Cycles of coherent sheaves and of closed subschemes, with flat pullback

Finiteness of the coefficient sum. Let $V$ be a $d$-dimensional integral closed subscheme of $X$ whose generic point lies in the support of $\mathcal{F}$. Put $R={\mathcal{O}}_{X,{\eta }_V}$ and $M={\mathcal{F}}_{{\eta }_V}$. The dimension bound makes $V$ a component of the support whenever $M\mathrm{\ne }0$: a strictly larger integral subvariety has strictly larger dimension on a finite type affine neighbourhood. Thus $M$ is finitely generated and its support in $\mathrm{Spec}R$ is only the maximal ideal $\mathfrak{m}$. Consequently $\sqrt{{\mathrm{Ann}}_R(M)}=\mathfrak{m}$. Choose finite generators of $\mathfrak{m}$; a power of each lies in the annihilator, so a sufficiently large power ${\mathfrak{m}}^N$ annihilates $M$. The finite filtration $M\supseteq \mathfrak{m}M\supseteq \cdots \supseteq {\mathfrak{m}}^{N}M=0$ has finite-dimensional residue-field quotients. Refining their finite vector-space filtrations proves finite length over $R$, without asserting that $R$ itself is a field. Hence every coefficient ${\mathrm{\ell }}_{{\mathcal{O}}_{X,{\eta }_V}}({\mathcal{F}}_{{\eta }_V})$ in the displayed sum is a well-defined nonnegative integer, and only the $d$-dimensional components of $\mathrm{Supp}\mathcal{F}$ carry nonzero coefficients, by [L4]. The family of $d$-dimensional components of a closed subset of a locally Noetherian scheme is locally finite, and finite when $X$ is quasi-compact, so the sum is locally finite in general and finite for finite type $X$; this defines $[\mathcal{F}{]}_d$ in the sense of [L1].

Additivity. Let $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of coherent sheaves, all supported in dimension at most $d$. For every $d$-dimensional integral closed subscheme $V$ with generic point ${\eta }_V$, localization at ${\eta }_V$ is exact, giving $0\to {\mathcal{F}}_{{\eta }_V}^{\mathrm{\prime }}\to {\mathcal{F}}_{{\eta }_V}\to {\mathcal{F}}_{{\eta }_V}^{\mathrm{\prime }\mathrm{\prime }}\to 0$; by [L3] the lengths add. Taking coefficients, $[\mathcal{F}{]}_d=[{\mathcal{F}}^{\mathrm{\prime }}{]}_d+[{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}{]}_d$. The common dimension bound is used here and cannot be dropped: on a smooth integral curve $C$ with a closed point $p$ the sequence $0\to {\mathcal{O}}_C(-p)\to {\mathcal{O}}_C\to {\mathcal{O}}_p\to 0$ has full-support cycles $[C],[C],[p]$, and $[C]\mathrm{\ne }[C]+[p]$ in $Z_{\ast }(C)$.

Fundamental cycles. For a closed subscheme $Y\subseteq X$, the stalk of ${\mathcal{O}}_Y$ at the generic point ${\eta }_Z$ of an irreducible component $Z$ of $Y$ is a $0$-dimensional Noetherian local ring, the local ring of $Y$ at its generic point, and its length is finite; setting the coefficient of $Z$ in $[Y]$ to that length makes $[Y]$ well defined by [L1]. If $Y$ is pure of dimension $d$, its components of dimension at most $d$ are exactly its irreducible components, so $[Y]=[{\mathcal{O}}_Y{]}_d$ by the definition of the latter and [L2]. If $Y$ is reduced, the local ring at the generic point of each component is a field, of length $1$, so every coefficient of $[Y]$ is one.

Flat pullback. Let $Y$ be pure of dimension $d$, and let $W$ be a component of $f^{-1}Y$ of dimension $d+r$ with generic point ${\eta }_W$; write $\zeta =f({\eta }_W)$. Because all fibres of $f$ have pure dimension $r$ by [L5], $\zeta$ is a generic point ${\eta }_Z$ of a component $Z$ of $Y$, the fibre of $f$ over $\zeta$ has pure dimension $r$, and ${\eta }_W$ is a generic point of that fibre. The local ring of the scheme-theoretic preimage at ${\eta }_W$ is ${\mathcal{O}}_{f^{-1}Y,{\eta }_W}\cong {\mathcal{O}}_{Y,{\eta }_Z}{\otimes }_{{\mathcal{O}}_{X,{\eta }_Z}}{\mathcal{O}}_{X^{\mathrm{\prime }},{\eta }_W}$: the preimage is $Y{\times }_X{X}^{\mathrm{\prime }}$ and since $Y\hookrightarrow X$ is closed, both sides are the quotient of ${\mathcal{O}}_{X^{\mathrm{\prime }},{\eta }_W}$ by the extended ideal of $Y$. Put $R={\mathcal{O}}_{X,{\eta }_Z}$ and $M={\mathcal{O}}_{Y,{\eta }_Z}$. The module $M$ has finite length by step 2.2 and therefore has a composition series whose factors are copies of $\kappa ({\eta }_Z)$, with length $m_Z={\mathrm{\ell }}_R(M)$. Tensoring this series with the flat $R$-algebra $S={\mathcal{O}}_{X^{\mathrm{\prime }},{\eta }_W}$ preserves exactness, and each factor contributes $S\mathrm{/}{\mathfrak{m}}_{Z}S$, which is the zero-dimensional Noetherian local ring of the fibre at its generic point. Its finite length $e_W={\mathrm{\ell }}_S(S\mathrm{/}{\mathfrak{m}}_{Z}S)$ is the generic multiplicity of $W$ in $[f^{-1}Z{]}_{d+r}$, and need not be one. By additivity of length, ${\mathrm{\ell }}_S({\mathcal{O}}_{f^{-1}Y,{\eta }_W})=m_Z{e}_W$. Thus this coefficient equals the coefficient of $W$ in $f^{\ast }[Y{]}_d={\sum }_Z{m}_Z[f^{-1}Z{]}_{d+r}$, where the sum is the linear extension of the assignment $[Z]\mapsto [f^{-1}Z{]}_{d+r}$ on integral cycles. Therefore $[f^{-1}Y{]}_{d+r}=f^{\ast }[Y{]}_d$.

Divisors. Let $Y$ be pure of dimension $d$ and let $D$ be an effective Cartier divisor whose local equation $t$ at each point of $Y$ is a nonzerodivisor on ${\mathcal{O}}_Y$. Then $D{\mathrm{\mid }}_Y$ is an effective Cartier divisor with local equation $t$, by [L5]. No irreducible component $Z$ of $Y$ is contained in $D$: otherwise $t$ would lie in the maximal ideal of ${\mathcal{O}}_{Y,{\eta }_Z}$ and be nilpotent there, hence a zerodivisor, contrary to the hypothesis. Consequently $Y\cap D=V(t)$ is pure of dimension $d-1$, and at the generic point $\eta$ of each of its components one has ${\mathcal{O}}_{Y\cap D,\eta }={\mathcal{O}}_{Y,\eta }\mathrm{/}t{\mathcal{O}}_{Y,\eta }$ with $t$ a nonzerodivisor, so this local ring has finite length and its length $\mathrm{\ell }({\mathcal{O}}_{Y,\eta }\mathrm{/}t{\mathcal{O}}_{Y,\eta })$ is the generic coefficient of the effective Cartier divisor $D{\mathrm{\mid }}_Y$. This uses the quotient length, without applying the domain order function to a possibly nonreduced local ring. Hence the cycle of $D{\mathrm{\mid }}_Y$ is the fundamental $(d-1)$-cycle of $Y\cap D$ as defined in step 2.2. The hypothesis cannot be weakened to $Y\subseteq ̸D$: for $X={\mathbb{A}}^2$, $Y=\mathrm{Spec}k[x,y]\mathrm{/}(y^2,xy)$, the $x$-axis with an embedded point at the origin, and $D=V(x)$, one has $Y\subseteq ̸D$ but $xy=0$ with $y\mathrm{\ne }0$ in ${\mathcal{O}}_Y$ shows that $x$ is a zerodivisor, so $D{\mathrm{\mid }}_Y$ is not an effective Cartier divisor at the embedded point.