smoothing-lemma-lift-twice

Let $R \to \Lambda$ be a ring map. Let $I \subset R$ be an ideal. Assume that \begin{enumerate} \item $I^2 = 0$, \item $\Lambda/I\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and \item $R \to \Lambda$ is flat. \end{enumerate} Let $\varphi : B \to \Lambda$ be an $R$-algebra map with $B$ smooth over $R$. Let $J \subset IB$ be a finitely generated ideal such that $\varphi(J) = 0$. Then there exists $R$-algebra maps $$ B \xrightarrow{\alpha} B' \xrightarrow{\beta} \Lambda $$ such that $B'$ is smooth over $R$, such that $\alpha(J) = 0$ and such that $\beta \circ \alpha = \varphi \bmod I\Lambda$.


smoothing-lemma-lifting

Let $R$ be a Noetherian ring. Let $\Lambda$ be an $R$-algebra. Let $\pi \in R$ and assume that $\text{Ann}_R(\pi) = \text{Ann}_R(\pi^2)$ and $\text{Ann}_\Lambda(\pi) = \text{Ann}_\Lambda(\pi^2)$. Suppose we have $R$-algebra maps $R/\pi^2R \to \bar C \to \Lambda/\pi^2\Lambda$ with $\bar C$ of finite presentation. Then there exists an $R$-algebra homomorphism $D \to \Lambda$ and a commutative diagram $$ \xymatrix{ R/\pi^2R \ar[r] \ar[d] & \bar C \ar[r] \ar[d] & \Lambda/\pi^2\Lambda \ar[d] \\ R/\pi R \ar[r] & D/\pi D \ar[r] & \Lambda/\pi \Lambda } $$ with the following properties \begin{enumerate} \item[(a)] $D$ is of finite presentation, \item[(b)] $R \to D$ is smooth at any prime $\mathfrak q$ with $\pi \not \in \mathfrak q$, \item[(c)] $R \to D$ is smooth at any prime $\mathfrak q$ with $\pi \in \mathfrak q$ lying over a prime of $\bar C$ where $R/\pi^2 R \to \bar C$ is smooth, and \item[(d)] $\bar C/\pi \bar C \to D/\pi D$ is smooth at any prime lying over a prime of $\bar C$ where $R/\pi^2R \to \bar C$ is smooth. \end{enumerate}


smoothing-lemma-desingularize

Let $R$ be a Noetherian ring. Let $\Lambda$ be an $R$-algebra. Let $\pi \in R$ and assume that $\text{Ann}_\Lambda(\pi) = \text{Ann}_\Lambda(\pi^2)$. Let $A \to \Lambda$ be an $R$-algebra map with $A$ of finite presentation. Assume \begin{enumerate} \item the image of $\pi$ is strictly standard in $A$ over $R$, and \item there exists a section $\rho : A/\pi^4 A \to R/\pi^4 R$ which is compatible with the map to $\Lambda/\pi^4 \Lambda$. \end{enumerate} Then we can find $R$-algebra maps $A \to B \to \Lambda$ with $B$ of finite presentation such that $\mathfrak a B \subset H_{B/R}$ where $\mathfrak a = \text{Ann}_R(\text{Ann}_R(\pi^2)/\text{Ann}_R(\pi))$.


smoothing-lemma-desingularize-strictly-standard

Let $R$ be a Noetherian ring. Let $\Lambda$ be an $R$-algebra. Let $\pi \in R$ and assume that $\text{Ann}_R(\pi) = \text{Ann}_R(\pi^2)$ and $\text{Ann}_\Lambda(\pi) = \text{Ann}_\Lambda(\pi^2)$. Let $A \to \Lambda$ and $D \to \Lambda$ be $R$-algebra maps with $A$ and $D$ of finite presentation. Assume \begin{enumerate} \item $\pi$ is strictly standard in $A$ over $R$, and \item there exists an $R$-algebra map $A/\pi^4 A \to D/\pi^4 D$ compatible with the maps to $\Lambda/\pi^4 \Lambda$. \end{enumerate} Then we can find an $R$-algebra map $B \to \Lambda$ with $B$ of finite presentation and $R$-algebra maps $A \to B$ and $D \to B$ compatible with the maps to $\Lambda$ such that $H_{D/R}B \subset H_{B/D}$ and $H_{D/R}B \subset H_{B/R}$.


smoothing-lemma-desingularize-lifting-apply

Let $R$ be a Noetherian ring. Let $\Lambda$ be an $R$-algebra. Let $\pi \in R$ and assume that $\text{Ann}_R(\pi) = \text{Ann}_R(\pi^2)$ and $\text{Ann}_\Lambda(\pi) = \text{Ann}_\Lambda(\pi^2)$. Let $A \to \Lambda$ be an $R$-algebra map with $A$ of finite presentation and assume $\pi$ is strictly standard in $A$ over $R$. Let $$ A/\pi^8A \to \bar C \to \Lambda/\pi^8\Lambda $$ be a factorization with $\bar C$ of finite presentation. Then we can find a factorization $A \to B \to \Lambda$ with $B$ of finite presentation such that $R_\pi \to B_\pi$ is smooth and such that $$ H_{\bar C/(R/\pi^8 R)} \cdot \Lambda/\pi^8\Lambda \subset \sqrt{H_{B/R} \Lambda} \bmod \pi^8\Lambda. $$


smoothing-lemma-product

Let $R_i \to \Lambda_i$, $i = 1, 2$ be as in Situation \ref{situation-global}. If PT holds for $R_i \to \Lambda_i$, $i = 1, 2$, then PT holds for $R_1 \times R_2 \to \Lambda_1 \times \Lambda_2$.


smoothing-lemma-delocalize-base

Let $R \to A \to \Lambda$ be ring maps with $A$ of finite presentation over $R$. Let $S \subset R$ be a multiplicative set. Let $S^{-1}A \to B' \to S^{-1}\Lambda$ be a factorization with $B'$ smooth over $S^{-1}R$. Then we can find a factorization $A \to B \to \Lambda$ such that some $s \in S$ maps to an elementary standard element (Definition \ref{definition-strictly-standard}) in $B$ over $R$.


smoothing-lemma-reduce-to-field

\begin{slogan} Proving Popescu approximation reduces to algebras over a field \end{slogan} If for every Situation \ref{situation-global} where $R$ is a field PT holds, then PT holds in general.


smoothing-lemma-lift-solution

Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation \ref{situation-local}. Let $r \geq 1$ and $\pi_1, \ldots, \pi_r \in R$ map to elements of $\mathfrak q$. Assume \begin{enumerate} \item for $i = 1, \ldots, r$ we have $$ \text{Ann}_{R/(\pi_1^8, \ldots, \pi_{i - 1}^8)R}(\pi_i) = \text{Ann}_{R/(\pi_1^8, \ldots, \pi_{i - 1}^8)R}(\pi_i^2) $$ and $$ \text{Ann}_{\Lambda/(\pi_1^8, \ldots, \pi_{i - 1}^8)\Lambda}(\pi_i) = \text{Ann}_{\Lambda/(\pi_1^8, \ldots, \pi_{i - 1}^8)\Lambda}(\pi_i^2) $$ \item for $i = 1, \ldots, r$ the element $\pi_i$ maps to a strictly standard element in $A$ over $R$. \end{enumerate} Then, if $$ R/(\pi_1^8, \ldots, \pi_r^8)R \to A/(\pi_1^8, \ldots, \pi_r^8)A \to \Lambda/(\pi_1^8, \ldots, \pi_r^8)\Lambda \supset \mathfrak q/(\pi_1^8, \ldots, \pi_r^8)\Lambda $$ can be resolved, so can $R \to A \to \Lambda \supset \mathfrak q$.


smoothing-lemma-delocalize-weak

\begin{reference} \cite[Lemma 12.2]{swan} or \cite[Lemma 2]{popescu-GND} \end{reference} Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation \ref{situation-local}. Let $\mathfrak p = R \cap \mathfrak q$. Assume that $\mathfrak q$ is minimal over $\mathfrak h_A$ and that $R_\mathfrak p \to A_\mathfrak p \to \Lambda_\mathfrak q \supset \mathfrak q\Lambda_\mathfrak q$ can be resolved. Then there exists a factorization $A \to C \to \Lambda$ with $C$ of finite presentation such that $H_{C/R} \Lambda \not \subset \mathfrak q$.


smoothing-lemma-delocalize-height-zero

Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation \ref{situation-local}. Let $\mathfrak p = R \cap \mathfrak q$. Assume \begin{enumerate} \item $\mathfrak q$ is minimal over $\mathfrak h_A$, \item $R_\mathfrak p \to A_\mathfrak p \to \Lambda_\mathfrak q \supset \mathfrak q\Lambda_\mathfrak q$ can be resolved, and \item $\dim(\Lambda_\mathfrak q) = 0$. \end{enumerate} Then $R \to A \to \Lambda \supset \mathfrak q$ can be resolved.


smoothing-lemma-ogoma

Let $A$ be a Noetherian ring and let $M$ be a finite $A$-module. Let $S \subset A$ be a multiplicative set. If $\pi \in A$ and $\Ker(\pi : S^{-1}M \to S^{-1}M) = \Ker(\pi^2 : S^{-1}M \to S^{-1}M)$ then there exists an $s \in S$ such that for any $n > 0$ we have $\Ker(s^n\pi : M \to M) = \Ker((s^n\pi)^2 : M \to M)$.


smoothing-lemma-find-sequence

Let $\Lambda$ be a Noetherian ring. Let $I \subset \Lambda$ be an ideal. Let $I \subset \mathfrak q$ be a prime. Let $n, e$ be positive integers Assume that $\mathfrak q^n\Lambda_\mathfrak q \subset I\Lambda_\mathfrak q$ and that $\Lambda_\mathfrak q$ is a regular local ring of dimension $d$. Then there exists an $n > 0$ and $\pi_1, \ldots, \pi_d \in \Lambda$ such that \begin{enumerate} \item $(\pi_1, \ldots, \pi_d)\Lambda_\mathfrak q = \mathfrak q\Lambda_\mathfrak q$, \item $\pi_1^n, \ldots, \pi_d^n \in I$, and \item for $i = 1, \ldots, d$ we have $$ \text{Ann}_{\Lambda/(\pi_1^e, \ldots, \pi_{i - 1}^e)\Lambda}(\pi_i) = \text{Ann}_{\Lambda/(\pi_1^e, \ldots, \pi_{i - 1}^e)\Lambda}(\pi_i^2). $$ \end{enumerate}


smoothing-lemma-resolve-special

Let $k \to A \to \Lambda \supset \mathfrak q$ be as in Situation \ref{situation-local} where \begin{enumerate} \item $k$ is a field, \item $\Lambda$ is Noetherian, \item $\mathfrak q$ is minimal over $\mathfrak h_A$, \item $\Lambda_\mathfrak q$ is a regular local ring, and \item the field extension $k \subset \kappa(\mathfrak q)$ is separable. \end{enumerate} Then $k \to A \to \Lambda \supset \mathfrak q$ can be resolved.


smoothing-lemma-helper

Let $k$ be a field of characteristic $p > 0$. Let $(\Lambda, \mathfrak m, K)$ be an Artinian local $k$-algebra. Assume that $\dim H_1(L_{K/k}) < \infty$. Then $\Lambda$ is a filtered colimit of Artinian local $k$-algebras $A$ with each map $A \to \Lambda$ flat, with $\mathfrak m_A \Lambda = \mathfrak m$, and with $A$ essentially of finite type over $k$.


smoothing-lemma-solution-modulo

Let $k$ be a field of characteristic $p > 0$. Let $\Lambda$ be a Noetherian geometrically regular $k$-algebra. Let $\mathfrak q \subset \Lambda$ be a prime ideal. Let $n \geq 1$ be an integer and let $E \subset \Lambda_\mathfrak q/\mathfrak q^n\Lambda_\mathfrak q$ be a finite subset. Then we can find $m \geq 0$ and $\varphi : k[y_1, \ldots, y_m] \to \Lambda$ with the following properties \begin{enumerate} \item setting $\mathfrak p = \varphi^{-1}(\mathfrak q)$ we have $\mathfrak q\Lambda_\mathfrak q = \mathfrak p \Lambda_\mathfrak q$ and $k[y_1, \ldots, y_m]_\mathfrak p \to \Lambda_\mathfrak q$ is flat, \item there is a factorization by homomorphisms of local Artinian rings $$ k[y_1, \ldots, y_m]_\mathfrak p/\mathfrak p^n k[y_1, \ldots, y_m]_\mathfrak p \to D \to \Lambda_\mathfrak q/\mathfrak q^n\Lambda_\mathfrak q $$ where the first arrow is essentially smooth and the second is flat, \item $E$ is contained in $D$ modulo $\mathfrak q^n\Lambda_\mathfrak q$. \end{enumerate}


smoothing-lemma-enlarge-solution-modulo

Let $\varphi : k[y_1, \ldots, y_m] \to \Lambda$, $n$, $\mathfrak q$, $\mathfrak p$ and $$ k[y_1, \ldots, y_m]_\mathfrak p/\mathfrak p^n \to D \to \Lambda_\mathfrak q/\mathfrak q^n \Lambda_\mathfrak q $$ be as in Lemma \ref{lemma-solution-modulo}. Then for any $\lambda \in \Lambda \setminus \mathfrak q$ there exists an integer $q > 0$ and a factorization $$ k[y_1, \ldots, y_m]_\mathfrak p/\mathfrak p^n \to D \to D' \to \Lambda_\mathfrak q/\mathfrak q^n \Lambda_\mathfrak q $$ such that $D \to D'$ is an essentially smooth map of local Artinian rings, the last arrow is flat, and $\lambda^q$ is in $D'$.


smoothing-lemma-resolve-general

Let $k \to A \to \Lambda \supset \mathfrak q$ be as in Situation \ref{situation-local} where \begin{enumerate} \item $k$ is a field of characteristic $p > 0$, \item $\Lambda$ is Noetherian and geometrically regular over $k$, \item $\mathfrak q$ is minimal over $\mathfrak h_A$. \end{enumerate} Then $k \to A \to \Lambda \supset \mathfrak q$ can be resolved.


smoothing-lemma-approximation-property-variant

Let $R$ be a Noetherian ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $f_1, \ldots, f_m \in R[x_1, \ldots, x_n]$. Suppose that $(a_1, \ldots, a_n) \in ((R_\mathfrak p)^\wedge)^n$ is a solution. If $R_\mathfrak p$ is a G-ring, then for every integer $N$ there exist \begin{enumerate} \item an \'etale ring map $R \to R'$, \item a prime ideal $\mathfrak p' \subset R'$ lying over $\mathfrak p$ \item a solution $(b_1, \ldots, b_n) \in (R')^n$ in $R'$ \end{enumerate} such that $\kappa(\mathfrak p) = \kappa(\mathfrak p')$ and $a_i - b_i \in (\mathfrak p')^N(R'_{\mathfrak p'})^\wedge$.


smoothing-lemma-henselian-pair

\begin{slogan} Approximation for henselian pairs. \end{slogan} Let $(A, I)$ be a henselian pair with $A$ Noetherian. Let $A^\wedge$ be the $I$-adic completion of $A$. Assume at least one of the following conditions holds \begin{enumerate} \item $A \to A^\wedge$ is a regular ring map, \item $A$ is a Noetherian G-ring, or \item $(A, I)$ is the henselization (More on Algebra, Lemma \ref{more-algebra-lemma-henselization}) of a pair $(B, J)$ where $B$ is a Noetherian G-ring. \end{enumerate} Given $f_1, \ldots, f_m \in A[x_1, \ldots, x_n]$ and $\hat{a}_1, \ldots, \hat{a}_n \in A^\wedge$ such that $f_j(\hat{a}_1, \ldots, \hat{a}_n) = 0$ for $j = 1, \ldots, m$, for every $N \geq 1$ there exist $a_1, \ldots, a_n \in A$ such that $\hat{a}_i - a_i \in I^N$ and such that $f_j(a_1, \ldots, a_n) = 0$ for $j = 1, \ldots, m$.


smoothing-proposition-lift-smooth

\begin{slogan} Smooth and syntomic algebras lift along surjections \end{slogan} Let $R \to R_0$ be a surjective ring map with kernel $I$. \begin{enumerate} \item If $R_0 \to A_0$ is a syntomic ring map, then there exists a syntomic ring map $R \to A$ such that $A/IA \cong A_0$. \item If $R_0 \to A_0$ is a smooth ring map, then there exists a smooth ring map $R \to A$ such that $A/IA \cong A_0$. \end{enumerate}


smoothing-proposition-lift

\begin{slogan} Ind-smoothness of an algebra is stable under infinitesimal deformations \end{slogan} Let $R \to \Lambda$ be a ring map. Let $I \subset R$ be an ideal. Assume that \begin{enumerate} \item $I$ is nilpotent, \item $\Lambda/I\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and \item $R \to \Lambda$ is flat. \end{enumerate} Then $\Lambda$ is a filtered colimit of smooth $R$-algebras.


chow-lemma-additivity-periodic-length

Let $R$ be a ring. Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3). $$


chow-lemma-finite-periodic-length

Let $R$ be a ring. If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length, then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$. In particular, if $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length, then $e_R(M, \varphi, \psi) = 0$.


chow-lemma-compare-periodic-lengths

Let $R$ be a ring. Let $f : (M, \varphi, \psi) \to (M', \varphi', \psi')$ be a map of $(2, 1)$-periodic complexes whose cohomology modules have finite length. If $\Ker(f)$ and $\Coker(f)$ have finite length, then $e_R(M, \varphi, \psi) = e_R(M', \varphi', \psi')$.


chow-lemma-length-multiplication

Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Let $x \in R$. Assume that \begin{enumerate} \item $\dim(\text{Supp}(M)) \leq 1$, and \item $\dim(\text{Supp}(M/xM)) \leq 0$. \end{enumerate} Write $\text{Supp}(M) = \{\mathfrak m, \mathfrak q_1, \ldots, \mathfrak q_t\}$. Then $$ e_R(M, 0, x) = \sum\nolimits_{i = 1, \ldots, t} \text{ord}_{R/\mathfrak q_i}(x) \text{length}_{R_{\mathfrak q_i}}(M_{\mathfrak q_i}). $$


chow-lemma-additivity-divisors-restricted

Let $R$ be a Noetherian local ring. Let $x \in R$. If $M$ is a finite Cohen-Macaulay module over $R$ with $\dim(\text{Supp}(M)) = 1$ and $\dim(\text{Supp}(M/xM)) = 0$, then $$ \text{length}_R(M/xM) = \sum\nolimits_i \text{length}_R(R/(x, \mathfrak q_i)) \text{length}_{R_{\mathfrak q_i}}(M_{\mathfrak q_i}). $$ where $\mathfrak q_1, \ldots, \mathfrak q_t$ are the minimal primes of the support of $M$. If $I \subset R$ is an ideal such that $x$ is a nonzerodivisor on $R/I$ and $\dim(R/I) = 1$, then $$ \text{length}_R(R/(x, I)) = \sum\nolimits_i \text{length}_R(R/(x, \mathfrak q_i)) \text{length}_{R_{\mathfrak q_i}}((R/I)_{\mathfrak q_i}) $$ where $\mathfrak q_1, \ldots, \mathfrak q_n$ are the minimal primes over $I$.


chow-lemma-powers-period-length-zero

Let $R$ be a ring. Let $M$ be an $R$-module. Let $\varphi : M \to M$ be an endomorphism and $n > 0$ such that $\varphi^n = 0$ and such that $\Ker(\varphi)/\Im(\varphi^{n - 1})$ has finite length as an $R$-module. Then $$ e_R(M, \varphi^i, \varphi^{n - i}) = 0 $$ for $i = 0, \ldots, n$.


chow-lemma-multiply-period-length

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $(M, \varphi, \psi)$ be a $(2, 1)$-periodic complex over $R$ with $M$ finite and with cohomology groups of finite length over $R$. Let $x \in R$ be such that $\dim(\text{Supp}(M/xM)) \leq 0$. Then $$ e_R(M, x\varphi, \psi) = e_R(M, \varphi, \psi) - e_R(\Im(\varphi), 0, x) $$ and $$ e_R(M, \varphi, x\psi) = e_R(M, \varphi, \psi) + e_R(\Im(\psi), 0, x) $$


chow-lemma-glue-at-max

Let $A$ be a Noetherian ring. Let $\mathfrak m_1, \ldots, \mathfrak m_r$ be pairwise distinct maximal ideals of $A$. For $i = 1, \ldots, r$ let $\varphi_i : A_{\mathfrak m_i} \to B_i$ be a ring map whose kernel and cokernel are annihilated by a power of $\mathfrak m_i$. Then there exists a ring map $\varphi : A \to B$ such that \begin{enumerate} \item the localization of $\varphi$ at $\mathfrak m_i$ is isomorphic to $\varphi_i$, and \item $\Ker(\varphi)$ and $\Coker(\varphi)$ are annihilated by a power of $\mathfrak m_1 \cap \ldots \cap \mathfrak m_r$. \end{enumerate} Moreover, if each $\varphi_i$ is finite, injective, or surjective then so is $\varphi$.


chow-lemma-Noetherian-domain-dim-1-two-elements

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $a, b \in R$ be nonzerodivisors. There exists a finite ring extension $R \subset R'$ with $R'/R$ annihilated by a power of $\mathfrak m$ and nonzerodivisors $t, a', b' \in R'$ such that $a = ta'$ and $b = tb'$ and $R' = a'R' + b'R'$.


chow-lemma-not-infinitely-divisible

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $a, b \in R$ be nonzerodivisors with $a \in \mathfrak m$. There exists an integer $n = n(R, a, b)$ such that for a finite ring extension $R \subset R'$ if $b = a^m c$ for some $c \in R'$, then $m \leq n$.


chow-lemma-prepare-tame-symbol

Let $(A, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $r \geq 2$ and let $a_1, \ldots, a_r \in A$ be nonzerodivisors not all units. Then there exist \begin{enumerate} \item a finite ring extension $A \subset B$ with $B/A$ annihilated by a power of $\mathfrak m$, \item for each of maximal ideal $\mathfrak m_j \subset B$ a nonzerodivisor $\pi_j \in B_j = B_{\mathfrak m_j}$, and \item factorizations $a_i = u_{i, j} \pi_j^{e_{i, j}}$ in $B_j$ with $u_{i, j} \in B_j$ units and $e_{i, j} \geq 0$. \end{enumerate}


chow-lemma-norm-down-tame-symbol

Let $(A, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $A \subset B$ be a finite ring extension with $B/A$ annihilated by a power of $\mathfrak m$ and $\mathfrak m$ not an associated prime of $B$. For $a, b \in A$ nonzerodivisors we have $$ \partial_A(a, b) = \prod \text{Norm}_{\kappa(\mathfrak m_j)/\kappa(\mathfrak m)}(\partial_{B_j}(a, b)) $$ where the product is over the maximal ideals $\mathfrak m_j$ of $B$ and $B_j = B_{\mathfrak m_j}$.


chow-lemma-tame-symbol-formally-smooth

Let $(A, \mathfrak m, \kappa) \to (A', \mathfrak m', \kappa')$ be a local homomorphism of Noetherian local rings of dimension $1$. If $A \to A'$ is flat, $\mathfrak m' = \mathfrak m A'$, and $\kappa'/\kappa$ is separable, then for $a_1, a_2 \in A$ nonzerodivisors the tame symbol $\partial_A(a_1, a_2)$ maps to $\partial_{A'}(a_1, a_2)$.


chow-lemma-perpare-key

Let $(A, \mathfrak m)$ be a $2$-dimensional Noetherian local ring. Let $t \in \mathfrak m$ be a nonzerodivisor. Say $V(t) = \{\mathfrak m, \mathfrak q_1, \ldots, \mathfrak q_r\}$. Let $A_{\mathfrak q_i} \subset B_i$ be a finite ring extension with $B_i/A_{\mathfrak q_i}$ annihilated by a power of $t$. Then there exists a finite extension $A \subset B$ of local rings identifying residue fields with $B_i \cong B_{\mathfrak q_i}$ and $B/A$ annihilated by a power of $t$.


chow-lemma-key-nonzerodivisors

Let $(A, \mathfrak m)$ be a $2$-dimensional Noetherian local ring. Let $a, b \in A$ be nonzerodivisors. Then we have $$ \sum \text{ord}_{A/\mathfrak q}(\partial_{A_{\mathfrak q}}(a, b)) = 0 $$ where the sum is over the height $1$ primes $\mathfrak q$ of $A$.


chow-lemma-milnor-gersten-low-degree

\begin{reference} When $A$ is an excellent ring this is \cite[Proposition 1]{Kato-Milnor-K}. \end{reference} Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$. Let $f, g \in K^*$. Let $\mathfrak q_1, \ldots, \mathfrak q_t$ be the height $1$ primes $\mathfrak q$ of $A$ such that either $f$ or $g$ is not an element of $A^*_{\mathfrak q}$. Then we have $$ \sum\nolimits_{i = 1, \ldots, t} \text{ord}_{A/\mathfrak q_i}(\partial_{A_{\mathfrak q_i}}(f, g)) = 0 $$ We can also write this as $$ \sum\nolimits_{\text{height}(\mathfrak q) = 1} \text{ord}_{A/\mathfrak q}(\partial_{A_{\mathfrak q}}(f, g)) = 0 $$ since at any height $1$ prime $\mathfrak q$ of $A$ where $f, g \in A^*_{\mathfrak q}$ we have $\partial_{A_{\mathfrak q}}(f, g) = 1$.


chow-lemma-multiplicity-finite

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. \begin{enumerate} \item Let $Z' \subset Z$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then $$ \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{O}_{Z, \xi} < \infty $$ \item If $\dim_\delta(Z) \leq k$ and $\xi \in Z$ with $\delta(\xi) = k$, then $\xi$ is a generic point of an irreducible component of $Z$. \end{enumerate}


chow-lemma-length-finite

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item The collection of irreducible components of the support of $\mathcal{F}$ is locally finite. \item Let $Z' \subset \text{Supp}(\mathcal{F})$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then $$ \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{F}_\xi < \infty $$ \item If $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$ and $\xi \in Z$ with $\delta(\xi) = k$, then $\xi$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F})$. \end{enumerate}


chow-lemma-cycle-closed-coherent

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. If $\dim_\delta(Z) \leq k$, then $[Z]_k = [{\mathcal O}_Z]_k$.


chow-lemma-additivity-sheaf-cycle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent sheaves on $X$. Assume that the $\delta$-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ is $\leq k$. Then $[\mathcal{G}]_k = [\mathcal{F}]_k + [\mathcal{H}]_k$.


chow-lemma-equal-dimension

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $X$, $Y$ integral and $\dim_\delta(X) = \dim_\delta(Y)$. Then either $f(X)$ is contained in a proper closed subscheme of $Y$, or $f$ is dominant and the extension of function fields $R(Y) \subset R(X)$ is finite.


chow-lemma-quasi-compact-locally-finite

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is quasi-compact, and $\{Z_i\}_{i \in I}$ is a locally finite collection of closed subsets of $X$. Then $\{\overline{f(Z_i)}\}_{i \in I}$ is a locally finite collection of closed subsets of $Y$.


chow-lemma-compose-pushforward

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$, and $Z$ be locally of finite type over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms. Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_k(X) \to Z_k(Z)$.


chow-lemma-exact-sequence-closed

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \subset X$ be closed subschemes such that $X = X_1 \cup X_2$ set theoretically. For every $k \in \mathbf{Z}$ the sequence of abelian groups $$ \xymatrix{ Z_k(X_1 \cap X_2) \ar[r] & Z_k(X_1) \oplus Z_k(X_2) \ar[r] & Z_k(X) \ar[r] & 0 } $$ is exact. Here $X_1 \cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$.


chow-lemma-cycle-push-sheaf

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a proper morphism of schemes which are locally of finite type over $S$. \begin{enumerate} \item Let $Z \subset X$ be a closed subscheme with $\dim_\delta(Z) \leq k$. Then $$ f_*[Z]_k = [f_*{\mathcal O}_Z]_k. $$ \item Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then $$ f_*[\mathcal{F}]_k = [f_*{\mathcal F}]_k. $$ \end{enumerate} Note that the statement makes sense since $f_*\mathcal{F}$ and $f_*\mathcal{O}_Z$ are coherent $\mathcal{O}_Y$-modules by Cohomology of Schemes, Proposition \ref{coherent-proposition-proper-pushforward-coherent}.


chow-lemma-flat-inverse-image-dimension

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. For any closed subset $Z \subset Y$ we have $$ \dim_\delta(f^{-1}(Z)) = \dim_\delta(Z) + r. $$ provided $f^{-1}(Z)$ is nonempty. If $Z$ is irreducible and $Z' \subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\dim_\delta(Z') = \dim_\delta(Z) + r$.


chow-lemma-inverse-image-locally-finite

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $\{Z_i\}_{i \in I}$ is a locally finite collection of closed subsets of $Y$. Then $\{f^{-1}(Z_i)\}_{i \in I}$ is a locally finite collection of closed subsets of $X$.


chow-lemma-exact-sequence-open

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $U \subset X$ be an open subscheme, and denote $i : Y = X \setminus U \to X$ as a reduced closed subscheme of $X$. For every $k \in \mathbf{Z}$ the sequence $$ \xymatrix{ Z_k(Y) \ar[r]^{i_*} & Z_k(X) \ar[r]^{j^*} & Z_k(U) \ar[r] & 0 } $$ is an exact complex of abelian groups.


chow-lemma-compose-flat-pullback

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X, Y, Z$ be locally of finite type over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of relative dimensions $r$ and $s$. Then $g \circ f$ is flat of relative dimension $r + s$ and $$ f^* \circ g^* = (g \circ f)^* $$ as maps $Z_k(Z) \to Z_{k + r + s}(X)$.


chow-lemma-pullback-coherent

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X, Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. \begin{enumerate} \item Let $Z \subset Y$ be a closed subscheme with $\dim_\delta(Z) \leq k$. Then we have $\dim_\delta(f^{-1}(Z)) \leq k + r$ and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$. \item Let $\mathcal{F}$ be a coherent sheaf on $Y$ with $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then we have $\dim_\delta(\text{Supp}(f^*\mathcal{F})) \leq k + r$ and $$ f^*[{\mathcal F}]_k = [f^*{\mathcal F}]_{k+r} $$ in $Z_{k + r}(X)$. \end{enumerate}


chow-lemma-flat-pullback-proper-pushforward

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a fibre product diagram of schemes locally of finite type over $S$. Assume $f : X \to Y$ proper and $g : Y' \to Y$ flat of relative dimension $r$. Then also $f'$ is proper and $g'$ is flat of relative dimension $r$. For any $k$-cycle $\alpha$ on $X$ we have $$ g^*f_*\alpha = f'_*(g')^*\alpha $$ in $Z_{k + r}(Y')$.


chow-lemma-finite-flat

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a finite locally free morphism of degree $d$ (see Morphisms, Definition \ref{morphisms-definition-finite-locally-free}). Then $f$ is both proper and flat of relative dimension $0$, and $$ f_*f^*\alpha = d\alpha $$ for every $\alpha \in Z_k(Y)$.


chow-lemma-divisor-delta-dimension

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral. \begin{enumerate} \item If $Z \subset X$ is an integral closed subscheme, then the following are equivalent: \begin{enumerate} \item $Z$ is a prime divisor, \item $Z$ has codimension $1$ in $X$, and \item $\dim_\delta(Z) = \dim_\delta(X) - 1$. \end{enumerate} \item If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\dim_\delta(Z) = \dim_\delta(X) - 1$. \end{enumerate}


chow-lemma-finite-in-codimension-one

Let $f : X \to Y$ be a morphism of schemes. Let $\xi \in Y$ be a point. Assume that \begin{enumerate} \item $X$, $Y$ are integral, \item $Y$ is locally Noetherian \item $f$ is proper, dominant and $R(Y) \subset R(X)$ is finite, and \item $\dim(\mathcal{O}_{Y, \xi}) = 1$. \end{enumerate} Then there exists an open neighbourhood $V \subset Y$ of $\xi$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.


chow-lemma-flat-pullback-principal-divisor

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \dim_\delta(Y)$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $g \in R(Y)^*$. Then $$ f^*(\text{div}_Y(g)) = \text{div}_X(g) $$ in $Z_{n + r - 1}(X)$.


chow-lemma-proper-pushforward-alteration

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \dim_\delta(X) = \dim_\delta(Y)$. Let $p : X \to Y$ be a dominant proper morphism. Let $f \in R(X)^*$. Set $$ g = \text{Nm}_{R(X)/R(Y)}(f). $$ Then we have $p_*\text{div}(f) = \text{div}(g)$.


chow-lemma-rational-function

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim_\delta(X)$. Let $f \in R(X)^*$. Let $U \subset X$ be a nonempty open such that $f$ corresponds to a section $f \in \Gamma(U, \mathcal{O}_X^*)$. Let $Y \subset X \times_S \mathbf{P}^1_S$ be the closure of the graph of $f : U \to \mathbf{P}^1_S$. Then \begin{enumerate} \item the projection morphism $p : Y \to X$ is proper, \item $p|_{p^{-1}(U)} : p^{-1}(U) \to U$ is an isomorphism, \item the pullbacks $Y_0 = q^{-1}D_0$ and $Y_\infty = q^{-1}D_\infty$ via the morphism $q : Y \to \mathbf{P}^1_S$ are defined (Divisors, Definition \ref{divisors-definition-pullback-effective-Cartier-divisor}), \item we have $$ \text{div}_Y(f) = [Y_0]_{n - 1} - [Y_\infty]_{n - 1} $$ \item we have $$ \text{div}_X(f) = p_*\text{div}_Y(f) $$ \item if we view $Y_0$ and $Y_\infty$ as closed subschemes of $X$ via the morphism $p$ then we have $$ \text{div}_X(f) = [Y_0]_{n - 1} - [Y_\infty]_{n - 1} $$ \end{enumerate}


chow-lemma-curve-principal-divisor

Let $K$ be any field. Let $X$ be a $1$-dimensional integral scheme endowed with a proper morphism $c : X \to \Spec(K)$. Let $f \in K(X)^*$ be an invertible rational function. Then $$ \sum\nolimits_{x \in X \text{ closed}} [\kappa(x) : K] \text{ord}_{\mathcal{O}_{X, x}}(f) = 0 $$ where $\text{ord}$ is as in Algebra, Definition \ref{algebra-definition-ord}. In other words, $c_*\text{div}(f) = 0$.


chow-lemma-restrict-to-open

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $U \subset X$ be an open subscheme, and denote $i : Y = X \setminus U \to X$ as a reduced closed subscheme of $X$. Let $k \in \mathbf{Z}$. Suppose $\alpha, \beta \in Z_k(X)$. If $\alpha|_U \sim_{rat} \beta|_U$ then there exist a cycle $\gamma \in Z_k(Y)$ such that $$ \alpha \sim_{rat} \beta + i_*\gamma. $$ In other words, the sequence $$ \xymatrix{ \CH_k(Y) \ar[r]^{i_*} & \CH_k(X) \ar[r]^{j^*} & \CH_k(U) \ar[r] & 0 } $$ is an exact complex of abelian groups.


chow-lemma-exact-sequence-closed-chow

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \subset X$ be closed subschemes such that $X = X_1 \cup X_2$ set theoretically. For every $k \in \mathbf{Z}$ the sequence of abelian groups $$ \xymatrix{ \CH_k(X_1 \cap X_2) \ar[r] & \CH_k(X_1) \oplus \CH_k(X_2) \ar[r] & \CH_k(X) \ar[r] & 0 } $$ is exact. Here $X_1 \cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$.


chow-lemma-prepare-flat-pullback-rational-equivalence

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Assume $Y$ integral with $\dim_\delta(Y) = k$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Then for $g \in R(Y)^*$ we have $$ f^*\text{div}_Y(g) = \sum n_j i_{j, *}\text{div}_{X_j}(g \circ f|_{X_j}) $$ as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible components $X_j$ of $X$ and $n_j$ is the multiplicity of $X_j$ in $X$.


chow-lemma-flat-pullback-rational-equivalence

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha \sim_{rat} \beta$ be rationally equivalent $k$-cycles on $Y$. Then $f^*\alpha \sim_{rat} f^*\beta$ as $(k + r)$-cycles on $X$.


chow-lemma-proper-pushforward-rational-equivalence

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $p : X \to Y$ be a proper morphism. Suppose $\alpha, \beta \in Z_k(X)$ are rationally equivalent. Then $p_*\alpha$ is rationally equivalent to $p_*\beta$.


chow-lemma-rational-equivalence-family

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $W \subset X \times_S \mathbf{P}^1_S$ be an integral closed subscheme of $\delta$-dimension $k + 1$. Assume $W \not = W_0$, and $W \not = W_\infty$. Then \begin{enumerate} \item $W_0$, $W_\infty$ are effective Cartier divisors of $W$, \item $W_0$, $W_\infty$ can be viewed as closed subschemes of $X$ and $$ [W_0]_k \sim_{rat} [W_\infty]_k, $$ \item for any locally finite family of integral closed subschemes $W_i \subset X \times_S \mathbf{P}^1_S$ of $\delta$-dimension $k + 1$ with $W_i \not = (W_i)_0$ and $W_i \not = (W_i)_\infty$ we have $\sum ([(W_i)_0]_k - [(W_i)_\infty]_k) \sim_{rat} 0$ on $X$, and \item for any $\alpha \in Z_k(X)$ with $\alpha \sim_{rat} 0$ there exists a locally finite family of integral closed subschemes $W_i \subset X \times_S \mathbf{P}^1_S$ as above such that $\alpha = \sum ([(W_i)_0]_k - [(W_i)_\infty]_k)$. \end{enumerate}


chow-lemma-closed-subscheme-cross-p1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z$ be a closed subscheme of $X \times \mathbf{P}^1$. Assume \begin{enumerate} \item $\dim_\delta(Z) \leq k + 1$, \item $\dim_\delta(Z_0) \leq k$, $\dim_\delta(Z_\infty) \leq k$, and \item for any embedded point $\xi$ (Divisors, Definition \ref{divisors-definition-embedded}) of $Z$ either $\xi \not \in Z_0 \cup Z_\infty$ or $\delta(\xi) < k$. \end{enumerate} Then $[Z_0]_k \sim_{rat} [Z_\infty]_k$ as $k$-cycles on $X$.


chow-lemma-cycles-k-group

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The maps $$ Z_k(X) \longrightarrow K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)), \quad \sum n_Z[Z] \mapsto \left[\bigoplus\nolimits_{n_Z > 0} \mathcal{O}_Z^{\oplus n_Z}\right] - \left[\bigoplus\nolimits_{n_Z < 0} \mathcal{O}_Z^{\oplus -n_Z}\right] $$ and $$ K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow Z_k(X),\quad \mathcal{F} \longmapsto [\mathcal{F}]_k $$ are mutually inverse isomorphisms.


chow-lemma-finite-cycles-k-group

Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. Then $\pi_* : \textit{Coh}(X) \to \textit{Coh}(Y)$ is an exact functor which sends $\textit{Coh}_{\leq k}(X)$ into $\textit{Coh}_{\leq k}(Y)$ and induces homomorphisms on $K_0$ of these categories and their quotients. The maps of Lemma \ref{lemma-cycles-k-group} fit into a commutative diagram $$ \xymatrix{ Z_k(X) \ar[d]^{\pi_*} \ar[r] & K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \ar[d]^{\pi_*} \ar[r] & Z_k(X) \ar[d]^{\pi_*} \\ Z_k(Y) \ar[r] & K_0(\textit{Coh}_{\leq k}(Y)/\textit{Coh}_{\leq k - 1}(Y)) \ar[r] & Z_k(Y) } $$


chow-lemma-from-chow-to-K

Let $X$ be a scheme locally of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. There is a canonical map $$ \CH_k(X) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) $$ induced by the map $Z_k(X) \to K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$ from Lemma \ref{lemma-cycles-k-group}.


chow-lemma-K-coherent-supported-on-closed

Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Denote $\textit{Coh}_Z(X) \subset \textit{Coh}(X)$ the Serre subcategory of coherent $\mathcal{O}_X$-modules whose set theoretic support is contained in $Z$. Then the exact inclusion functor $\textit{Coh}(Z) \to \textit{Coh}_Z(X)$ induces an isomorphism $$ K'_0(Z) = K_0(\textit{Coh}(Z)) \longrightarrow K_0(\textit{Coh}_Z(X)) $$


chow-lemma-compute-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim_\delta(X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a nonzero global section. Then $$ \text{div}_\mathcal{L}(s) = [Z(s)]_{n - 1} $$ in $Z_{n - 1}(X)$ and $$ c_1(\mathcal{L}) \cap [X] = [Z(s)]_{n - 1} $$ in $\CH_{n - 1}(X)$.


chow-lemma-c1-cap-additive

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be an invertible sheaves on $X$. Then $$ c_1(\mathcal{L}) \cap \alpha  + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}) \cap \alpha $$ in $\CH_k(X)$ for every $\alpha \in Z_{k + 1}(X)$. Moreover, $c_1(\mathcal{O}_X) \cap \alpha = 0$ for all $\alpha$.


chow-lemma-prepare-geometric-cap

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $Y$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module. Let $s \in \Gamma(Y, \mathcal{L})$. Assume \begin{enumerate} \item $\dim_\delta(Y) \leq k + 1$, \item $\dim_\delta(Z(s)) \leq k$, and \item for every generic point $\xi$ of an irreducible component of $Z(s)$ of $\delta$-dimension $k$ the multiplication by $s$ induces an injection $\mathcal{O}_{Y, \xi} \to \mathcal{L}_\xi$. \end{enumerate} Write $[Y]_{k + 1} = \sum n_i[Y_i]$ where $Y_i \subset Y$ are the irreducible components of $Y$ of $\delta$-dimension $k + 1$. Set $s_i = s|_{Y_i} \in \Gamma(Y_i, \mathcal{L}|_{Y_i})$. Then \begin{equation} \label{equation-equal-as-cycles} [Z(s)]_k =  \sum n_i[Z(s_i)]_k \end{equation} as $k$-cycles on $Y$.


chow-lemma-geometric-cap

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y \subset X$ be a closed subscheme. Let $s \in \Gamma(Y, \mathcal{L}|_Y)$. Assume \begin{enumerate} \item $\dim_\delta(Y) \leq k + 1$, \item $\dim_\delta(Z(s)) \leq k$, and \item for every generic point $\xi$ of an irreducible component of $Z(s)$ of $\delta$-dimension $k$ the multiplication by $s$ induces an injection $\mathcal{O}_{Y, \xi} \to (\mathcal{L}|_Y)_\xi$\footnote{For example, this holds if $s$ is a regular section of $\mathcal{L}|_Y$.}. \end{enumerate} Then $$ c_1(\mathcal{L}) \cap [Y]_{k + 1} = [Z(s)]_k $$ in $\CH_k(X)$.


chow-lemma-prepare-flat-pullback-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \dim_\delta(Y)$. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Then we have $$ f^*\text{div}_\mathcal{L}(s) = \sum n_i\text{div}_{f^*\mathcal{L}|_{X_i}}(s_i) $$ in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_i \subset X$ of $\delta$-dimension $n + r$, the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and $n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$.


chow-lemma-flat-pullback-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_1(\mathcal{L}) \cap \alpha) = c_1(f^*\mathcal{L}) \cap f^*\alpha $$ in $\CH_{k + r - 1}(X)$.


chow-lemma-equal-c1-as-cycles

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a proper morphism. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $s$ be a nonzero meromorphic section $s$ of $\mathcal{L}$ on $Y$. Assume $X$, $Y$ integral, $f$ dominant, and $\dim_\delta(X) = \dim_\delta(Y)$. Then $$ f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) = [R(X) : R(Y)]\text{div}_\mathcal{L}(s). $$ as cycles on $Y$. In particular $$ f_*(c_1(f^*\mathcal{L}) \cap [X]) = c_1(\mathcal{L}) \cap [Y]. $$


chow-lemma-pushforward-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \to Y$ be a proper morphism. Let $\alpha \in Z_{k + 1}(X)$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ p_*(c_1(p^*\mathcal{L}) \cap \alpha) = c_1(\mathcal{L}) \cap p_*\alpha $$ in $\CH_k(Y)$.


chow-lemma-key-formula

In the situation above the cycle $$ \sum (Z_i \to X)_*\left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \right) $$ is equal to the cycle $$ \sum (Z_i \to X)_*\text{div}(\partial_{B_i}(f_i, g_i)) $$


chow-lemma-commutativity-on-integral

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}$ and a nonzero meromorphic section $t$ of $\mathcal{N}$. Set $\alpha = \text{div}_\mathcal{L}(s)$ and $\beta = \text{div}_\mathcal{N}(t)$. Then $$ c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L}) \cap \beta $$ in $\CH_{n - 2}(X)$.


chow-lemma-factors

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be invertible on $X$. The operation $\alpha \mapsto c_1(\mathcal{L}) \cap \alpha$ factors through rational equivalence to give an operation $$ c_1(\mathcal{L}) \cap - : \CH_{k + 1}(X) \to \CH_k(X) $$


chow-lemma-cap-commutative

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. For any $\alpha \in \CH_{k + 2}(X)$ we have $$ c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha $$ as elements of $\CH_k(X)$.


chow-lemma-support-cap-effective-Cartier

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Let $\alpha$ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\alpha = c_1(\mathcal{L}) \cap \alpha$ in $\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \cdot \alpha = c_1(\mathcal{O}_X(D)) \cap \alpha$.


chow-lemma-easy-gysin

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. \begin{enumerate} \item Let $Z \subset X$ be a closed subscheme such that $\dim_\delta(Z) \leq k + 1$ and such that $D \cap Z$ is an effective Cartier divisor on $Z$. Then $i^*[Z]_{k + 1} = [D \cap Z]_k$. \item Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k + 1$ and $s : \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}$ is injective. Then $$ i^*[\mathcal{F}]_{k + 1} = [i^*\mathcal{F}]_k $$ in $\CH_k(D)$. \end{enumerate}


chow-lemma-closed-in-X-gysin

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X' \to X$ be a proper morphism of schemes locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Form the diagram $$ \xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X } $$ as in Remark \ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\alpha'$ on $X'$ we have $i^*f_*\alpha' = g_*(i')^*\alpha'$ in $\CH_k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).


chow-lemma-gysin-flat-pullback

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of schemes locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Form the diagram $$ \xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X } $$ as in Remark \ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\alpha$ on $X$ we have $(i')^*f^*\alpha = g^*i^*\alpha'$ in $\CH_{k + r}(D)$ (this makes sense as $f^*$ is defined on the level of cycles).


chow-lemma-gysin-factors-general

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \dim_\delta(X)$. Let $i : D \to X$ be an effective Cartier divisor. Let $\mathcal{N}$ be an invertible $\mathcal{O}_X$-module and let $t$ be a nonzero meromorphic section of $\mathcal{N}$. Then $i^*\text{div}_\mathcal{N}(t) = c_1(\mathcal{N}) \cap [D]_{n - 1}$ in $\CH_{n - 2}(D)$.


chow-lemma-gysin-factors

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \CH_{k + 1}(X) \to \CH_k(D)$.


chow-lemma-gysin-back

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Then $i^*i_* : \CH_k(D) \to \CH_{k - 1}(D)$ sends $\alpha$ to $c_1(\mathcal{L}|_D) \cap \alpha$.


chow-lemma-gysin-commutes-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. Let $\mathcal{N}$ be an invertible $\mathcal{O}_X$-module. Then $i^*(c_1(\mathcal{N}) \cap \alpha) = c_1(i^*\mathcal{N}) \cap i^*\alpha$ in $\CH_{k - 2}(D)$ for all $\alpha \in \CH_k(X)$.


chow-lemma-gysin-commutes-gysin

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ and $(\mathcal{L}', s', i' : D' \to X)$ be two triples as in Definition \ref{definition-gysin-homomorphism}. Then the diagram $$ \xymatrix{ \CH_k(X) \ar[r]_{i^*} \ar[d]_{(i')^*} & \CH_{k - 1}(D) \ar[d]^{j^*} \\ \CH_{k - 1}(D') \ar[r]^{(j')^*} & \CH_{k - 2}(D \cap D') } $$ commutes where each of the maps is a gysin map.


chow-lemma-relative-effective-cartier

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \to Y$ be a flat morphism of relative dimension $r$. Let $i : D \to X$ be a relative effective Cartier divisor (Divisors, Definition \ref{divisors-definition-relative-effective-Cartier-divisor}). Let $\mathcal{L} = \mathcal{O}_X(D)$. For any $\alpha \in \CH_{k + 1}(Y)$ we have $$ i^*p^*\alpha = (p|_D)^*\alpha $$ in $\CH_{k + r}(D)$ and $$ c_1(\mathcal{L}) \cap p^*\alpha = i_* ((p|_D)^*\alpha) $$ in $\CH_{k + r}(X)$.


chow-lemma-pullback-affine-fibres-surjective

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Assume that for every $y \in Y$, there exists an open neighbourhood $U \subset Y$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is identified with the morphism $U \times \mathbf{A}^r \to U$. Then $f^* : \CH_k(Y) \to \CH_{k + r}(X)$ is surjective for all $k \in \mathbf{Z}$.


chow-lemma-linebundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $$ p : L = \underline{\Spec}(\text{Sym}^*(\mathcal{L})) \longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \CH_k(X) \to \CH_{k + 1}(L)$ is an isomorphism for all $k$.


chow-lemma-linebundle-formulae

In the situation of Lemma \ref{lemma-linebundle} denote $o : X \to L$ the zero section (see proof of the lemma). Then we have \begin{enumerate} \item $o(X)$ is the zero scheme of a regular global section of $p^*\mathcal{L}^{\otimes -1}$, \item $o_* : \CH_k(X) \to \CH_k(L)$ as $o$ is a closed immersion, \item $o^* : \CH_{k + 1}(L) \to \CH_k(X)$ as $o(X)$ is an effective Cartier divisor, \item $o^* p^* : \CH_k(X) \to \CH_k(X)$ is the identity map, \item $o_*\alpha = - p^*(c_1(\mathcal{L}) \cap \alpha)$ for any $\alpha \in \CH_k(X)$, and \item $o^* o_* : \CH_k(X) \to \CH_{k - 1}(X)$ is equal to the map $\alpha \mapsto - c_1(\mathcal{L}) \cap \alpha$. \end{enumerate}


chow-lemma-decompose-section

Let $Y$ be a scheme. Let $\mathcal{L}_i$, $i = 1, 2$ be invertible $\mathcal{O}_Y$-modules. Let $s$ be a global section of $\mathcal{L}_1 \otimes_{\mathcal{O}_X} \mathcal{L}_2$. Denote $i : D \to X$ the zero scheme of $s$. Then there exists a commutative diagram $$ \xymatrix{ D_1 \ar[r]_{i_1} \ar[d]_{p_1} & L \ar[d]^p & D_2 \ar[l]^{i_2} \ar[d]^{p_2} \\ D \ar[r]^i & Y & D \ar[l]_i } $$ and sections $s_i$ of $p^*\mathcal{L}_i$ such that the following hold: \begin{enumerate} \item $p^*s = s_1 \otimes s_2$, \item $p$ is of finite type and flat of relative dimension $1$, \item $D_i$ is the zero scheme of $s_i$, \item $D_i \cong \underline{\Spec}(\text{Sym}^*(\mathcal{L}_{1 - i}^{\otimes -1})|_D))$ over $D$ for $i = 1, 2$, \item $p^{-1}D = D_1 \cup D_2$ (scheme theoretic union), \item $D_1 \cap D_2$ (scheme theoretic intersection) maps isomorphically to $D$, and \item $D_1 \cap D_2 \to D_i$ is the zero section of the line bundle $D_i \to D$ for $i = 1, 2$. \end{enumerate} Moreover, the formation of this diagram and the sections $s_i$ commutes with arbitrary base change.


chow-lemma-normal-cone-effective-Cartier

In Situation \ref{situation-setup} let $X$ be a scheme locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. There exists a commutative diagram $$ \xymatrix{ D' \ar[r]_{i'} \ar[d]_p & X' \ar[d]^g \\ D \ar[r]^i & X } $$ such that \begin{enumerate} \item $p$ and $g$ are of finite type and flat of relative dimension $1$, \item $p^* : \CH_k(D) \to \CH_{k + 1}(D')$ is injective for all $k$, \item $D' \subset X'$ is the zero scheme of a global section $s' \in \Gamma(X', \mathcal{O}_{X'})$, \item $p^*i^* = (i')^*g^*$ as maps $\CH_k(X) \to \CH_k(D')$. \end{enumerate} Moreover, these properties remain true after arbitrary base change by morphisms $Y \to X$ which are locally of finite type.


chow-lemma-flat-pullback-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a flat morphism of relative dimension $r$ between schemes locally of finite type over $S$. Then the rule that to $Y' \to Y$ assigns $(f')^* : \CH_k(Y') \to \CH_{k + r}(X')$ where $X' = X \times_Y Y'$ is a bivariant class of degree $-r$.


chow-lemma-gysin-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. Then the rule that to $f : X' \to X$ assigns $(i')^* : \CH_k(X') \to \CH_{k - 1}(D')$ where $D' = D \times_X X'$ is a bivariant class of degree $1$.


chow-lemma-push-proper-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes locally of finite type over $S$. Let $c \in A^p(X \to Z)$ and assume $f$ is proper. Then the rule that to $Z' \to Z$ assigns $\alpha \longmapsto f'_*(c \cap \alpha)$ is a bivariant class denoted $f_* \circ c \in A^p(Y \to Z)$.


chow-lemma-cap-c1-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then the rule that to $f : X' \to X$ assigns $c_1(f^*\mathcal{L}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')$ is a bivariant class of degree $1$.


chow-lemma-c1-center

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then \begin{enumerate} \item $c_1(\mathcal{L}) \in A^1(X)$ is in the center of $A^*(X)$ and \item if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_1(\mathcal{L}) = c_1(f^*\mathcal{L}) \circ c$. \end{enumerate}


chow-lemma-vanish-above-dimension

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a finite type scheme over $S$ which has an ample invertible sheaf. Assume $d = \dim(X) < \infty$ (here we really mean dimension and not $\delta$-dimension). Then for any invertible sheaves $\mathcal{L}_1, \ldots, \mathcal{L}_{d + 1}$ on $X$ we have $c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_{d + 1}) = 0$ in $A^{d + 1}(X)$.


chow-lemma-bivariant-zero

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $c \in A^p(X \to Y)$. For $Y'' \to Y' \to Y$ set $X'' = Y'' \times_Y X$ and $X' = Y' \times_Y X$. The following are equivalent \begin{enumerate} \item $c$ is zero, \item $c \cap [Y'] = 0$ in $\CH_*(X')$ for every integral scheme $Y'$ locally of finite type over $Y$, and \item for every integral scheme $Y'$ locally of finite type over $Y$, there exists a proper birational morphism $Y'' \to Y'$ such that $c \cap [Y''] = 0$ in $\CH_*(X'')$. \end{enumerate}


chow-lemma-disjoint-decomposition-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Assume we have disjoint union decompositions $X = \coprod_{i \in I} X_i$ and $Y = \coprod_{j \in J} Y_j$ by open and closed subschemes and a map $a : I \to J$ of sets such that $f(X_i) \subset Y_{a(i)}$. Then $$ A^p(X \to Y) = \prod\nolimits_{i \in I} A^p(X_i \to Y_{a(i)}) $$


chow-lemma-cap-projective-bundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. For any $\alpha \in \CH_k(X)$ the element $$ \pi_*\left( c_1(\mathcal{O}_P(1))^s \cap \pi^*\alpha \right) \in \CH_{k + r - 1 - s}(X) $$ is $0$ if $s < r - 1$ and is equal to $\alpha$ when $s = r - 1$.


chow-lemma-chow-ring-projective-bundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. The map $$ \bigoplus\nolimits_{i = 0}^{r - 1} \CH_{k + i}(X) \longrightarrow \CH_{k + r - 1}(P), $$ $$ (\alpha_0, \ldots, \alpha_{r-1}) \longmapsto \pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1} $$ is an isomorphism.


chow-lemma-vectorbundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $$ p : E = \underline{\Spec}(\text{Sym}^*(\mathcal{E})) \longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \CH_k(X) \to \CH_{k + r}(E)$ is an isomorphism for all $k$.


chow-lemma-determine-intersections

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. For $\alpha \in Z_k(X)$ the elements $c_j(\mathcal{E}) \cap \alpha$ are the unique elements $\alpha_j$ of $\CH_{k - j}(X)$ such that $\alpha_0 = \alpha$ and $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*(\alpha_{r - i}) = 0 $$ holds in the Chow group of $P$.


chow-lemma-cap-chern-class-factors-rational-equivalence

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. If $\alpha \sim_{rat} \beta$ are rationally equivalent $k$-cycles on $X$ then $c_j(\mathcal{E}) \cap \alpha = c_j(\mathcal{E}) \cap \beta$ in $\CH_{k - j}(X)$.


chow-lemma-pushforward-cap-cj

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $p : X \to Y$ be a proper morphism. Let $\alpha$ be a $k$-cycle on $X$. Let $\mathcal{E}$ be a finite locally free sheaf on $Y$. Then $$ p_*(c_j(p^*\mathcal{E}) \cap \alpha) = c_j(\mathcal{E}) \cap p_*\alpha $$


chow-lemma-flat-pullback-cap-cj

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_j(\mathcal{E}) \cap \alpha) = c_j(f^*\mathcal{E}) \cap f^*\alpha $$


chow-lemma-cap-chern-class-commutes-with-gysin

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Then $c_j(\mathcal{E}|_D) \cap i^*\alpha = i^*(c_j(\mathcal{E}) \cap \alpha)$ for all $\alpha \in \CH_k(X)$.


chow-lemma-cap-cp-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$. Let $0 \leq p \leq r$. Then the rule that to $f : X' \to X$ assigns $c_p(f^*\mathcal{E}) \cap - : \CH_k(X') \to \CH_{k - p}(X')$ is a bivariant class of degree $p$.


chow-lemma-cap-commutative-chern

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$. Then \begin{enumerate} \item $c_j(\mathcal{E}) \in A^j(X)$ is in the center of $A^*(X)$ and \item if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_j(\mathcal{E}) = c_j(f^*\mathcal{E}) \circ c$. \end{enumerate} In particular, if $\mathcal{F}$ is a second locally free $\mathcal{O}_X$-module on $X$ of rank $s$, then $$ c_i(\mathcal{E}) \cap c_j(\mathcal{F}) \cap \alpha = c_j(\mathcal{F}) \cap c_i(\mathcal{E}) \cap \alpha $$ as elements of $\CH_{k - i - j}(X)$ for all $\alpha \in \CH_k(X)$.


chow-lemma-chern-classes-E-tensor-L

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then we have \begin{equation} \label{equation-twist} c_i({\mathcal E} \otimes {\mathcal L}) = \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c_{i - j}({\mathcal E}) c_1({\mathcal L})^j \end{equation} in $A^*(X)$.


chow-lemma-get-rid-of-trivial-subbundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{F} \to 0 $$ Then we have $$ c_r(\mathcal{E}) = 0, \quad c_j(\mathcal{E}) = c_j(\mathcal{F}), \quad j = 0, \ldots, r - 1 $$ in $A^*(X)$.


chow-lemma-additivity-invertible-subsheaf

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \to \mathcal{L} \to \mathcal{E} \to \mathcal{F} \to 0 $$ where $\mathcal{L}$ is an invertible sheaf. Then $$ c(\mathcal{E}) = c(\mathcal{L}) c(\mathcal{F}) $$ in $A^*(X)$.


chow-lemma-additivity-chern-classes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Suppose that ${\mathcal E}$ sits in an exact sequence $$ 0 \to {\mathcal E}_1 \to {\mathcal E} \to {\mathcal E}_2 \to 0 $$ of finite locally free sheaves $\mathcal{E}_i$ of rank $r_i$. The total Chern classes satisfy $$ c({\mathcal E}) = c({\mathcal E}_1) c({\mathcal E}_2) $$ in $A^*(X)$.


chow-lemma-chern-filter-by-linebundles

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let ${\mathcal L}_i$, $i = 1, \ldots, r$ be invertible $\mathcal{O}_X$-modules on $X$. Let $\mathcal{E}$ be a locally free rank $\mathcal{O}_X$-module endowed with a filtration $$ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E} $$ such that $\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i$. Set $c_1({\mathcal L}_i) = x_i$. Then $$ c(\mathcal{E}) = \prod\nolimits_{i = 1}^r (1 + x_i) $$ in $A^*(X)$.


chow-lemma-degrees-and-numerical-intersections

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \subset X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \deg( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z]_d) $$ where the left hand side is defined in Varieties, Definition \ref{varieties-definition-intersection-number}. In particular, $$ \deg_\mathcal{L}(Z) = \deg(c_1(\mathcal{L})^d \cap [Z]_d) $$ if $\mathcal{L}$ is an ample invertible $\mathcal{O}_X$-module.


chow-lemma-locally-equidimensional

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Write $\delta = \delta_{X/S}$ as in Section \ref{section-setup}. The following are equivalent \begin{enumerate} \item There exists a decomposition $X = \coprod_{n \in \mathbf{Z}} X_n$ into open and closed subschemes such that $\delta(\xi) = n$ whenever $\xi \in X_n$ is a generic point of an irreducible component of $X_n$. \item For all $x \in X$ there exists an open neighbourhood $U \subset X$ of $x$ and an integer $n$ such that $\delta(\xi) = n$ whenever $\xi \in U$ is a generic point of an irreducible component of $U$. \item For all $x \in X$ there exists an integer $n_x$ such that $\delta(\xi) = n_x$ for any generic point $\xi$ of an irreducible component of $X$ containing $x$. \end{enumerate} The conditions are satisfied if $X$ is either normal or Cohen-Macaulay\footnote{In fact, it suffices if $X$ is $(S_2)$. Compare with Local Cohomology, Lemma \ref{local-cohomology-lemma-catenary-S2-equidimensional}.}.


chow-lemma-splitting-principle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}_i$ be a finite collection of locally free $\mathcal{O}_X$-modules of rank $r_i$. There exists a projective flat morphism $\pi : P \to X$ of relative dimension $d$ such that \begin{enumerate} \item for any morphism $f : Y \to X$ the map $\pi_Y^* : \CH_*(Y) \to \CH_{* + d}(Y \times_X P)$ is injective, and \item each $\pi^*\mathcal{E}_i$ has a filtration whose successive quotients $\mathcal{L}_{i, 1}, \ldots, \mathcal{L}_{i, r_i}$ are invertible ${\mathcal O}_P$-modules. \end{enumerate} Moreover, when (1) holds the restriction map $A^*(X) \to A^*(P)$ (Remark \ref{remark-pullback-cohomology}) is injective.


chow-lemma-chern-classes-dual

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module with dual $\mathcal{E}^\vee$. Then $$ c_i(\mathcal{E}^\vee) = (-1)^i c_i(\mathcal{E}) $$ in $A^i(X)$.


chow-lemma-chern-classes-tensor-product

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ and $\mathcal{F}$ be a finite locally free $\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have $$ c_1(\mathcal{E} \otimes \mathcal{F}) = r c_1(\mathcal{F}) + s c_1(\mathcal{E}) $$ $$ c_2(\mathcal{E} \otimes \mathcal{F}) = r c_2(\mathcal{F}) + s c_2(\mathcal{E}) + {r \choose 2} c_1(\mathcal{F})^2 + (rs - 1) c_1(\mathcal{F})c_1(\mathcal{E}) + {s \choose 2} c_1(\mathcal{E})^2 $$ and so on in $A^*(X)$.


chow-lemma-top-chern-class

In the situation described just above assume $\dim_\delta(X') = n$, that $f^*\mathcal{E}$ has constant rank $r$, that $\dim_\delta(Z(s)) \leq n - r$, and that for every generic point $\xi \in Z(s)$ with $\delta(\xi) = n - r$ the ideal of $Z(s)$ in $\mathcal{O}_{X', \xi}$ is generated by a regular sequence of length $r$. Then $$ c_r(\mathcal{E}) \cap [X']_n = [Z(s)]_{n - r} $$ in $\CH_*(X')$.


chow-lemma-easy-virtual-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $$ 0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0 $$ be a short exact sequence of finite locally free $\mathcal{O}_X$-modules. Consider the closed embedding $$ i : N' = \underline{\text{Spec}}_X(\text{Sym}((\mathcal{N}')^\vee)) \longrightarrow N = \underline{\text{Spec}}_X(\text{Sym}(\mathcal{N}^\vee)) $$ For $\alpha \in \CH_k(X)$ we have $$ i_*(p')^*\alpha = p^*(c_{top}(\mathcal{E}) \cap \alpha) $$ where $p' : N' \to X$ and $p : N \to X$ are the structure morphisms.


chow-lemma-chern-character-additive

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $ 0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0 $ be a short exact sequence of finite locally free $\mathcal{O}_X$-modules. Then we have the equality $$ ch(\mathcal{E}) = ch(\mathcal{E}_1) + ch(\mathcal{E}_2) $$ in $A^*(X) \otimes \mathbf{Q}$. More precisely, we have $P_p(\mathcal{E}) = P_p(\mathcal{E}_1) + P_p(\mathcal{E}_2)$ in $A^p(X)$ where $P_p$ is as in Example \ref{example-power-sum}.


chow-lemma-chern-character-multiplicative

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be finite locally free $\mathcal{O}_X$-modules. Then we have the equality $$ ch(\mathcal{E}_1 \otimes_{\mathcal{O}_X} \mathcal{E}_2) = ch(\mathcal{E}_1) ch(\mathcal{E}_2) $$ in $A^*(X) \otimes \mathbf{Q}$. More precisely, we have $$ P_p(\mathcal{E}_1 \otimes_{\mathcal{O}_X} \mathcal{E}_2) = \sum\nolimits_{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(\mathcal{E}_1) P_{p_2}(\mathcal{E}_2) $$ in $A^p(X)$ where $P_p$ is as in Example \ref{example-power-sum}.


chow-lemma-commutative-chern-perfect

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_X)$ be perfect. If the Chern classes of $E$ are defined then \begin{enumerate} \item $c_p(E)$ is in the center of the algebra $A^*(X)$ and \item if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_j(E) = c_j(Lf^*E) \circ c$. \end{enumerate}


chow-lemma-additivity-on-perfect

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $$ E_1 \to E_2 \to E_3 \to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\mathcal{O}_X)$. If $E_1 \to E_2$ can be represented be a map of bounded complexes of finite locally free $\mathcal{O}_X$-modules, then we have $c(E_2) = c(E_1) c(E_3)$, $ch(E_2) = ch(E_1) + ch(E_3)$, and $P_p(E_2) = P_p(E_1) + P_p(E_3)$.


chow-lemma-chern-class-perfect-tensor-invertible

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_X)$ whose Chern classes are defined. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $$ c_i(E \otimes \mathcal{L}) = \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c_{i - j}(E) c_1(\mathcal{L})^j $$ provided $E$ has constant rank $r \in \mathbf{Z}$.


chow-lemma-chern-classes-perfect-tensor-product

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E$ and $F$ be perfect objects of $D(\mathcal{O}_X)$ whose Chern classes are defined. Then we have $$ c_1(E \otimes_{\mathcal{O}_X}^\mathbf{L} F) = r(E) c_1(\mathcal{F}) + r(F) c_1(\mathcal{E}) $$ and for $c_2(E \otimes_{\mathcal{O}_X}^\mathbf{L} F)$ we have the expression $$ r(E) c_2(F) + r(F) c_2(E) + {r(E) \choose 2} c_1(F)^2 + (r(E)r(F) - 1) c_1(F)c_1(E) + {r(F) \choose 2} c_1(E)^2 $$ and so on for higher Chern classes in $A^*(X)$. Similarly, we have $ch(E \otimes_{\mathcal{O}_X}^\mathbf{L} F) = ch(E) ch(F)$ in $A^*(X) \otimes \mathbf{Q}$. More precisely, we have $$ P_p(E \otimes_{\mathcal{O}_X}^\mathbf{L} F) = \sum\nolimits_{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(E) P_{p_2}(F) $$ in $A^p(X)$.


chow-lemma-silly

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E_2 \in D(\mathcal{O}_{X_2})$ be a perfect object. Assume \begin{enumerate} \item Chern classes of $E_2$ are defined, \item the restriction $E_2|_{X_1 \cap X_2}$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{X_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$. \end{enumerate} Then there is a canonical bivariant class $$ P'_p(E_2),\text{ resp. }c'_p(E_2) \in A^p(X_2 \to X) $$ characterized by the property $$ P'_p(E_2) \cap i_{2, *} \alpha_2 = P_p(E_2) \cap \alpha_2 \quad\text{and}\quad P'_p(E_2) \cap i_{1, *} \alpha_1 = 0, $$ respectively $$ c'_p(E_2) \cap i_{2, *} \alpha_2 = c_p(E_2) \cap \alpha_2 \quad\text{and}\quad c'_p(E_2) \cap i_{1, *} \alpha_1 = 0 $$ for $\alpha_i \in \CH_k(X_i)$ and similarly after any base change $X' \to X$ locally of finite type.


chow-lemma-silly-independent

In Lemma \ref{lemma-silly} the bivariant class $P'_p(E_2)$, resp.\ $c'_p(E_2)$ in $A^p(X_2 \to X)$ does not depend on the choice of $X_1$.


chow-lemma-silly-silly

In Lemma \ref{lemma-silly} say $E_2$ is the restriction of a perfect $E \in D(\mathcal{O}_X)$ such that $E|_{X_1}$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. If Chern classes of $E$ are defined, then $i_{2, *} \circ P'_p(E_2) = P_p(E)$, resp.\ $i_{2, *} \circ c'_p(E_2) = c_p(E)$ (with $\circ$ as in Lemma \ref{lemma-push-proper-bivariant}).


chow-lemma-silly-shrink

In Lemma \ref{lemma-silly} suppose we have closed subschemes $X'_2 \subset X_2$ and $X_1 \subset X'_1 \subset X$ such that $X = X'_1 \cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \cap X_2}$ is zero, resp.\ isomorphic to a finite locally free module of rank $< p$ placed in degree $0$. Then we have $(X'_2 \to X_2)_* \circ P'_p(E_2|_{X'_2}) = P'_p(E_2)$, resp.\  $(X'_2 \to X_2)_* \circ c'_p(E_2|_{X'_2}) = c_p(E_2)$ (with $\circ$ as in Lemma \ref{lemma-push-proper-bivariant}).


chow-lemma-silly-commutes

In Lemma \ref{lemma-silly} let $f : Y \to X$ be locally of finite type and say $c \in A^*(Y \to X)$. Then $$ c \circ P'_p(E_2) = P'_p(Lf_2^*E_2) \circ c \quad\text{resp.}\quad c \circ c'_p(E_2) = c'_p(Lf_2^*E_2) \circ c $$ in $A^*(Y_2 \to Y)$ where $f_2 : Y_2 \to X_2$ is the base change of $f$.


chow-lemma-silly-compose

In Lemma \ref{lemma-silly} assume $E_2|_{X_1 \cap X_2}$ is zero. Then \begin{align*} P'_1(E_2) & = c'_1(E_2), \\ P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\ P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\ P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), \end{align*} and so on with multiplication as in Remark \ref{remark-ring-loc-classes}.


chow-lemma-silly-sum-c

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_X)$ be perfect objects. Assume \begin{enumerate} \item Chern classes of $E$ and $F$ are defined, \item the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are isomorphic to a finite locally free $\mathcal{O}_{X_1}$-modules of rank $< p$ and $< q$ sitting in cohomological degree $0$. \end{enumerate} With notation as in Remark \ref{remark-ring-loc-classes} set $$ c^{(p)}(E) = 1 + c_1(E) + \ldots + c_{p - 1}(E) + c'_p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \ldots \in A^{(p)}(X_2 \to X) $$ with $c'_p(E|_{X_2})$ as in Lemma \ref{lemma-silly}. Similarly for $c^{(q)}(F)$ and $c^{(p + q)}(E \oplus F)$. Then $c^{(p + q)}(E \oplus F) = c^{(p)}(E)c^{(q)}(F)$ in $A^{(p + q)}(X_2 \to X)$.


chow-lemma-silly-sum-P

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_{X_2})$ be perfect objects. Assume \begin{enumerate} \item Chern classes of $E$ and $F$ are defined, \item the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are zero, \end{enumerate} Denote $P'_p(E), P'_p(F), P'_p(E \oplus F) \in A^p(X_2 \to X)$ for $p \geq 0$ the classes constructed in Lemma \ref{lemma-silly}. Then $P'_p(E \oplus F) = P'_p(E) + P'_p(F)$.


chow-lemma-silly-tensor-product

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $$ X = X_1 \cup X_2 = X'_1 \cup X'_2 $$ be two ways of writing $X$ as a set theoretic union of closed subschemes. Let $E$, $E'$ be perfect objects of $D(\mathcal{O}_X)$ whose Chern classes are defined. Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero\footnote{Presumably there is a variant of this lemma where we only assume these restrictions are isomorphic to a finite locally free modules of rank $< p$ and $< p'$.} for $i = 1, 2$. Denote \begin{enumerate} \item $r = P'_0(E) \in A^0(X_2 \to X)$ and $r' = P'_0(E') \in A^0(X'_2 \to X)$, \item $\gamma_p = c'_p(E|_{X_2}) \in A^p(X_2 \to X)$ and $\gamma'_p = c'_p(E'|_{X'_2}) \in A^p(X'_2 \to X)$, \item $\chi_p = P'_p(E|_{X_2}) \in A^p(X_2 \to X)$ and $\chi'_p = P'_p(E'|_{X'_2}) \in A^p(X'_2 \to X)$ \end{enumerate} the classes constructed in Lemma \ref{lemma-silly}. Then we have $$ c'_1((E \otimes_{\mathcal{O}_X}^\mathbf{L} E')|_{X_2 \cap X'_2}) = r \gamma'_1 + r' \gamma_1 $$ in $A^1(X_2 \cap X'_2 \to X)$ and $$ c'_2((E \otimes_{\mathcal{O}_X}^\mathbf{L} E')|_{X_2 \cap X'_2}) = r \gamma'_2 + r' \gamma_2 + {r \choose 2} (\gamma'_1)^2 + (rr' - 1) \gamma'_1\gamma_1 + {r' \choose 2} \gamma_1^2 $$ in $A^2(X_2 \cap X'_2 \to X)$ and so on for higher Chern classes. Similarly, we have $$ P'_p((E \otimes_{\mathcal{O}_X}^\mathbf{L} E')|_{X_2 \cap X'_2}) = \sum\nolimits_{p_1 + p_2 = p} {p \choose p_1} \chi_{p_1} \chi'_{p_2} $$ in $A^p(X_2 \cap X'_2 \to X)$.


chow-lemma-gysin-at-infty

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $b : W \to \mathbf{P}^1_X$ be a proper morphism of schemes which is an isomorphism over $\mathbf{A}^1_X$. Denote $i_\infty : W_\infty \to W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_X$ with complement $\mathbf{A}^1_X$. Then there is a canonical bivariant class $$ C \in A^0(W_\infty \to X) $$ with the property that $i_{\infty, *}(C \cap \alpha) = i_{0, *}\alpha$ for $\alpha \in \CH_k(X)$ and similarly after any base change by $X' \to X$ locally of finite type.


chow-lemma-gysin-at-infty-independent

In Lemma \ref{lemma-gysin-at-infty} let $g : W' \to W$ be a proper morphism which is an isomorphism over $\mathbf{A}^1_X$. Let $C' \in A^0(W'_\infty \to X)$ and $C \in A^0(W_\infty \to X)$ be the classes constructed in Lemma \ref{lemma-gysin-at-infty}. Then $g_{\infty, *} \circ C' = C$ in $A^0(W_\infty \to X)$.


chow-lemma-homomorphism-pre

In Lemma \ref{lemma-gysin-at-infty} we have $C \circ (W_\infty \to X)_* \circ i_\infty^* = i_\infty^*$.


chow-lemma-gysin-at-infty-commutes

In Lemma \ref{lemma-gysin-at-infty} let $f : Y \to X$ be a morphism locally of finite type and $c \in A^*(Y \to X)$. Then $C \circ c = c \circ C$ in $A^*(W_\infty \times_X Y)$.


chow-lemma-localized-chern-pre

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. Let $$ b : W \longrightarrow \mathbf{P}^1_X $$ be a proper morphism of schemes. Let $Q \in D(\mathcal{O}_W)$ be a perfect object. Denote $W_\infty \subset W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_X$ with complement $\mathbf{A}^1_X$. We assume \begin{enumerate} \item[(A0)] Chern classes of $Q$ are defined (Section \ref{section-pre-derived}), \item[(A1)] $b$ is an isomorphism over $\mathbf{A}^1_X$, \item[(A2)] there exists a closed subscheme $T \subset W_\infty$ containing all points of $W_\infty$ lying over $X \setminus Z$ such that $Q|_T$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p$ sitting in cohomological degree $0$. \end{enumerate} Then there exists a canonical bivariant class $$ P'_p(Q),\text{ resp. }c'_p(Q) \in A^p(Z \to X) $$ with $(Z \to X)_* \circ P'_p(Q) = P_p(Q|_{X \times \{0\}})$, resp.\ $(Z \to X)_* \circ c'_p(Q) = c_p(Q|_{X \times \{0\}})$.


chow-lemma-localized-chern-pre-independent

In Lemma \ref{lemma-localized-chern-pre} the bivariant class $P'_p(Q)$, resp.\ $c'_p(Q)$ is independent of the choice of the closed subscheme $T$. Moreover, given a proper morphism $g : W' \to W$ which is an isomorphism over $\mathbf{A}^1_X$, then setting $Q' = g^*Q$ we have $P'_p(Q) = P'_p(Q')$, resp.\ $c'_p(Q) = c'_p(Q')$.


chow-lemma-homomorphism

In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p$. Denote $C \in A^0(W_\infty \to X)$ the class of Lemma \ref{lemma-gysin-at-infty}. Then $$ C \circ c_p(Q|_{X \times \{0\}}) = C \circ (Z \to X)_* \circ c'_p(Q) = c_p(Q|_{W_\infty}) \circ C $$


chow-lemma-homomorphism-commute

In Lemma \ref{lemma-localized-chern-pre} let $Y \to X$ be a morphism locally of finite type and let $c \in A^*(Y \to X)$ be a bivariant class. Then $$ P'_p(Q) \circ c = c \circ P'_p(Q) \quad\text{resp.}\quad c'_p(Q) \circ c = c \circ c'_p(Q) $$ in $A^*(Y \times_X Z \to X)$.


chow-lemma-localized-chern-pre-compose

In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. In $A^*(Z \to X)$ we have \begin{align*} P'_1(Q) & = c'_1(Q), \\ P'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), \\ P'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), \\ P'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) + 4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q), \end{align*} and so on with multiplication as in Remark \ref{remark-ring-loc-classes}.


chow-lemma-localized-chern-pre-sum-c

In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p$. Assume we have another perfect object $Q' \in D(\mathcal{O}_W)$ whose Chern classes are defined with $Q'|_T$ isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p'$ placed in cohomological degree $0$. With notation as in Remark \ref{remark-ring-loc-classes} set $$ c^{(p)}(Q) = 1 + c_1(Q|_{X \times \{0\}}) + \ldots + c_{p - 1}(Q|_{X \times \{0\}}) + c'_{p}(Q) + c'_{p + 1}(Q) + \ldots $$ in $A^{(p)}(Z \to X)$ with $c'_i(Q)$ for $i \geq p$ as in Lemma \ref{lemma-localized-chern-pre}. Similarly for $c^{(p')}(Q')$ and $c^{(p + p')}(Q \oplus Q')$. Then $c^{(p + p')}(Q \oplus Q') = c^{(p)}(Q)c^{(p')}(Q')$ in $A^{(p + p')}(Z \to X)$.


chow-lemma-localized-chern-pre-sum-P

In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. Assume we have another perfect object $Q' \in D(\mathcal{O}_W)$ whose Chern classes are defined such that the restriction $Q'|_T$ is zero. In this case the classes $P'_p(Q), P'_p(Q'), P'_p(Q \oplus Q') \in A^p(Z \to X)$ constructed in Lemma \ref{lemma-localized-chern-pre} satisfy $P'_p(Q \oplus Q') = P'_p(Q) + P'_p(Q')$.


chow-lemma-base-change-loc-chern

In the situation above let $f : X' \to X$ be a morphism of schemes which is locally of finite type. Denote $E' = Lf^*E$ and $Z' = f^{-1}(Z)$. Then the bivariant class $$ P_p(Z' \to X', E') \in A^p(Z' \to X'), \quad\text{resp.}\quad c_p(Z' \to X', E') \in A^p(Z' \to X') $$ constructed above using $X', Z', E'$ is the restriction (Remark \ref{remark-restriction-bivariant}) of the bivariant class $P_p(Z \to X, E) \in A^p(Z \to X)$, resp.\ $c_p(Z \to X, E) \in A^p(Z \to X)$.


chow-lemma-loc-chern-after-pushforward

In the situation above we have $$ P_p(Z \to X, E) \cap i_*\alpha = P_p(E|_Z) \cap \alpha, \quad\text{resp.}\quad c_p(Z \to X, E) \cap i_*\alpha = c_p(E|_Z) \cap \alpha $$ in $\CH_*(Z)$ for any $\alpha \in \CH_*(Z)$.


chow-lemma-loc-chern-disjoint

In the situation of Definition \ref{definition-localized-chern} if $\alpha \in \CH_k(X)$ has support disjoint from $Z$, then $P_p(Z \to X, E) \cap \alpha = 0$, resp.\ $c_p(Z \to X, E) \cap \alpha = 0$.


chow-lemma-loc-chern-shrink-Z

In the situation of Definition \ref{definition-localized-chern} assume $Z \subset Z' \subset X$ where $Z'$ is a closed subscheme of $X$. Then $P_p(Z' \to X, E) = (Z \to Z')_* \circ P_p(Z \to X, E)$, resp.\ $c_p(Z' \to X, E) = (Z \to Z')_* \circ c_p(Z \to X, E)$ (with $\circ$ as in Lemma \ref{lemma-push-proper-bivariant}).


chow-lemma-loc-chern-agree

In Lemma \ref{lemma-silly} say $E_2$ is the restriction of a perfect $E \in D(\mathcal{O}_X)$ whose Chern classes are defined and whose restriction to $X_1$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. Then the class $P'_p(E_2)$, resp.\ $c'_p(E_2)$ of Lemma \ref{lemma-silly} agrees with $P_p(X_2 \to X, E)$, resp.\ $c_p(X_2 \to X, E)$ of Definition \ref{definition-localized-chern}.


chow-lemma-homomorphism-final

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $b : W \longrightarrow \mathbf{P}^1_X$ be a proper morphism of schemes. Let $n \geq 1$. For $i = 1, \ldots, n$ let $Z_i \subset X$ be a closed subscheme, let $Q_i \in D(\mathcal{O}_W)$, be a perfect object, let $p_i \geq 0$ be an integer, and let $T_i \subset W_\infty$, $i = 1, \ldots, n$ be closed. Denote $W_i = b^{-1}(\mathbf{P}^1_{Z_i})$. Assume \begin{enumerate} \item for $i = 1, \ldots, n$ the assumption of Lemma \ref{lemma-localized-chern-pre} hold for $b, Z_i, Q_i, T_i, p_i$, \item $Q_i|_{W \setminus W_i}$ is zero, resp.\ isomorphic to a finite locally free module of rank $< p_i$ placed in cohomological degree $0$. \end{enumerate} Then $P'_{p_n}(Q_n) \circ \ldots \circ P'_{p_1}(Q_1)$ is equal to $$ (W_{n, \infty} \cap \ldots \cap W_{1, \infty} \to Z_n \cap \ldots \cap Z_1)_* \circ P'_{p_n}(Q_n|_{W_{n, \infty}}) \circ \ldots \circ P'_{p_1}(Q_1|_{W_{1, \infty}}) \circ C $$ in $A^{p_n + \ldots + p_1}(Z_n \cap \ldots \cap Z_1 \to X)$, resp.\ $c'_{p_n}(Q_n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to $$ (W_{n, \infty} \cap \ldots \cap W_{1, \infty} \to Z_n \cap \ldots \cap Z_1)_* \circ c'_{p_n}(Q_n|_{W_{n, \infty}}) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty}}) \circ C $$ in $A^{p_n + \ldots + p_1}(Z_n \cap \ldots \cap Z_1 \to X)$.


chow-lemma-independent-loc-chern-bQ

Assume $(S, \delta), X, Z, b : W \to \mathbf{P}^1_X, Q, T, p$ satisfy all the assumptions of Lemma \ref{lemma-localized-chern-pre}. Finally, let $F \in D(\mathcal{O}_X)$ be a perfect object whose Chern classes are defined such that \begin{enumerate} \item the restriction of $Q$ to $b^{-1}(\mathbf{A}^1_X)$ is isomorphic to the pullback of $F$, and \item $F|_{X \setminus Z}$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$. \end{enumerate} Then the class $P'_p(Q)$, resp.\ $c'_p(Q)$ in $A^p(Z \to X)$ constructed in Lemma \ref{lemma-localized-chern-pre} is equal to $P_p(Z \to X, F)$, resp.\ $c_p(Z \to X, F)$.


chow-lemma-loc-chern-character

In the situation of Definition \ref{definition-localized-chern} assume $E|_{X \setminus Z}$ is zero. Then \begin{align*} P_1(Z \to X, E) & = c_1(Z \to X, E), \\ P_2(Z \to X, E) & = c_1(Z \to X, E)^2 - 2c_2(Z \to X, E), \\ P_3(Z \to X, E) & = c_1(Z \to X, E)^3 - 3c_1(Z \to X, E)c_2(Z \to X, E) + 3c_3(Z \to X, E), \end{align*} and so on where the products are taken in the algebra $A^{(1)}(Z \to X)$ of Remark \ref{remark-ring-loc-classes}.


chow-lemma-additivity-loc-chern-c

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \to X$ be a closed immersion. Let $$ E_1 \to E_2 \to E_3 \to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\mathcal{O}_X)$. Assume \begin{enumerate} \item $E_3 \to E_1[1]$ can be represented be a map of bounded complexes of finite locally free $\mathcal{O}_X$-modules, and \item the restrictions $E_1|_{X \setminus Z}$ and $E_3|_{X \setminus Z}$ are isomorphic to finite locally free $\mathcal{O}_{X \setminus Z}$-modules of rank $< p_1$ and $< p_3$ placed in degree $0$. \end{enumerate} With notation as in Remark \ref{remark-loc-chern-classes} we have $$ c^{(p_1 + p_3)}(Z \to X, E_2) = c^{(p_1)}(Z \to X, E_1)c^{(p_3)}(Z \to X, E_3) $$ in $A^{(p_1 + p_3)}(Z \to X)$.


chow-lemma-additivity-loc-chern-P

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \to X$ be a closed immersion. Let $$ E_1 \to E_2 \to E_3 \to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\mathcal{O}_X)$. Assume \begin{enumerate} \item $E_3 \to E_1[1]$ can be represented be a map of bounded complexes of finite locally free $\mathcal{O}_X$-modules, and \item the restrictions $E_1|_{X \setminus Z}$ and $E_3|_{X \setminus Z}$ are zero. \end{enumerate} Then we have $$ P_p(Z \to X, E_2) = P_p(Z \to X, E_1) + P_p(Z \to X, E_3) $$ for all $p \in \mathbf{Z}$ and consequently $ch(Z \to X, E_2) = ch(Z \to X, E_1) + ch(Z \to X, E_3)$.


chow-lemma-loc-chern-tensor-product

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $Z_i \subset X$, $i = 1, 2$ be closed subschemes. Let $F_i$, $i = 1, 2$ be perfect objects of $D(\mathcal{O}_X)$ whose Chern classes are defined. Assume that $F_i|_{X \setminus Z_i}$ is zero\footnote{Presumably there is a variant of this lemma where we only assume $F_i|_{X \setminus Z_i}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z_i}$-module of rank $< p_i$.} for $i = 1, 2$. Denote $r_i = P_0(Z_i \to X, F_i) \in A^0(Z_i \to X)$. Then we have $$ c_1(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) = r_1 c_1(Z_2 \to X, F_2) + r_2 c_1(Z_1 \to X, F_1) $$ in $A^1(Z_1 \cap Z_2 \to X)$ and \begin{align*} c_2(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) & = r_1 c_2(Z_2 \to X, F_2) + r_2 c_2(Z_1 \to X, F_1) + \\ & {r_1 \choose 2} c_1(Z_2 \to X, F_2)^2 + \\ & (r_1r_2 - 1) c_1(Z_2 \to X, F_2)c_1(Z_1 \to X, F_1) + \\ & {r_2 \choose 2} c_1(Z_1 \to X, F_1)^2 \end{align*} in $A^2(Z_1 \cap Z_2 \to X)$ and so on for higher Chern classes. Similarly, we have $$ ch(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) = ch(Z_1 \to X, F_1) ch(Z_2 \to X, F_2) $$ in $A^*(Z_1 \cap Z_2 \to X) \otimes \mathbf{Q}$. More precisely, we have $$ P_p(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) = \sum\nolimits_{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(Z_1 \to X, F_1) P_{p_2}(Z_2 \to X, F_2) $$ in $A^p(Z_1 \cap Z_2 \to X)$.


chow-lemma-pullback-virtual-normal-sheaf

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $$ \xymatrix{ Z' \ar[r] \ar[d]_g & X' \ar[d]^f \\ Z \ar[r] & X } $$ be a cartesian diagram of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. If $\mathcal{N}$ is a virtual normal sheaf for $Z$ in $X$, then $\mathcal{N}' = g^*\mathcal{N}$ is a virtual normal sheaf for $Z'$ in $X'$.


chow-lemma-construction-gysin

The construction above defines a bivariant class\footnote{The notation $A^*(Z \to X)^\wedge$ is discussed in Remark \ref{remark-completion-bivariant}. If $X$ is quasi-compact, then $A^*(Z \to X)^\wedge = A^*(Z \to X)$.} $$ c(Z \to X, \mathcal{N}) \in A^*(Z \to X)^\wedge $$ and moreover the construction is compatible with base change as in Lemma \ref{lemma-pullback-virtual-normal-sheaf}. If $\mathcal{N}$ has constant rank $r$, then $c(Z \to X, \mathcal{N}) \in A^r(Z \to X)$.


chow-lemma-gysin-decompose

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Suppose that we have a short exact sequence $0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0$ of finite locally free $\mathcal{O}_Z$-modules such that the given surjection $\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ factors through a map $\sigma' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/X}$. Then $$ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ c(Z \to X, \mathcal{N}') $$ as bivariant classes.


chow-lemma-gysin-excess

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider a cartesian diagram $$ \xymatrix{ Z' \ar[r] \ar[d]_g & X' \ar[d]^f \\ Z \ar[r] & X } $$ of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. Let $\mathcal{N}$, resp.\ $\mathcal{N}'$ be a virtual normal sheaf for $Z \subset X$, resp.\ $Z' \to X'$. Assume given a short exact sequence $0 \to \mathcal{N}' \to g^*\mathcal{N} \to \mathcal{E} \to 0$ of finite locally free modules on $Z'$ such that the diagram $$ \xymatrix{ g^*\mathcal{N}^\vee \ar[r] \ar[d] & (\mathcal{N}')^\vee \ar[d] \\ g^*\mathcal{C}_{Z/X} \ar[r] & \mathcal{C}_{Z'/X'} } $$ commutes. Then we have $$ res(c(Z \to X, \mathcal{N})) = c_{top}(\mathcal{E}) \circ c(Z' \to X', \mathcal{N}') $$ in $A^*(Z' \to X')^\wedge$.


chow-lemma-gysin-fundamental

In the situation described just above assume $\dim_\delta(Y) = n$ and that $\mathcal{C}_{Y \times_X Z/Z}$ has constant rank $r$. Then $$ c(Z \to X, \mathcal{N}) \cap [Y]_n = c_{top}(\mathcal{E}) \cap [Z \times_X Y]_{n - r} $$ in $\CH_*(Z \times_X Y)$.


chow-lemma-gysin-easy

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Let $Y \to X$ be a morphism which is locally of finite type. Given integers $r$, $n$ assume \begin{enumerate} \item $\mathcal{N}$ is locally free of rank $r$, \item every irreducible component of $Y$ has $\delta$-dimension $n$, \item $\dim_\delta(Z \times_X Y) \leq n - r$, and \item for $\xi \in Z \times_X Y$ with $\delta(\xi) = n - r$ the local ring $\mathcal{O}_{Y, \xi}$ is Cohen-Macaulay. \end{enumerate} Then $c(Z \to X, \mathcal{N}) \cap [Y]_n = [Z \times_X Y]_{n - r}$ in $\CH_{n - r}(Z \times_X Y)$.


chow-lemma-gysin-agrees

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. The gysin homomorphism $i^*$ viewed as an element of $A^1(D \to X)$ (see Lemma \ref{lemma-gysin-bivariant}) is the same as the bivariant class $c(D \to X, \mathcal{N}) \in A^1(D \to X)$ constructed using $\mathcal{N} = i^*\mathcal{L}$ viewed as a virtual normal sheaf for $D$ in $X$.


chow-lemma-gysin-commutes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme with virtual normal sheaf $\mathcal{N}$. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $c$ and $c(Z \to X, \mathcal{N})$ commute (Remark \ref{remark-bivariant-commute}).


chow-lemma-relation-normal-cones

With notation as above we have $$ o^*[C_ZX]_n = [C_Z Y]_{n - 1} $$ in $\CH_{n - 1}(Y \times_{o, C_Y X} C_ZX)$.


chow-lemma-gysin-composition

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $Z \subset Y \subset X$ be closed subschemes of a scheme locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal sheaf for $Z \subset X$. Let $\mathcal{N}'$ be a virtual normal sheaf for $Z \subset Y$. Let $\mathcal{N}''$ be a virtual normal sheaf for $Y \subset X$. Assume there is a commutative diagram $$ \xymatrix{ (\mathcal{N}'')^\vee|_Z \ar[r] \ar[d] & \mathcal{N}^\vee \ar[r] \ar[d] & (\mathcal{N}')^\vee \ar[d] \\ \mathcal{C}_{Y/X}|_Z \ar[r] & \mathcal{C}_{Z/X} \ar[r] & \mathcal{C}_{Z/Y} } $$ where the sequence at the bottom is from More on Morphisms, Lemma \ref{more-morphisms-lemma-transitivity-conormal} and the top sequence is a short exact sequence. Then $$ c(Z \to X, \mathcal{N}) = c(Z \to Y, \mathcal{N}') \circ c(Y \to X, \mathcal{N}'') $$ in $A^*(Z \to X)^\wedge$.


chow-lemma-compute-koszul

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$. Then $$ \prod\nolimits_{n = 0, \ldots, r} c(\wedge^n \mathcal{E})^{(-1)^n} = 1 - (r - 1)! c_r(\mathcal{E}) + \ldots $$


chow-lemma-agreement-with-loc-chern

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $i : Z \to X$ be a regular closed immersion of codimension $r$ between schemes locally of finite type over $S$. Let $\mathcal{N} = \mathcal{C}_{Z/X}^\vee$ be the normal sheaf. If $X$ is quasi-compact and has the resolution property, then $c_t(Z \to X, i_*\mathcal{O}_Z) = 0$ for $t = 1, \ldots, r - 1$ and $$ c_r(Z \to X, i_*\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N}) \quad\text{in}\quad A^r(Z \to X) $$ where $c_t(Z \to X, i_*\mathcal{O}_Z)$ is the localized Chern class of Definition \ref{definition-localized-chern}.


chow-lemma-actual-computation

In the situation of Lemma \ref{lemma-agreement-with-loc-chern} say $\dim_\delta(X) = n$. Then we have \begin{enumerate} \item $c_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0$ for $t = 1, \ldots, r - 1$, \item $c_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = (-1)^{r - 1}(r - 1)![Z]_{n - r}$, \item $ch_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0$ for $t = 0, \ldots, r - 1$, and \item $ch_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = [Z]_{n - r}$. \end{enumerate}


chow-lemma-second-adams-operator

Let $X$ be a scheme. There is a ring map $$ \psi^2 : K_0(\textit{Vect}(X)) \longrightarrow K_0(\textit{Vect}(X)) $$ which sends $[\mathcal{L}]$ to $[\mathcal{L}^{\otimes 2}]$ when $\mathcal{L}$ is invertible and is compatible with pullbacks.


chow-lemma-minus-adams-operator

Let $X$ be a scheme. There is a ring map $\psi^{-1} : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X))$ which sends $[\mathcal{E}]$ to $[\mathcal{E}^\vee]$ when $\mathcal{E}$ is finite locally free and is compatible with pullbacks.


chow-lemma-perf-Z-regular

Let $X$ be a Noetherian regular scheme of finite dimension. Let $Z \subset X$ be a closed subschemes. The maps constructed in Remarks \ref{remark-perf-Z-cohomology-K} and \ref{remark-perf-Z-regular} are mutually inverse and we get $K'_0(Z) = K_0(D_{Z, perf}(\mathcal{O}_X))$.


chow-lemma-K-tensor-Q

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a quasi-compact regular scheme of finite type over $S$ with affine diagonal and $\delta_{X/S} : X \to \mathbf{Z}$ bounded. Then the composition $$ K_0(\textit{Vect}(X)) \otimes \mathbf{Q} \longrightarrow A^*(X) \otimes \mathbf{Q} \longrightarrow \CH_*(X) \otimes \mathbf{Q} $$ of the map $ch$ from Remark \ref{remark-chern-character-K} and the map $c \mapsto c \cap [X]$ is an isomorphism.


chow-lemma-composition-regular-immersion

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $i : X \to Y$ and $j : Y \to Z$ be regular immersions of schemes locally of finite type over $S$. Then $j \circ i$ is a regular immersion and $(j \circ i)^! = i^! \circ j^!$.


chow-lemma-lci-gysin-flat

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. If the gysin map exists for $f$ and $f$ is flat, then $f^!$ is equal to the bivariant class of Lemma \ref{lemma-flat-pullback-bivariant}.


chow-lemma-lci-gysin-composition

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ and $g : Y \to Z$ be local complete intersection morphisms of schemes locally of finite type over $S$. Assume the gysin map exists for $g \circ f$ and $g$. Then the gysin map exists for $f$ and $(g \circ f)^! = f^! \circ g^!$.


chow-lemma-lci-gysin-commutes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider a commutative diagram $$ \xymatrix{ X'' \ar[d] \ar[r] & X' \ar[d] \ar[r] & X \ar[d]^f \\ Y'' \ar[r] & Y' \ar[r] & Y } $$ of schemes locally of finite type over $S$ with both square cartesian. Assume $f : X \to Y$ is a local complete intersection morphism such that the gysin map exists for $f$. Let $c \in A^*(Y'' \to Y')$. Denote $res(f^!) \in A^*(X' \to Y')$ the restriction of $f^!$ to $Y'$ (Remark \ref{remark-restriction-bivariant}). Then $c$ and $res(f^!)$ commute (Remark \ref{remark-bivariant-commute}).


chow-lemma-lci-gysin-easy

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider a cartesian diagram $$ \xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^f \\ Y' \ar[r] & Y } $$ of schemes locally of finite type over $S$. Assume \begin{enumerate} \item $f$ is a local complete intersection morphism and the gysin map exists for $f$, \item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of Lemma \ref{lemma-locally-equidimensional}, \item for $x' \in X'$ with images $x$, $y'$, and $y$ in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$ where $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and \item for every generic point $\xi \in X'$ the local ring $\mathcal{O}_{Y', f'(\xi)}$ is Cohen-Macaulay. \end{enumerate} Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in Remark \ref{remark-fundamental-class}.