $\phi _X = \rho _X \circ {\operatorname {\textrm {cyc}}}_X$ is injective by Theorem 1.2.

6.3. Proof of Theorem 1.5

Let X be as in Theorem 1.5 and let $f \colon \widetilde {X} \to X$ be a resolution of singularities with reduced exceptional divisor E. By Corollary 2.6, we can assume that X is normal. Since $f^* \colon {\textrm {CH}}^{LW}_0(X) \to {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )$ is clearly surjective for all $m \ge 1$ , we only need to show that this map is injective for all $m \gg 1$ . The latter is equivalent to showing that the map $f^* \colon {\textrm {CH}}^{LW}_0(X)^0 \to {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )^0$ is injective for all $m \gg 1$ .

We let $C(X^o) = {\underset {m}\varprojlim } {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )$ and let $C(X^o)^0$ denote the kernel of the degree map $C(X^o) \to {\mathbb Z}$ . It was shown in [Reference Kerz and Saito38, Proposition 3.2] that the Frobenius substitution associated to closed points in $X^o$ defines a reciprocity map $\phi _{X^o} \colon C(X^o) \to \pi ^{\textrm {ab}}_1(X^o)$ such that one has a commutative diagram

(6.7)

If we restrict this diagram to the degree $0$ subgroups, then Theorem 1.2 says that the top horizontal arrow is an isomorphism. On the other hand, [Reference Kerz and Saito38, Theorem III] (if ${\textrm {char}}(k) \neq 2$ ) and [Reference Binda, Krishna and Saito5, Theorem 8.5] (in general) say that the bottom horizontal arrow is an isomorphism. It follows that the map $f^* \colon {\textrm {CH}}^{LW}_0(X)^0 \to C(X^o)^0$ is an isomorphism. In particular, $C(X^o)^0$ is finite.

Since $C(X^o) \twoheadrightarrow {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )$ for every $m \ge 1$ , it follows that $C(X^o)^0 \twoheadrightarrow {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )^0$ for every $m \ge 1$ . We conclude that $\left \{{\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )^0\right \}_{m \ge 1}$ is an inverse system of abelian groups whose transition maps are all surjective and whose limit $C(X^o)^0$ is finite. But this implies that this inverse system is eventually constant. That is, the map $C(X^o)^0 \to {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )^0$ is an isomorphism for all $m \gg 1$ . It follows that the map $f^* \colon {\textrm {CH}}^{LW}_0(X)^0 \to {\textrm {CH}}_0\left (\widetilde {X}\mid mE\right )^0$ is an isomorphism for all $m \gg 1$ .

6.4. Necessity of $R_1$ -condition

We show by an example that it is necessary to assume the $R_1$ -condition in Theorem 1.2. Let C be the projective plane curve over a finite field k which has a simple cusp along the origin and is regular elsewhere. Its local ring at the singular point is analytically isomorphic to $k\left [\left [t^2, t^3\right ]\right ]$ , which is canonically a subring of its normalization $k[[t]]$ . Let $\pi \colon {\mathbb P}^1_k \to C$ denote the normalization map. Let $S \cong {\textrm {Spec}}\left ({k\left [t^2,t^3\right ]}/{\left (t^2,t^3\right )}\right )$ denote the reduced conductor and $\widetilde {S} \cong {\textrm {Spec}}\left ({k[t]}/{\left (t^2\right )}\right )$ its scheme-theoretic inverse image in ${\mathbb P}^1_k$ . We have a commutative diagram with exact rows:

(6.8)

The isomorphism of the middle vertical map follows from the known result that the double relative K-groups $K_0\left (C, {\mathbb P}^1_k, mS\right )$ and $K_{-1}\left (C, {\mathbb P}^1_k, mS\right )$ vanish.

It is easy to check from the K-theory localization sequence that ${\textrm {Pic}}\left ({\mathbb P}^1_k, m\widetilde {S}\right ) \xrightarrow {\cong } K_0\left ({\mathbb P}^1_k, m\widetilde {S}\right )$ . On the other hand, the known class field theory for curves (with modulus) tells us that there is a canonical isomorphism ${\underset {m}\varprojlim } {\textrm {Pic}}^0\left ({\mathbb P}^1_k, m\widetilde {S}\right ) \xrightarrow {\cong } \pi ^{\textrm {ab}}(C^o)^0$ . It follows that there are isomorphisms $(1 + tk[[t]])^{\times } \xrightarrow {\cong } {\mathbb W}(k) \xrightarrow {\cong } \pi ^{\textrm {ab}}(C^o)^0$ . On the other hand, ${\textrm {CH}}^{LW}_0(C)^0 \cong {\textrm {Pic}}^0(C) \cong k$ . This shows that there is no reciprocity map ${\textrm {CH}}^{LW}_0(C)^0 \to \pi ^{\textrm {ab}}(C^o)^0$ , and the two can not be isomorphic.

7.1. Generalized Albanese of a smooth variety

Let $X \in {\operatorname {\mathbf {Sch}}}_k$ be an integral projective $R_1$ -scheme of dimension $d \ge 1$ . Let $U \subset X^o$ be a nonempty open subscheme and set $Z = X \setminus U$ , endowed with the reduced induced closed subscheme structure. In this case, Serre gave an explicit description of ${\textrm {Alb}}_S(U)$ in [Reference Serre69]. We recall this description. We remark here that even if we do not assume X to be smooth, the exposition of [Reference Serre69] remains valid in the present case with no modification.

Let ${\textrm {Div}}(X)$ denote the free abelian group of Weil divisors on X. Let $\Lambda ^1_{U}(X)$ denote the image of the push-forward map ${\mathcal Z}_{d-1}(Z) \to {\mathcal Z}_{d-1}(X) = {\textrm {Div}}(X)$ . There is thus a canonical homomorphism $\iota _U: \Lambda ^1_{U}(X) \to \frac {{\textrm {Cl}}(X)}{{\textrm {Cl}}^0(X)} = {\textrm {NS}}(X)$ , where ${\textrm {NS}}(X)$ is the Néron–Severi group of X. Let $\Lambda _U(X)$ denote the kernel of the canonical map $\Lambda ^1_{U}(X) \xrightarrow {\iota _U} {\textrm {NS}}(X)$ so that the quotient ${\textrm {Div}}(X) \twoheadrightarrow {\textrm {Cl}}(X)$ induces a homomorphism $\Lambda _U(X) \to {\textrm {Cl}}^0(X)$ . It was shown by Serre [Reference Serre69] that ${\textrm {Alb}}_S(U)$ is the Cartier dual of the $1$ -motive $[\Lambda _U(X) \to {\textrm {Pic}}_W(X)]$ (see [Reference Deligne13] for the definitions of $1$ -motives and their Cartier duals).

We thus have a canonical exact sequence of algebraic groups

(7.1) $$ \begin{align} 0 \to \Lambda_U(X)^{\vee} \to {\textrm{Alb}}_S(U) \to {\textrm{Alb}}_W(X) \to 0, \end{align} $$

where $\Lambda _U(X)^{\vee }$ is the Cartier dual of the constant group scheme over k associated to the lattice $\Lambda _U(X)$ . In particular, $\Lambda _U(X)^{\vee }$ is a split torus of rank equal to the rank of the lattice $\Lambda _U(X)$ .

7.2. A Lefschetz theorem for ${\textrm {Alb}}_S(U)$

We shall now prove a Lefschetz theorem for the generalized Albanese variety. We let $X \subset {\mathbb P}^N_k$ be an integral normal projective scheme of dimension $d \ge 3$ over k which is an $R_2$ -scheme. Let $U \subset X^o$ be a nonempty open subscheme. We let $Z = X \setminus U$ with the reduced closed subscheme structure. We let $H \subset {\mathbb P}^N_k$ be a hypersurface and $Y = X \cap H$ the scheme theoretic intersection. We shall say that Y is ‘Z-admissible’ if the following hold:

1. (1) Y is good (see §5).

2. (2) For every irreducible component $Z'$ of Z of dimension $d-1$ , the scheme-theoretic intersection $Y \cap Z'$ is integral of dimension $d-2$ .

Let $\iota \colon Y \hookrightarrow X$ be the inclusion of a Z-admissible hypersurface section of X. Then the construction of the pullback map on algebraic cycles in [Reference Fulton17, Chapter 2, §4] yields a homomorphism $\iota ^* \colon {\textrm {Div}}(X) \to {\textrm {Div}}(Y)$ . Furthermore, it easily follows from the proof of [Reference Fulton17, Corollary 2.4.1] that it induces the pullback maps $\iota ^* \colon {\textrm {Cl}}(X) \to {\textrm {Cl}}(Y)$ and $\iota ^* \colon {\textrm {Cl}}^0(X) \to {\textrm {Cl}}^0(X)$ . Taking the quotients, we get a pullback map $\iota ^* \colon {\textrm {NS}}(X) \to {\textrm {NS}}(Y)$ . We shall follow the notations of §5.

Proof. Since X is normal and Y is good, the latter is also normal. It follows therefore from [Reference Fulton17, Example 10.3.4] that the pullback maps $j^* \colon {\textrm {NS}}(X) \to {\textrm {NS}}(X^o)$ and $\tilde {j}^* \colon {\textrm {NS}}(Y) \to {\textrm {NS}}(Y^o)$ are isomorphisms. Hence, the lemma is equivalent to the statement that the map $(\iota ^o)^* \colon {\textrm {NS}}(X^o)_{{\textrm {tor}}} \to {\textrm {NS}}(Y^o)_{{\textrm {tor}}}$ is injective.

Since ${\textrm {Pic}}_W(X)(k)$ and ${\textrm {Pic}}_W(Y)(k)$ are divisible, there is a commutative diagram of short exact sequences

(7.2)

Since $Y \subset X$ is a general hypersurface section, it follows from the Lefschetz theorem for the Weil–Albanese variety [Reference Lang46, Chapter VII, Theorem 5] that the canonical map ${\textrm {Alb}}_W(Y) \to {\textrm {Alb}}_W(X)$ is an isogeny of abelian varieties whose kernel is isomorphic to the finite infinitesimal group scheme $\alpha _{p^r}$ for some $r \ge 0$ . Considering the induced map between the dual abelian varieties, we see that the pullback morphism ${\textrm {Pic}}_W(X) \to {\textrm {Pic}}_W(Y)$ is an isogeny of abelian varieties [Reference Cornell and Silverman10, Theorem 11.1]. It is then an easy exercise to show that the map ${\textrm {Cl}}^0(X)_{{\textrm {tor}}} \to {\textrm {Cl}}^0(Y)_{{\textrm {tor}}}$ is surjective. Using a diagram chase in diagram (7.2), the lemma is now reduced to showing that the map ${{\textrm {Pic}}}(X^o)_{{\textrm {tor}}} \to {{\textrm {Pic}}}(Y^o)_{{\textrm {tor}}}$ is injective.

We first fix a prime-to-p integer n. Since k is algebraically closed, we can identify $\mu _n$ with ${{\mathbb Z}}/n$ . Since $H^0_{{\acute {e}t}}\left (X^o, {\mathcal O}^{\times }_{X^o}\right ) \cong k^{\times }$ (because X is $R_1$ ) and the latter is a divisible group, one observes using the Kummer sequence that ${}_n{{\textrm {Pic}}}(X^o) \cong H^1_{{\acute {e}t}}(X^o, {{\mathbb Z}}/n)$ . By the same token, we have ${}_n{{\textrm {Pic}}}(Y^o) \cong H^1_{{\acute {e}t}}(Y^o, {{\mathbb Z}}/n)$ . It follows therefore from Proposition 5.3 that the map ${}_n{{\textrm {Pic}}}(X^o) \to {}_n{{\textrm {Pic}}}(Y^o)$ is injective. We note here that this part of Proposition 5.3 does not require X to be $R_2$ .

We now let $n = p^r$ for some $r \ge 1$ . Using the short exact sequence of étale sheaves

(7.3) $$ \begin{align} 0 \to {\mathcal O}^{\times}_{X^o} \xrightarrow{n} {\mathcal O}^{\times}_{X^o} \to {{\mathcal O}^{\times}_{X^o}}/{n} \to 0, \end{align} $$

we see that ${}_n{{\textrm {Pic}}}(X^o) \cong H^0_{{\acute {e}t}}\left (X^o, {{\mathcal O}^{\times }_{X^o}}/{n}\right )$ . Using the similar isomorphism for $Y^o$ , we need to show that the map $H^0_{{\acute {e}t}}\left (X^o, {{\mathcal O}^{\times }_{X^o}}/{n}\right ) \to H^0_{{\acute {e}t}}\left (Y^o, {{\mathcal O}^{\times }_{Y^o}}/{n}\right )$ is injective. Comparing the exact sequence (7.3) with the similar sequence for $Y^o$ , this injectivity is equivalent to showing that $H^0_{{\acute {e}t}}\left (X^o, {{\mathcal K}_{1, X^o\mid Y^o}}/n\right ) = 0$ , where we let ${\mathcal K}_{1, X^o\mid Y^o} = {\textrm {Ker}}\left ({\mathcal O}^{\times }_{X^o} \twoheadrightarrow (\iota ^o)^*\left ({\mathcal O}^{\times }_{Y^o}\right )\right )$ . Note here that ${{\mathcal K}_{1, X^o\mid Y^o}}/n = {\textrm {Ker}}\left ({{\mathcal K}_{1, X^o}}/n \to (\iota ^o)^*\left ({{\mathcal O}^{\times }_{Y^o}}/n\right )\right )$ , since ${\mathcal O}^{\times }_{Y^o}$ is p-torsion free.

We let $W_r\Omega ^\bullet _{X^o}$ be the p-typical de Rham–Witt complex of $X^o$ (see, for example, [Reference Illusie30]) and let $W_r\Omega ^i_{X^o, {\operatorname {log}}}$ be the image of the Bloch–Gabber–Kato homomorphism ${\textrm {dlog}} \colon {\mathcal K}^M_{i, X^o} \to W_r\Omega ^i_{X^o}$ . This map is given by ${\textrm {dlog}}(\{x_1, \dotsc , x_i\}) = {\textrm {dlog}}[x_1]_r \wedge \dotsb \wedge {\textrm {dlog}}[x_i]_r$ , where $[\cdot ]_r$ denotes the Teichmüller homomorphism $[\cdot ]_r \colon {\mathcal O}^{\times }_{X^o} \to \left (W_r{\mathcal O}_{X^o}\right )^{\times }$ . The Bloch–Gabber–Kato homomorphism induces an isomorphism ${\textrm {dlog}} \colon {{\mathcal K}^M_{i, X^o}}/{n} \xrightarrow {\cong } W_r\Omega ^i_{X^o, {\operatorname {log}}}$ .

We let $W_r\Omega ^1_{{X^o}/{Y^o}, {\operatorname {log}}}$ denote the image of ${\textrm {dlog}} \colon {{\mathcal K}_{1, X^o\mid Y^o}}/n \hookrightarrow W_r\Omega ^1_{X^o}$ . It suffices to show that $H^0_{{\acute {e}t}}\left (X^o, W_r\Omega ^1_{X^o\mid Y^o, {\operatorname {log}}}\right ) = 0$ . Using the short exact sequence [Reference Jannsen, Saito and Zhao32, Theorem 1.1.6]

(7.4) $$ \begin{align} 0 \to W_{r-1}\Omega^1_{X^o\mid Y^o, {\operatorname{log}}} \xrightarrow{\underline{p}} W_r\Omega^1_{X^o\mid Y^o, {\operatorname{log}}} \to W_1\Omega^1_{X^o\mid Y^o, {\operatorname{log}}} \to 0 \end{align} $$

and induction on r, it suffices to show that $H^0_{{\acute {e}t}}\left (X^o, W_1\Omega ^1_{X^o\mid Y^o, {\operatorname {log}}}\right ) = 0$ .

One easily checks that the image of the composite inclusion

$$ \begin{align*} {{\mathcal K}_{1, X^o\mid Y^o}}/p \hookrightarrow \Omega^1_{X^o} \hookrightarrow \Omega^1_{X^o}({\operatorname{log}} Y^o)\end{align*} $$

lies in the ${\mathcal O}_{X^o}$ -submodule $\Omega ^1_{X^o\mid Y^o} := \Omega ^1_{X^o}({\operatorname {log}} Y^o)(-Y^o)$ [Reference Jannsen, Saito and Zhao32, Theorem 1.2.1]. Hence, it suffices to show that $H^0_{{\acute {e}t}}\left (X^o, \Omega ^1_{X^o\mid Y^o}\right ) = 0$ .

To show this, we use the exact sequence

(7.5) $$ \begin{align} 0 \to \Omega^1_{X^o}(-Y^o) \to \Omega^1_{X^o\mid Y^o} \xrightarrow{{\textrm{Res}}} {\mathcal O}_{Y^o}(-Y^o) \to 0, \end{align} $$

where ${\textrm {Res}}$ is the Poincaré residue map twisted by ${\mathcal O}_{X^o}(-Y^o)$ . It suffices therefore to show that the left and the right terms of this sequence have no global sections. Since ${\textrm {char}}(k) = p> 0$ , we are finally reduced to showing that

(7.6) $$ \begin{align} H^0_{{\textrm{zar}}}\left(Y^o, {\mathcal O}_{Y^o}(-Y^o)\right) = H^0_{{\textrm{zar}}}\left(X^o, \Omega^1_{{X^o}/k}(-Y^o)\right) = 0, \end{align} $$

where we note that these Zariski cohomologies coincide with the corresponding étale cohomologies.

Now, we first note that ${\mathcal O}_{Y}(Y)$ is very ample on Y. This already implies that $H^0_{{\textrm {zar}}}\left (Y, {\mathcal O}_{Y}(-Y)\right ) = 0$ [Reference Hartshorne29, Exercise III.7.1]. Since Y is normal, we conclude from Corollary 4.7 that $H^0_{{\textrm {zar}}}\left (Y^o, {\mathcal O}_{Y^o}(-Y^o)\right ) = 0$ . On the other hand, since $\Omega ^1_{{X^o}/k}$ is locally free and $m \gg 0$ , it follows from Lemma 5.2 that $H^0_{{\textrm {zar}}}\left (X^o, \Omega ^1_{{X^o}/k}(-Y^o)\right ) = 0$ . This concludes the proof of the lemma.