chow-lemma-compare-gysin-base-change

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider a cartesian square $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes locally of finite type over $S$. Assume \begin{enumerate} \item both $f$ and $f'$ are local complete intersection morphisms, and \item the gysin map exists for $f$ \end{enumerate} Then $\mathcal{C} = \Ker(H^{-1}((g')^*\NL_{X/Y}) \to H^{-1}(\NL_{X'/Y'}))$ is a finite locally free $\mathcal{O}_{X'}$-module, the gysin map exists for $f'$, and we have $$ res(f^!) = c_{top}(\mathcal{C}^\vee) \circ (f')^! $$ in $A^*(X' \to Y')$.


chow-lemma-blow-up-formula

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $i : Z \to X$ be a regular closed immersion of schemes locally of finite type over $S$. Let $b : X' \to X$ be the blowing up with center $Z$. Picture $$ \xymatrix{ E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\ Z \ar[r]^i & X } $$ Assume that the gysin map exists for $b$. Then we have $$ res(b^!) = c_{top}(\mathcal{F}^\vee) \circ \pi^* $$ in $A^*(E \to Z)$ where $\mathcal{F}$ is the kernel of the canonical map $\pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$.


chow-lemma-diagonal-identity

In the situation above we have $\Delta^! \circ \text{pr}_i^! = 1$ in $A^0(X)$.


chow-lemma-chow-cohomology-towards-point

Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Then we have a canonical identification $$ A^p(X \to \Spec(k)) = \CH_{-p}(X) $$ for all $p \in \mathbf{Z}$.


chow-lemma-chow-cohomology-towards-point-commutes

Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $c \in A^p(X \to \Spec(k))$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \in A^q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times_k Y \to X \times_k Z)$.


chow-lemma-exterior-product-associative

Exterior product is associative. More precisely, let $k$ be a field, let $X, Y, Z$ be schemes locally of finite type over $k$, let $\alpha \in \CH_*(X)$, $\beta \in \CH_*(Y)$, $\gamma \in \CH_*(Z)$. Then $(\alpha \times \beta) \times \gamma = \alpha \times (\beta \times \gamma)$ in $\CH_*(X \times_k Y \times_k Z)$.


chow-lemma-associative

The product defined above is associative. More precisely, let $k$ be a field, let $X$ be smooth over $k$, let $Y, Z, W$ be schemes locally of finite type over $X$, let $\alpha \in \CH_*(Y)$, $\beta \in \CH_*(Z)$, $\gamma \in \CH_*(W)$. Then $(\alpha \cdot \beta) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma)$ in $\CH_*(Y \times_X Z \times_X W)$.


chow-lemma-identify-chow-for-smooth

Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional of dimension $d$. The map $$ A^p(X) \longrightarrow \CH_{d - p}(X),\quad c \longmapsto c \cap [X]_d $$ is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.


chow-lemma-lci-gysin-product

Let $k$ be a field. Let $f : X \to Y$ be a morphism of schemes smooth over $k$. Then the gysin map is defined for $f$ and $f^!(\alpha \cdot \beta) = f^!\alpha \cdot f^!\beta$.


chow-lemma-projection-formula

Let $k$ be a field. Let $f : X \to Y$ be a proper morphism of schemes smooth over $k$. Then the gysin map is defined for $f$ and $f_*(\alpha \cdot f^!\beta) = f_*\alpha \cdot \beta$.


chow-lemma-intersect-properly

Let $k$ be a field. Let $X$ be an integral scheme smooth over $k$. Let $Y, Z \subset X$ be integral closed subschemes. Set $d = \dim(Y) + \dim(Z) - \dim(X)$. Assume \begin{enumerate} \item $\dim(Y \cap Z) \leq d$, and \item $\mathcal{O}_{Y, \xi}$ and $\mathcal{O}_{Z, \xi}$ are Cohen-Macaulay for every $\xi \in Y \cap Z$ with $\delta(\xi) = d$. \end{enumerate} Then $[Y] \cdot [Z] = [Y \cap Z]_d$ in $\CH_d(X)$.


chow-lemma-intersect-regularly-embedded

Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \to X$ be a regular closed immersion. Let $\alpha \in \CH_*(X)$. If $Y$ is equidimensional of dimension $e$, then $\alpha \cdot [Y]_e = i_*(i^!(\alpha))$ in $\CH_*(X)$.


chow-lemma-exterior-product-well-defined-dim-1

The map $\times : \CH_n(X) \otimes_{\mathbf{Z}} \CH_m(Y) \to \CH_{n + m - 1}(X \times_S Y)$ is well defined.


chow-lemma-chow-cohomology-towards-base-dim-1

Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Then we have a canonical identification $$ A^p(X \to S) = \CH_{1 - p}(X) $$ for all $p \in \mathbf{Z}$.


chow-lemma-chow-cohomology-towards-base-dim-1-commutes

Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $c \in A^p(X \to S)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \in A^q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times_S Y \to X \times_S Z)$.


chow-lemma-exterior-product-associative-dim-1

Exterior product is associative. More precisely, let $(S, \delta)$ be as above, let $X, Y, Z$ be schemes locally of finite type over $S$, let $\alpha \in \CH_*(X)$, $\beta \in \CH_*(Y)$, $\gamma \in \CH_*(Z)$. Then $(\alpha \times \beta) \times \gamma = \alpha \times (\beta \times \gamma)$ in $\CH_*(X \times_S Y \times_S Z)$.


chow-lemma-dimension-base-change

In Situation \ref{situation-setup-base-change} let $X \to S$ be locally of finite type. Denote $X' \to S'$ the base change by $S' \to S$. If $X$ is integral with $\dim_\delta(X) = k$, then every irreducible component $Z'$ of $X'$ has $\dim_{\delta'}(Z') = k + c$,


chow-lemma-pullback-coherent-base-change

In Situation \ref{situation-setup-base-change} let $X \to S$ locally of finite type and let $X' \to S$ be the base change by $S' \to S$. \begin{enumerate} \item Let $Z \subset X$ be a closed subscheme with $\dim_\delta(Z) \leq k$ and base change $Z' \subset X'$. Then we have $\dim_{\delta'}(Z')) \leq k + c$ and $[Z']_{k + c} = g^*[Z]_k$ in $Z_{k + c}(X')$. \item Let $\mathcal{F}$ be a coherent sheaf on $X$ with $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$ and base change $\mathcal{F}'$ on $X'$. Then we have $\dim_\delta(\text{Supp}(\mathcal{F}')) \leq k + c$ and $g^*[\mathcal{F}]_k = [\mathcal{F}']_{k + c}$ in $Z_{k + c}(X')$. \end{enumerate}


chow-lemma-pullback-base-change

In Situation \ref{situation-setup-base-change} let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. The map $g^* : Z_k(X) \to Z_{k + c}(X')$ above factors through rational equivalence to give a map $$ g^* : \CH_k(X) \longrightarrow \CH_{k + c}(X') $$ of chow groups.


chow-lemma-pullback-base-change-pullback

In Situation \ref{situation-setup-base-change} let $Y \to X \to S$ be locally of finite type and let $Y' \to X' \to S'$ be the base change by $S' \to S$. Assume $f : Y \to X$ is flat of relative dimension $r$. Then $f' : Y' \to X'$ is flat of relative dimension $r$ and the diagram $$ \xymatrix{ \CH_{k + r}(Y) \ar[r]_{g^*} & \CH_{k + c + r}(Y') \\ \CH_k(X) \ar[r]^{g^*} \ar[u]^{(f')^*} & \CH_{k + c}(X') \ar[u]_{f^*} } $$ of chow groups commutes.


chow-lemma-pullback-base-change-pushforward

In Situation \ref{situation-setup-base-change} let $Y \to X \to S$ be locally of finite type and let $Y' \to X' \to S'$ be the base change by $S' \to S$. Assume $f : Y \to X$ is proper. Then $f' : Y' \to X'$ is proper and the diagram $$ \xymatrix{ \CH_k(Y) \ar[r]_{g^*} \ar[d]_{f_*} & \CH_{k + c}(Y') \ar[d]^{f'_*} \\ \CH_k(X) \ar[r]^{g^*} & \CH_{k + c}(X') } $$ of chow groups commutes.


chow-lemma-pullback-base-change-c1

In Situation \ref{situation-setup-base-change} let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module with base change $\mathcal{L}'$ on $X'$. Then the diagram $$ \xymatrix{ \CH_k(X) \ar[r]_{g^*} \ar[d]_{c_1(\mathcal{L}) \cap -} & \CH_{k + c}(X') \ar[d]^{c_1(\mathcal{L}') \cap -} \\ \CH_{k - 1}(X) \ar[r]^{g^*} & \CH_{k + c - 1}(X') } $$ of chow groups commutes.


chow-lemma-pullback-base-change-chern-classes

In Situation \ref{situation-setup-base-change} let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module of rank $r$ with base change $\mathcal{E}'$ on $X'$. Then the diagram $$ \xymatrix{ \CH_k(X) \ar[r]_{g^*} \ar[d]_{c_i(\mathcal{E}) \cap -} & \CH_{k + c}(X') \ar[d]^{c_i(\mathcal{E}') \cap -} \\ \CH_{k - i}(X) \ar[r]^{g^*} & \CH_{k + c - i}(X') } $$ of chow groups commutes for all $i$.


chow-lemma-compose-base-change

Let $(S, \delta)$, $(S', \delta')$, $(S'', \delta'')$ be as in Situation \ref{situation-setup}. Let $g : S' \to S$ and $g' : S'' \to S'$ be flat morphisms of schemes and let $c, c' \in \mathbf{Z}$ be integers such that $S, \delta, S', \delta', g, c$ and $S', \delta', S'', g', c'$ are as in Situation \ref{situation-setup-base-change}. Let $X \to S$ be locally of finite type and denote $X' \to S'$ and $X'' \to S''$ the base changes by $S' \to S$ and $S'' \to S$. Then $S, \delta, S'', \delta'', g \circ g', c + c'$ is as in Situation \ref{situation-setup-base-change} and the maps $g^* : \CH_k(X) \to \CH_{k + c}(X')$ and $(g')^* : \CH_{k + c}(X') \to \CH_{k + c + c'}(X'')$ of Lemma \ref{lemma-pullback-base-change} compose to give the map $(g \circ g')^* : \CH_k(X) \to \CH_{k + c + c'}(X'')$ of Lemma \ref{lemma-pullback-base-change}.


chow-lemma-chow-limit

In Situation \ref{situation-setup-base-change} assume $c = 0$ and assume that $S' = \lim_{i \in I} S_i$ is a filtered limit of schemes $S_i$ affine over $S$ such that \begin{enumerate} \item with $\delta_i$ equal to $S_i \to S \xrightarrow{\delta} \mathbf{Z}$ the pair $(S_i, \delta_i)$ is as in Situation \ref{situation-setup}, \item $S_i, \delta_i, S, \delta, S \to S_i, c = 0$ is as in Situation \ref{situation-setup-base-change}, \item $S_i, \delta_i, S_{i'}, \delta_{i'}, S_i \to S_{i'}, c = 0$  for $i \geq i'$ is as in Situation \ref{situation-setup-base-change}. \end{enumerate} Then for a quasi-compact scheme $X$ of finite type over $S$ with base change $X'$ and $X_i$ by $S' \to S$ and $S_i \to S$ we have $\CH_k(X') = \colim \CH_k(X_i)$.


chow-lemma-dimension-at-most-one

With notations as above we have $\dim_\kappa(\det_\kappa(M)) \leq 1$.


chow-lemma-compare-det

Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa$. Let $M$ be a finite length $R$-module which is annihilated by $\mathfrak m$. Let $l = \dim_\kappa(M)$. Then the map $$ \det\nolimits_\kappa(M) \longrightarrow \wedge^l_\kappa(M), \quad [e_1, \ldots, e_l] \longmapsto e_1 \wedge \ldots \wedge e_l $$ is an isomorphism.


chow-lemma-determinant-dimension-one

Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa$. Let $M$ be a finite length $R$-module. The determinant $\det_\kappa(M)$ defined above is a $\kappa$-vector space of dimension $1$. It is generated by the symbol $[f_1, \ldots, f_l]$ for any admissible sequence such that $\langle f_1, \ldots f_l \rangle = M$.


chow-lemma-det-exact-sequences

Let $(R, \mathfrak m, \kappa)$ be a local ring. For every short exact sequence $$ 0 \to K \to L \to M \to 0 $$ of finite length $R$-modules there exists a canonical isomorphism $$ \gamma_{K \to L \to M} : \det\nolimits_\kappa(K) \otimes_\kappa \det\nolimits_\kappa(M) \longrightarrow \det\nolimits_\kappa(L) $$ defined by the rule on nonzero symbols $$ [e_1, \ldots, e_k] \otimes [\overline{f}_1, \ldots, \overline{f}_m] \longrightarrow [e_1, \ldots, e_k, f_1, \ldots, f_m] $$ with the following properties: \begin{enumerate} \item For every isomorphism of short exact sequences, i.e., for every commutative diagram $$ \xymatrix{ 0 \ar[r] & K \ar[r] \ar[d]^u & L \ar[r] \ar[d]^v & M \ar[r] \ar[d]^w & 0 \\ 0 \ar[r] & K' \ar[r] & L' \ar[r] & M' \ar[r] & 0 } $$ with short exact rows and isomorphisms $u, v, w$ we have $$ \gamma_{K' \to L' \to M'} \circ (\det\nolimits_\kappa(u) \otimes \det\nolimits_\kappa(w)) = \det\nolimits_\kappa(v) \circ \gamma_{K \to L \to M}, $$ \item for every commutative square of finite length $R$-modules with exact rows and columns $$ \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar[r] & A \ar[r] \ar[d] & B \ar[r] \ar[d] & C \ar[r] \ar[d] & 0 \\ 0 \ar[r] & D \ar[r] \ar[d] & E \ar[r] \ar[d] & F \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G \ar[r] \ar[d] & H \ar[r] \ar[d] & I \ar[r] \ar[d] & 0 \\ & 0  & 0  & 0  & } $$ the following diagram is commutative $$ \xymatrix{ \det\nolimits_\kappa(A) \otimes \det\nolimits_\kappa(C) \otimes \det\nolimits_\kappa(G) \otimes \det\nolimits_\kappa(I) \ar[dd]_{\epsilon} \ar[rrr]_-{\gamma_{A \to B \to C} \otimes \gamma_{G \to H \to I}} & & & \det\nolimits_\kappa(B) \otimes \det\nolimits_\kappa(H) \ar[d]^{\gamma_{B \to E \to H}} \\ & & & \det\nolimits_\kappa(E) \\ \det\nolimits_\kappa(A) \otimes \det\nolimits_\kappa(G) \otimes \det\nolimits_\kappa(C) \otimes \det\nolimits_\kappa(I) \ar[rrr]^-{\gamma_{A \to D \to G} \otimes \gamma_{C \to F \to I}} & & & \det\nolimits_\kappa(D) \otimes \det\nolimits_\kappa(F) \ar[u]_{\gamma_{D \to E \to F}} } $$ where $\epsilon$ is the switch of the factors in the tensor product times $(-1)^{cg}$ with $c = \text{length}_R(C)$ and $g = \text{length}_R(G)$, and \item the map $\gamma_{K \to L \to M}$ agrees with the usual isomorphism if $0 \to K \to L \to M \to 0$ is actually a short exact sequence of $\kappa$-vector spaces. \end{enumerate}


chow-lemma-uniqueness-det

Let $(R, \mathfrak m, \kappa)$ be any local ring. The functor $$ \det\nolimits_\kappa : \left\{ \begin{matrix} \text{finite length }R\text{-modules} \\ \text{with isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} 1\text{-dimensional }\kappa\text{-vector spaces} \\ \text{with isomorphisms} \end{matrix} \right\} $$ endowed with the maps $\gamma_{K \to L \to M}$ is characterized by the following properties \begin{enumerate} \item its restriction to the subcategory of modules annihilated by $\mathfrak m$ is isomorphic to the usual determinant functor (see Lemma \ref{lemma-compare-det}), and \item (1), (2) and (3) of Lemma \ref{lemma-det-exact-sequences} hold. \end{enumerate}


chow-lemma-determinant-quotient-ring

Let $(R', \mathfrak m') \to (R, \mathfrak m)$ be a local ring homomorphism which induces an isomorphism on residue fields $\kappa$. Then for every finite length $R$-module the restriction $M_{R'}$ is a finite length $R'$-module and there is a canonical isomorphism $$ \det\nolimits_{R, \kappa}(M) \longrightarrow \det\nolimits_{R', \kappa}(M_{R'}) $$ This isomorphism is functorial in $M$ and compatible with the isomorphisms $\gamma_{K \to L \to M}$ of Lemma \ref{lemma-det-exact-sequences} defined for $\det_{R, \kappa}$ and $\det_{R', \kappa}$.


chow-lemma-times-u-determinant

Let $R$ be a local ring with residue field $\kappa$. Let $u \in R^*$ be a unit. Let $M$ be a module of finite length over $R$. Denote $u_M : M \to M$ the map multiplication by $u$. Then $$ \det\nolimits_\kappa(u_M) : \det\nolimits_\kappa(M) \longrightarrow \det\nolimits_\kappa(M) $$ is multiplication by $\overline{u}^l$ where $l = \text{length}_R(M)$ and $\overline{u} \in \kappa^*$ is the image of $u$.


chow-lemma-periodic-determinant-shift

Let $R$ be a local ring with residue field $\kappa$. Let $(M, \varphi, \psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \varphi, \psi)$ is exact. Then $$ \det\nolimits_\kappa(M, \varphi, \psi) \det\nolimits_\kappa(M, \psi, \varphi) = 1. $$


chow-lemma-periodic-determinant-sign

Let $R$ be a local ring with residue field $\kappa$. Let $(M, \varphi, \varphi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \varphi, \varphi)$ is exact. Then $\text{length}_R(M) = 2 \text{length}_R(\Im(\varphi))$ and $$ \det\nolimits_\kappa(M, \varphi, \varphi) = (-1)^{\text{length}_R(\Im(\varphi))} = (-1)^{\frac{1}{2}\text{length}_R(M)} $$


chow-lemma-periodic-determinant-easy-case

Let $R$ be a local ring with residue field $\kappa$. Let $M$ be a finite length $R$-module. \begin{enumerate} \item if $\varphi : M \to M$ is an isomorphism then $\det_\kappa(M, \varphi, 0) = \det_\kappa(\varphi)$. \item if $\psi : M \to M$ is an isomorphism then $\det_\kappa(M, 0, \psi) = \det_\kappa(\psi)^{-1}$. \end{enumerate}


chow-lemma-periodic-determinant

Let $R$ be a local ring with residue field $\kappa$. Suppose that we have a short exact sequence of $(2, 1)$-periodic complexes $$ 0 \to (M_1, \varphi_1, \psi_1) \to (M_2, \varphi_2, \psi_2) \to (M_3, \varphi_3, \psi_3) \to 0 $$ with all $M_i$ of finite length, and each $(M_1, \varphi_1, \psi_1)$ exact. Then $$ \det\nolimits_\kappa(M_2, \varphi_2, \psi_2) = \det\nolimits_\kappa(M_1, \varphi_1, \psi_1) \det\nolimits_\kappa(M_3, \varphi_3, \psi_3). $$ in $\kappa^*$.


chow-lemma-multiplicativity-determinant

Let $R$ be a local ring with residue field $\kappa$. Let $M$ be a finite length $R$-module. Let $\alpha, \beta, \gamma$ be endomorphisms of $M$. Assume that \begin{enumerate} \item $I_\alpha = K_{\beta\gamma}$, and similarly for any permutation of $\alpha, \beta, \gamma$, \item $K_\alpha = I_{\beta\gamma}$, and similarly for any permutation of $\alpha, \beta, \gamma$. \end{enumerate} Then \begin{enumerate} \item The triple $(M, \alpha, \beta\gamma)$ is an exact $(2, 1)$-periodic complex. \item The triple $(I_\gamma, \alpha, \beta)$ is an exact $(2, 1)$-periodic complex. \item The triple $(M/K_\beta, \alpha, \gamma)$ is an exact $(2, 1)$-periodic complex. \item We have $$ \det\nolimits_\kappa(M, \alpha, \beta\gamma) = \det\nolimits_\kappa(I_\gamma, \alpha, \beta) \det\nolimits_\kappa(M/K_\beta, \alpha, \gamma). $$ \end{enumerate}


chow-lemma-tricky

Let $R$ be a local ring with residue field $\kappa$. Let $\alpha : (M, \varphi, \psi) \to (M', \varphi', \psi')$ be a morphism of $(2, 1)$-periodic complexes over $R$. Assume \begin{enumerate} \item $M$, $M'$ have finite length, \item $(M, \varphi, \psi)$, $(M', \varphi', \psi')$ are exact, \item the maps $\varphi$, $\psi$ induce the zero map on $K = \Ker(\alpha)$, and \item the maps $\varphi$, $\psi$ induce the zero map on $Q = \Coker(\alpha)$. \end{enumerate} Denote $N = \alpha(M) \subset M'$. We obtain two short exact sequences of $(2, 1)$-periodic complexes $$ \begin{matrix} 0 \to (N, \varphi', \psi') \to (M', \varphi', \psi') \to (Q, 0, 0) \to 0 \\ 0 \to (K, 0, 0) \to (M, \varphi, \psi) \to (N, \varphi', \psi') \to 0 \end{matrix} $$ which induce two isomorphisms $\alpha_i : Q \to K$, $i = 0, 1$. Then $$ \det\nolimits_\kappa(M, \varphi, \psi) = \det\nolimits_\kappa(\alpha_0^{-1} \circ \alpha_1) \det\nolimits_\kappa(M', \varphi', \psi') $$ In particular, if $\alpha_0 = \alpha_1$, then $\det\nolimits_\kappa(M, \varphi, \psi) = \det\nolimits_\kappa(M', \varphi', \psi')$.


chow-lemma-pre-symbol

Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Assume $\varphi, \psi : M \to M$ are two injective $A$-module maps, and assume $\varphi(\psi(M)) = \psi(\varphi(M))$, for example if $\varphi$ and $\psi$ commute. Then $\text{length}_R(M/\varphi\psi M) < \infty$ and $(M/\varphi\psi M, \varphi, \psi)$ is an exact $(2, 1)$-periodic complex.


chow-lemma-multiplicativity-symbol

Let $A$ be a Noetherian local ring. Let $a, b, c \in A$. Let $M$ be a finite $A$-module with $\dim(\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$. Then $$ d_M(a, bc) = d_M(a, b) d_M(a, c) $$ and $d_M(a, b)d_M(b, a) = 1$.


chow-lemma-symbol-when-equal

Let $A$ be a Noetherian local ring and $M$ a finite $A$-module of dimension $1$. Let $a \in A$ be a nonzerodivisor on $M$. Then $d_M(a, a) = (-1)^{\text{length}_A(M/aM)}$.


chow-lemma-symbol-when-one-is-a-unit

Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Let $b \in A$ be a nonzerodivisor on $M$, and let $u \in A^*$. Then $$ d_M(u, b) = u^{\text{length}_A(M/bM)} \bmod \mathfrak m_A. $$ In particular, if $M = A$, then $d_A(u, b) = u^{\text{ord}_A(b)} \bmod \mathfrak m_A$.


chow-lemma-symbol-short-exact-sequence

Let $A$ be a Noetherian local ring. Let $a, b \in A$. Let $$ 0 \to M \to M' \to M'' \to 0 $$ be a short exact sequence of $A$-modules of dimension $1$ such that $a, b$ are nonzerodivisors on all three $A$-modules. Then $$ d_{M'}(a, b) = d_M(a, b) d_{M''}(a, b) $$ in $\kappa^*$.


chow-lemma-symbol-compare-modules

Let $A$ be a Noetherian local ring. Let $\alpha : M \to M'$ be a homomorphism of finite $A$-modules of dimension $1$. Let $a, b \in A$. Assume \begin{enumerate} \item $a$, $b$ are nonzerodivisors on both $M$ and $M'$, and \item $\dim(\Ker(\alpha)), \dim(\Coker(\alpha)) \leq 0$. \end{enumerate} Then $d_M(a, b) = d_{M'}(a, b)$.


chow-lemma-compute-symbol-M

Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module with $\dim(\text{Supp}(M)) = 1$. Let $a, b \in A$ nonzerodivisors on $M$. Let $\mathfrak q_1, \ldots, \mathfrak q_t$ be the minimal primes in the support of $M$. Then $$ d_M(a, b) = \prod\nolimits_{i = 1, \ldots, t} d_{A/\mathfrak q_i}(a, b)^{ \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i})} $$ as elements of $\kappa^*$.


chow-lemma-symbol-is-steinberg-prepare

Let $A$ be a Noetherian local ring. Let $a, b \in A$. Let $M$ be a finite $A$-module of dimension $1$ on which each of $a$, $b$, $b - a$ are nonzerodivisors. Then $$ d_M(a, b - a)d_M(b, b) = d_M(b, b - a)d_M(a, b) $$ in $\kappa^*$.


chow-lemma-key-lemma

Let $R$ be a Noetherian local ring. Let $\mathfrak q \subset R$ be a prime with $\dim(R/\mathfrak q) = 1$. Let $\varphi : M \to N$ be a homomorphism of finite $R$-modules. Assume there exist $x_1, \ldots, x_l \in M$ and $y_1, \ldots, y_l \in M$ with the following properties \begin{enumerate} \item $M = \langle x_1, \ldots, x_l\rangle$, \item $\langle x_1, \ldots, x_i\rangle / \langle x_1, \ldots, x_{i - 1}\rangle \cong R/\mathfrak q$ for $i = 1, \ldots, l$, \item $N = \langle y_1, \ldots, y_l\rangle$, and \item $\langle y_1, \ldots, y_i\rangle / \langle y_1, \ldots, y_{i - 1}\rangle \cong R/\mathfrak q$ for $i = 1, \ldots, l$. \end{enumerate} Then $\varphi$ is injective if and only if $\varphi_{\mathfrak q}$ is an isomorphism, and in this case we have $$ \text{length}_R(\Coker(\varphi)) = \text{ord}_{R/\mathfrak q}(f) $$ where $f \in \kappa(\mathfrak q)$ is the element such that $$ [\varphi(x_1), \ldots, \varphi(x_l)] = f [y_1, \ldots, y_l] $$ in $\det_{\kappa(\mathfrak q)}(N_{\mathfrak q})$.


chow-lemma-good-sequence-exists

Let $R$ be a local Noetherian ring. Let $\mathfrak q \subset R$ be a prime ideal. Let $M$ be a finite $R$-module such that $\mathfrak q$ is one of the minimal primes of the support of $M$. Then there exist $x_1, \ldots, x_l \in M$ such that \begin{enumerate} \item the support of $M / \langle x_1, \ldots, x_l\rangle$ does not contain $\mathfrak q$, and \item $\langle x_1, \ldots, x_i\rangle / \langle x_1, \ldots, x_{i - 1}\rangle \cong R/\mathfrak q$ for $i = 1, \ldots, l$. \end{enumerate} Moreover, in this case $l = \text{length}_{R_\mathfrak q}(M_\mathfrak q)$.


chow-lemma-maps-between-coherent-sheaves

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $$ \xymatrix{ \ldots \ar[r] & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r]^\psi & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r] & \ldots } $$ be a complex as in Homology, Equation (\ref{homology-equation-cyclic-complex}). Assume that \begin{enumerate} \item $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k + 1$. \item $\dim_\delta(\text{Supp}(H^i(\mathcal{F}, \varphi, \psi))) \leq k$ for $i = 0, 1$. \end{enumerate} Then we have $$ [H^0(\mathcal{F}, \varphi, \psi)]_k \sim_{rat} [H^1(\mathcal{F}, \varphi, \psi)]_k $$ as $k$-cycles on $X$.


chow-lemma-cycles-rational-equivalence-K-group

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The map $$ \CH_k(X) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) $$ from Lemma \ref{lemma-from-chow-to-K} induces a bijection from $\CH_k(X)$ onto the image $B_k(X)$ of the map $$ K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)). $$


chow-lemma-no-embedded-points-modules

Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $a, b \in A$. Assume \begin{enumerate} \item $\dim(A) = 1$, \item both $a$ and $b$ are nonzerodivisors in $A$, \item $A$ has no embedded primes, \item $M$ has no embedded associated primes, \item $\text{Supp}(M) = \Spec(A)$. \end{enumerate} Let $I = \{x \in A \mid x(a/b) \in A\}$. Let $\mathfrak q_1, \ldots, \mathfrak q_t$ be the minimal primes of $A$. Then $(a/b)IM \subset M$ and $$ \text{length}_A(M/(a/b)IM) - \text{length}_A(M/IM) = \sum\nolimits_i \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i}) \text{ord}_{A/\mathfrak q_i}(a/b) $$


chow-lemma-no-embedded-points

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 $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{K}_X(\mathcal{L}))$ be a meromorphic section of $\mathcal{L}$. Assume \begin{enumerate} \item $\dim_\delta(X) \leq k + 1$, \item $X$ has no embedded points, \item $\mathcal{F}$ has no embedded associated points, \item the support of $\mathcal{F}$ is $X$, and \item the section $s$ is regular meromorphic. \end{enumerate} In this situation let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal of denominators of $s$, see Divisors, Definition \ref{divisors-definition-regular-meromorphic-ideal-denominators}. Then we have the following: \begin{enumerate} \item there are short exact sequences $$ \begin{matrix} 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow{1} & \mathcal{F} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow{s} & \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix} $$ \item the coherent sheaves $\mathcal{Q}_1$, $\mathcal{Q}_2$ are supported in $\delta$-dimension $\leq k$, \item the section $s$ restricts to a regular meromorphic section $s_i$ on every irreducible component $X_i$ of $X$ of $\delta$-dimension $k + 1$, and \item writing $[\mathcal{F}]_{k + 1} = \sum m_i[X_i]$ we have $$ [\mathcal{Q}_2]_k - [\mathcal{Q}_1]_k = \sum m_i(X_i \to X)_*\text{div}_{\mathcal{L}|_{X_i}}(s_i) $$ in $Z_k(X)$, in particular $$ [\mathcal{Q}_2]_k - [\mathcal{Q}_1]_k = c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1} $$ in $\CH_k(X)$. \end{enumerate}


chow-proposition-K-tensor-Q

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Assume given a closed immersion $X \to Y$ of schemes locally of finite type over $S$ with $Y$ regular, quasi-compact, affine diagonal, and $\delta_{Y/S} : Y \to \mathbf{Z}$ bounded. Then the composition $$ K'_0(X) \to K_0(D_{X, perf}(\mathcal{O}_Y)) \to A^*(X \to Y) \to \CH_*(X) $$ of the map $\mathcal{F} \mapsto \mathcal{F}[0]$ from Remark \ref{remark-perf-Z-regular}, the map $ch(X \to Y, -)$ from Remark \ref{remark-localized-chern-character-K}, and the map $c \mapsto c \cap [Y]$ induces an isomorphism $$ K'_0(X) \otimes \mathbf{Q} \longrightarrow \CH_*(X) \otimes \mathbf{Q} $$ which depends on the choice of $Y$. Moreover, the canonical map $$ \CH_k(X) \otimes \mathbf{Q} \longrightarrow \text{gr}_k K'_0(X) \otimes \mathbf{Q} $$ (see above) is an isomorphism of $\mathbf{Q}$-vector spaces for all $k \in \mathbf{Z}$.


chow-proposition-compute-bivariant

\begin{reference} \cite[Proposition 17.4.2]{F} \end{reference} 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 $S$. If $g$ is smooth of relative dimension $d$, then $A^p(X \to Y) = A^{p - d}(X \to Z)$.


chow-proposition-length-determinant-periodic-complex

Let $R$ be a local Noetherian ring with residue field $\kappa$. Suppose that $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex over $R$. Assume \begin{enumerate} \item $M$ is a finite $R$-module, \item the cohomology modules of $(M, \varphi, \psi)$ are of finite length, and \item $\dim(\text{Supp}(M)) = 1$. \end{enumerate} Let $\mathfrak q_i$, $i = 1, \ldots, t$ be the minimal primes of the support of $M$. Then we have\footnote{ Obviously we could get rid of the minus sign by redefining $\det_\kappa(M, \varphi, \psi)$ as the inverse of its current value, see Definition \ref{definition-periodic-determinant}.} $$ - e_R(M, \varphi, \psi) = \sum\nolimits_{i = 1, \ldots, t} \text{ord}_{R/\mathfrak q_i}\left( \det\nolimits_{\kappa(\mathfrak q_i)} (M_{\mathfrak q_i}, \varphi_{\mathfrak q_i}, \psi_{\mathfrak q_i}) \right) $$


flat-theorem-finite-type-flat

\begin{slogan} The flat locus is open (non-Noetherian version). \end{slogan} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X \to S$ is locally of finite presentation, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type, and \item the set of weakly associated points of $S$ is locally finite in $S$. \end{enumerate} Then $U = \{x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\}$ is open in $X$ and $\mathcal{F}|_U$ is an $\mathcal{O}_U$-module of finite presentation and flat over $S$.


flat-theorem-flattening-map

In Situation \ref{situation-iso} assume \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite presentation, flat over $S$, and pure relative to $S$, and \item $u$ is surjective. \end{enumerate} Then $F_{iso}$ is representable by a closed immersion $Z \to S$. Moreover $Z \to S$ is of finite presentation if $\mathcal{G}$ is of finite presentation.


flat-theorem-flattening-local

In Situation \ref{situation-flat-at-point} assume $A$ is henselian, $B$ is essentially of finite type over $A$, and $M$ is a finite $B$-module. Then there exists an ideal $I \subset A$ such that $A/I$ corepresents the functor $F_{lf}$ on the category $\mathcal{C}$. In other words given a local homomorphism of local rings $\varphi : A \to A'$ with $B' = B \otimes_A A'$ and $M' = M \otimes_A A'$ the following are equivalent: \begin{enumerate} \item $\forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_B B') \subset \Spec(B') : M'_{\mathfrak q}\text{ is flat over }A'$, and \item $\varphi(I) = 0$. \end{enumerate} If $B$ is essentially of finite presentation over $A$ and $M$ of finite presentation over $B$, then $I$ is a finitely generated ideal.


flat-theorem-check-flatness-at-associated-points

Let $f : X \to S$ be locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $x \in X$ with image $s \in S$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is flat at $x$ over $S$, and \item for every $x' \in \text{Ass}_{X_s}(\mathcal{F}_s)$ which specializes to $x$ we have that $\mathcal{F}$ is flat at $x'$ over $S$. \end{enumerate}


flat-theorem-flat-dimension-n-representable

In Situation \ref{situation-flat-dimension-n}. Assume moreover that $f$ is of finite presentation, that $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and that $\mathcal{F}$ is pure relative to $S$. Then $F_n$ is representable by a monomorphism $Z_n \to S$ of finite presentation.


flat-theorem-flatten-module

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume \begin{enumerate} \item $X$ is quasi-compact, \item $X$ is locally of finite presentation over $S$, \item $\mathcal{F}$ is a module of finite type, \item $\mathcal{F}_U$ is of finite presentation, and \item $\mathcal{F}_U$ is flat over $U$. \end{enumerate} Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is an $\mathcal{O}_{X \times_S S'}$-module of finite presentation and flat over $S'$.


flat-theorem-nagata

\begin{reference} See \cite{Lutkebohmert}, \cite{Conrad-Nagata}, \cite{Nagata-1}, \cite{Nagata-2}, \cite{Nagata-3}, and \cite{Nagata-4} \end{reference} Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a separated, finite type morphism. Then $X$ has a compactification over $S$.


flat-lemma-lift-etale

Let $i : Z \to X$ be a closed immersion of affine schemes. Let $Z' \to Z$ be an \'etale morphism with $Z'$ affine. Then there exists an \'etale morphism $X' \to X$ with $X'$ affine such that $Z' \cong Z \times_X X'$ as schemes over $Z$.


flat-lemma-etale-at-point

Let $$ \xymatrix{ X \ar[d] & X' \ar[l] \ar[d] \\ S & S' \ar[l] } $$ be a commutative diagram of schemes with $X' \to X$ and $S' \to S$ \'etale. Let $s' \in S'$ be a point. Then $$ X' \times_{S'} \Spec(\mathcal{O}_{S', s'}) \longrightarrow X \times_S \Spec(\mathcal{O}_{S', s'}) $$ is \'etale.


flat-lemma-etale-flat-up-down

Let $X \to T \to S$ be morphisms of schemes with $T \to S$ \'etale. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$ be a point. Then $$ \mathcal{F}\text{ flat over }S\text{ at }x \Leftrightarrow \mathcal{F}\text{ flat over }T\text{ at }x $$ In particular $\mathcal{F}$ is flat over $S$ if and only if $\mathcal{F}$ is flat over $T$.


flat-lemma-etale-flat-up-down-local-ring

Let $T \to S$ be an \'etale morphism. Let $t \in T$ with image $s \in S$. Let $M$ be a $\mathcal{O}_{T, t}$-module. Then $$ M\text{ flat over }\mathcal{O}_{S, s} \Leftrightarrow M\text{ flat over }\mathcal{O}_{T, t}. $$


flat-lemma-flat-up-down-henselization

Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^h$ (resp.\ $\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\ strict henselization), see Algebra, Definition \ref{algebra-definition-henselization}. Let $M^{sh}$ be a $\mathcal{O}_{S, s}^{sh}$-module. The following are equivalent \begin{enumerate} \item $M^{sh}$ is flat over $\mathcal{O}_{S, s}$, \item $M^{sh}$ is flat over $\mathcal{O}_{S, s}^h$, and \item $M^{sh}$ is flat over $\mathcal{O}_{S, s}^{sh}$. \end{enumerate} If $M^{sh} = M^h \otimes_{\mathcal{O}_{S, s}^h} \mathcal{O}_{S, s}^{sh}$ this is also equivalent to \begin{enumerate} \item[(4)] $M^h$ is flat over $\mathcal{O}_{S, s}$, and \item[(5)] $M^h$ is flat over $\mathcal{O}_{S, s}^h$. \end{enumerate} If $M^h = M \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^h$ this is also equivalent to \begin{enumerate} \item[(6)] $M$ is flat over $\mathcal{O}_{S, s}$. \end{enumerate}


flat-lemma-tor-amplitude-up-down-henselization

Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^h$ (resp.\ $\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\ strict henselization), see Algebra, Definition \ref{algebra-definition-henselization}. Let $M^{sh}$ be an object of $D(\mathcal{O}_{S, s}^{sh})$. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$, \item $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^h$, and \item $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^{sh}$. \end{enumerate} If $M^{sh} = M^h \otimes_{\mathcal{O}_{S, s}^h}^\mathbf{L} \mathcal{O}_{S, s}^{sh}$ for $M^h \in D(\mathcal{O}_{S, s}^h)$ this is also equivalent to \begin{enumerate} \item[(4)] $M^h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$, and \item[(5)] $M^h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^h$. \end{enumerate} If $M^h = M \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S, s}^h$ for $M \in D(\mathcal{O}_{S, s})$ this is also equivalent to \begin{enumerate} \item[(6)] $M$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$. \end{enumerate}


flat-lemma-finite-flat-weak-assassin-up-down

Let $g : T \to S$ be a finite flat morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $t \in T$ be a point with image $s \in S$. Then $$ t \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G}) $$


flat-lemma-etale-weak-assassin-up-down

Let $h : U \to S$ be an \'etale morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $u \in U$ be a point with image $s \in S$. Then $$ u \in \text{WeakAss}(h^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G}) $$


flat-lemma-weakly-associated-henselization

Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^h$ (resp.\ $\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\ strict henselization), see Algebra, Definition \ref{algebra-definition-henselization}. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. The following are equivalent \begin{enumerate} \item $s$ is a weakly associated point of $\mathcal{F}$, \item $\mathfrak m_s$ is a weakly associated prime of $\mathcal{F}_s$, \item $\mathfrak m_s^h$ is a weakly associated prime of $\mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^h$, and \item $\mathfrak m_s^{sh}$ is a weakly associated prime of $\mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$. \end{enumerate}


flat-lemma-sheaf-lives-on-subscheme

Let $f : X \to S$ be a finite type morphism of affine schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Set $\mathcal{F}_s = \mathcal{F}|_{X_s}$. Then there exist a closed immersion $i : Z \to X$ of finite presentation, and a quasi-coherent finite type $\mathcal{O}_Z$-module $\mathcal{G}$ such that $i_*\mathcal{G} = \mathcal{F}$ and $Z_s = \text{Supp}(\mathcal{F}_s)$.


flat-lemma-elementary-devissage

Let $f : X \to S$ be morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Set $\mathcal{F}_s = \mathcal{F}|_{X_s}$ and $n = \dim_x(\text{Supp}(\mathcal{F}_s))$. Then we can construct \begin{enumerate} \item elementary \'etale neighbourhoods $g : (X', x') \to (X, x)$, $e : (S', s') \to (S, s)$, \item a commutative diagram $$ \xymatrix{ X \ar[dd]_f & X' \ar[dd] \ar[l]^g & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] } $$ \item a point $z' \in Z'$ with $i(z') = x'$, $y' = \pi(z')$, $h(y') = s'$, \item a finite type quasi-coherent $\mathcal{O}_{Z'}$-module $\mathcal{G}$, \end{enumerate} such that the following properties hold \begin{enumerate} \item $X'$, $Z'$, $Y'$, $S'$ are affine schemes, \item $i$ is a closed immersion of finite presentation, \item $i_*(\mathcal{G}) \cong g^*\mathcal{F}$, \item $\pi$ is finite and $\pi^{-1}(\{y'\}) = \{z'\}$, \item the extension $\kappa(s') \subset \kappa(y')$ is purely transcendental, \item $h$ is smooth of relative dimension $n$ with geometrically integral fibres. \end{enumerate}


flat-lemma-devissage-finite-presentation

Assumptions and notation as in Lemma \ref{lemma-elementary-devissage}. If $f$ is locally of finite presentation then $\pi$ is of finite presentation. In this case the following are equivalent \begin{enumerate} \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation in a neighbourhood of $x$, \item $\mathcal{G}$ is an $\mathcal{O}_{Z'}$-module of finite presentation in a neighbourhood of $z'$, and \item $\pi_*\mathcal{G}$ is an $\mathcal{O}_{Y'}$-module of finite presentation in a neighbourhood of $y'$. \end{enumerate} Still assuming $f$ locally of finite presentation the following are equivalent to each other \begin{enumerate} \item[(a)] $\mathcal{F}_x$ is an $\mathcal{O}_{X, x}$-module of finite presentation, \item[(b)] $\mathcal{G}_{z'}$ is an $\mathcal{O}_{Z', z'}$-module of finite presentation, and \item[(c)] $(\pi_*\mathcal{G})_{y'}$ is an $\mathcal{O}_{Y', y'}$-module of finite presentation. \end{enumerate}


flat-lemma-devissage-flat

Assumptions and notation as in Lemma \ref{lemma-elementary-devissage}. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$, \item $\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and \item $\pi_*\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$. \end{enumerate} The following are equivalent also \begin{enumerate} \item[(a)] $\mathcal{F}_x$ is flat over $\mathcal{O}_{S, s}$, \item[(b)] $\mathcal{G}_{z'}$ is flat over $\mathcal{O}_{S', s'}$, and \item[(c)] $(\pi_*\mathcal{G})_{y'}$ is flat over $\mathcal{O}_{S', s'}$. \end{enumerate}


flat-lemma-elementary-devissage-variant

Let $f : X \to S$ be morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Then there exists a commutative diagram of pointed schemes $$ \xymatrix{ (X, x) \ar[d]_f & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] \\ } $$ such that $(S', s') \to (S, s)$ and $(X', x') \to (X, x)$ are elementary \'etale neighbourhoods, and such that $g^*\mathcal{F}/X'/S'$ has a one step d\'evissage at $x'$.


flat-lemma-base-change-one-step

Let $S$, $X$, $\mathcal{F}$, $s$ be as in Definition \ref{definition-one-step-devissage}. Let $(Z, Y, i, \pi, \mathcal{G})$ be a one step d\'evissage of $\mathcal{F}/X/S$ over $s$. Let $(S', s') \to (S, s)$ be any morphism of pointed schemes. Given this data let $X', Z', Y', i', \pi'$ be the base changes of $X, Z, Y, i, \pi$ via $S' \to S$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X'$ and let $\mathcal{G}'$ be the pullback of $\mathcal{G}$ to $Z'$. If $S'$ is affine, then $(Z', Y', i', \pi', \mathcal{G}')$ is a one step d\'evissage of $\mathcal{F}'/X'/S'$ over $s'$.


flat-lemma-base-change-one-step-at-x

Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition \ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \pi, \mathcal{G}, z, y)$ be a one step d\'evissage of $\mathcal{F}/X/S$ at $x$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes which induces an isomorphism $\kappa(s) = \kappa(s')$. Let $(Z', Y', i', \pi', \mathcal{G}')$ be as constructed in Lemma \ref{lemma-base-change-one-step} and let $x' \in X'$ (resp.\ $z' \in Z'$, $y' \in Y'$) be the unique point mapping to both $x \in X$ (resp.\ $z \in Z$, $y \in Y$) and $s' \in S'$. If $S'$ is affine, then $(Z', Y', i', \pi', \mathcal{G}', z', y')$ is a one step d\'evissage of $\mathcal{F}'/X'/S'$ at $x'$.


flat-lemma-shrink

With assumption and notation as in Definition \ref{definition-shrink} we have: \begin{enumerate} \item \label{item-shrink-base} If $S' \subset S$ is a standard open neighbourhood of $s$, then setting $X' = X_{S'}$, $Z' = Z_{S'}$ and $Y' = Y_{S'}$ we obtain a standard shrinking. \item \label{item-shrink-on-Y} Let $W \subset Y$ be a standard open neighbourhood of $y$. Then there exists a standard shrinking with $Y' = W \times_S S'$. \item \label{item-shrink-on-X} Let $U \subset X$ be an open neighbourhood of $x$. Then there exists a standard shrinking with $X' \subset U$. \end{enumerate}


flat-lemma-elementary-etale-neighbourhood

Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition \ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \pi, \mathcal{G}, z, y)$ be a one step d\'evissage of $\mathcal{F}/X/S$ at $x$. Let $$ \xymatrix{ (Y, y) \ar[d] & (Y', y') \ar[l] \ar[d] \\ (S, s) & (S', s') \ar[l] } $$ be a commutative diagram of pointed schemes such that the horizontal arrows are elementary \'etale neighbourhoods. Then there exists a commutative diagram $$ \xymatrix{ & & (X'', x'') \ar[lld] \ar[d] & (Z'', z'') \ar[l] \ar[lld] \ar[d] \\ (X, x) \ar[d] & (Z, z) \ar[l] \ar[d] & (S'', s'') \ar[lld] & (Y'', y'') \ar[lld] \ar[l] \\ (S, s) & (Y, y) \ar[l] } $$ of pointed schemes with the following properties: \begin{enumerate} \item $(S'', s'') \to (S', s')$ is an elementary \'etale neighbourhood and the morphism $S'' \to S$ is the composition $S'' \to S' \to S$, \item $Y''$ is an open subscheme of $Y' \times_{S'} S''$, \item $Z'' = Z \times_Y Y''$, \item $(X'', x'') \to (X, x)$ is an elementary \'etale neighbourhood, and \item $(Z'', Y'', i'', \pi'', \mathcal{G}'', z'', y'')$ is a one step d\'evissage at $x''$ of the sheaf $\mathcal{F}''$. \end{enumerate} Here $\mathcal{F}''$ (resp.\ $\mathcal{G}''$) is the pullback of $\mathcal{F}$ (resp.\ $\mathcal{G}$) via the morphism $X'' \to X$ (resp.\ $Z'' \to Z$) and $i'' : Z'' \to X''$ and $\pi'' : Z'' \to Y''$ are as in the diagram.


flat-lemma-existence-alpha

Let $S$, $X$, $\mathcal{F}$, $s$ be as in Definition \ref{definition-one-step-devissage}. Let $(Z, Y, i, \pi, \mathcal{G})$ be a one step d\'evissage of $\mathcal{F}/X/S$ over $s$. Let $\xi \in Y_s$ be the (unique) generic point. Then there exists an integer $r > 0$ and an $\mathcal{O}_Y$-module map $$ \alpha : \mathcal{O}_Y^{\oplus r} \longrightarrow \pi_*\mathcal{G} $$ such that $$ \alpha : \kappa(\xi)^{\oplus r} \longrightarrow (\pi_*\mathcal{G})_\xi \otimes_{\mathcal{O}_{Y, \xi}} \kappa(\xi) $$ is an isomorphism. Moreover, in this case we have $$ \dim(\text{Supp}(\Coker(\alpha)_s)) < \dim(\text{Supp}(\mathcal{F}_s)). $$


flat-lemma-base-change-complete

Let $S$, $X$, $\mathcal{F}$, $s$ be as in Definition \ref{definition-complete-devissage}. Let $(S', s') \to (S, s)$ be any morphism of pointed schemes. Let $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}$ be a complete d\'evissage of $\mathcal{F}/X/S$ over $s$. Given this data let $X', Z'_k, Y'_k, i'_k, \pi'_k$ be the base changes of $X, Z_k, Y_k, i_k, \pi_k$ via $S' \to S$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X'$ and let $\mathcal{G}'_k$ be the pullback of $\mathcal{G}_k$ to $Z'_k$. Let $\alpha'_k$ be the pullback of $\alpha_k$ to $Y'_k$. If $S'$ is affine, then $(Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k)_{k = 1, \ldots, n}$ is a complete d\'evissage of $\mathcal{F}'/X'/S'$ over $s'$.


flat-lemma-base-change-complete-at-x

Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition \ref{definition-complete-devissage-at-x}. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes which induces an isomorphism $\kappa(s) = \kappa(s')$. Let $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 1, \ldots, n}$ be a complete d\'evissage of $\mathcal{F}/X/S$ at $x$. Let $(Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k)_{k = 1, \ldots, n}$ be as constructed in Lemma \ref{lemma-base-change-complete} and let $x' \in X'$ (resp.\ $z'_k \in Z'$, $y'_k \in Y'$) be the unique point mapping to both $x \in X$ (resp.\ $z_k \in Z_k$, $y_k \in Y_k$) and $s' \in S'$. If $S'$ is affine, then $(Z'_k, Y'_k, i'_k, \pi'_k, \mathcal{G}'_k, \alpha'_k, z'_k, y'_k)_{k = 1, \ldots, n}$ is a complete d\'evissage of $\mathcal{F}'/X'/S'$ at $x'$.


flat-lemma-shrink-complete

With assumption and notation as in Definition \ref{definition-shrink-complete} we have: \begin{enumerate} \item \label{item-shrink-base-complete} If $S' \subset S$ is a standard open neighbourhood of $s$, then setting $X' = X_{S'}$, $Z'_k = Z_{S'}$ and $Y'_k = Y_{S'}$ we obtain a standard shrinking. \item \label{item-shrink-on-Y-complete} Let $W \subset Y_n$ be a standard open neighbourhood of $y$. Then there exists a standard shrinking with $Y'_n = W \times_S S'$. \item \label{item-shrink-on-X-complete} Let $U \subset X$ be an open neighbourhood of $x$. Then there exists a standard shrinking with $X' \subset U$. \end{enumerate}


flat-lemma-existence-complete

Let $X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$ be a point. There exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and \'etale morphisms $h_i : Y_i \to X_{S'}$, $i = 1, \ldots, n$ such that for each $i$ there exists a complete d\'evissage of $\mathcal{F}_i/Y_i/S'$ over $s'$, where $\mathcal{F}_i$ is the pullback of $\mathcal{F}$ to $Y_i$ and such that $X_s = (X_{S'})_{s'} \subset \bigcup h_i(Y_i)$.


flat-lemma-existence-algebra

Let $R \to S$ be a finite type ring map. Let $M$ be a finite $S$-module. Let $\mathfrak q$ be a prime ideal of $S$. There exists an elementary \'etale localization $R' \to S', \mathfrak q', \mathfrak p'$ of the ring map $R \to S$ at $\mathfrak q$ such that there exists a complete d\'evissage of $(M \otimes_S S')/S'/R'$ at $\mathfrak q'$.


flat-lemma-homothety-spectrum

Let $R \to S$ be a ring map. Let $N$ be a $S$-module. Assume \begin{enumerate} \item $R$ is a local ring with maximal ideal $\mathfrak m$, \item $\overline{S} = S/\mathfrak m S$ is Noetherian, and \item $\overline{N} = N/\mathfrak m_R N$ is a finite $\overline{S}$-module. \end{enumerate} Let $\Sigma \subset S$ be the multiplicative subset of elements which are not a zerodivisor on $\overline{N}$. Then $\Sigma^{-1}S$ is a semi-local ring whose spectrum consists of primes $\mathfrak q \subset S$ contained in an element of $\text{Ass}_S(\overline{N})$. Moreover, any maximal ideal of $\Sigma^{-1}S$ corresponds to an associated prime of $\overline{N}$ over $\overline{S}$.


flat-lemma-homothety-universally-injective

Assumption and notation as in Lemma \ref{lemma-homothety-spectrum}. Assume moreover that \begin{enumerate} \item $S$ is local and $R \to S$ is a local homomorphism, \item $S$ is essentially of finite presentation over $R$, \item $N$ is finitely presented over $S$, and \item $N$ is flat over $R$. \end{enumerate} Then each $s \in \Sigma$ defines a universally injective $R$-module map $s : N \to N$, and the map $N \to \Sigma^{-1}N$ is $R$-universally injective.


flat-lemma-base-change-universally-flat-local

Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Let $S \to S'$ be a ring map. Assume \begin{enumerate} \item $R \to S$ is a local homomorphism of local rings \item $S$ is essentially of finite presentation over $R$, \item $N$ is of finite presentation over $S$, \item $N$ is flat over $R$, \item $S \to S'$ is flat, and \item the image of $\Spec(S') \to \Spec(S)$ contains all primes $\mathfrak q$ of $S$ lying over $\mathfrak m_R$ such that $\mathfrak q$ is an associated prime of $N/\mathfrak m_R N$. \end{enumerate} Then $N \to N \otimes_S S'$ is $R$-universally injective.


flat-lemma-base-change-universally-flat

Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Let $S \to S'$ be a ring map. Assume \begin{enumerate} \item $R \to S$ is of finite presentation and $N$ is of finite presentation over $S$, \item $N$ is flat over $R$, \item $S \to S'$ is flat, and \item the image of $\Spec(S') \to \Spec(S)$ contains all primes $\mathfrak q$ such that $\mathfrak q$ is an associated prime of $N \otimes_R \kappa(\mathfrak p)$ where $\mathfrak p$ is the inverse image of $\mathfrak q$ in $R$. \end{enumerate} Then $N \to N \otimes_S S'$ is $R$-universally injective.


flat-lemma-universally-injective-local

Let $(R, \mathfrak m)$ be a local ring. Let $u : M \to N$ be an $R$-module map. If $M$ is a projective $R$-module, $N$ is a flat $R$-module, and $\overline{u} : M/\mathfrak mM \to N/\mathfrak mN$ is injective then $u$ is universally injective.


flat-lemma-invert-universally-injective

Assumption and notation as in Lemma \ref{lemma-homothety-spectrum}. Assume moreover that $N$ is projective as an $R$-module. Then each $s \in \Sigma$ defines a universally injective $R$-module map $s : N \to N$, and the map $N \to \Sigma^{-1}N$ is $R$-universally injective.


flat-lemma-completed-direct-sum-ML

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian and complete with respect to $I$. The completion $(\bigoplus\nolimits_{\alpha \in A} R)^\wedge$ is flat and Mittag-Leffler.


flat-lemma-lift-ML

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. Assume \begin{enumerate} \item $R$ is Noetherian and $I$-adically complete, \item $M$ is flat over $R$, and \item $M/IM$ is a projective $R/I$-module. \end{enumerate} Then the $I$-adic completion $M^\wedge$ is a flat Mittag-Leffler $R$-module.


flat-lemma-universally-injective-to-completion

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map, and $N$ an $S$-module. Assume \begin{enumerate} \item $R$ is a Noetherian ring, \item $S$ is a Noetherian ring, \item $N$ is a finite $S$-module, and \item for any finite $R$-module $Q$, any $\mathfrak q \in \text{Ass}_S(Q \otimes_R N)$ satisfies $IS + \mathfrak q \not = S$. \end{enumerate} Then the map $N \to N^\wedge$ of $N$ into the $I$-adic completion of $N$ is universally injective as a map of $R$-modules.


flat-lemma-universally-injective-to-completion-flat

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map, and $N$ an $S$-module. Assume \begin{enumerate} \item $R$ is a Noetherian ring, \item $S$ is a Noetherian ring, \item $N$ is a finite $S$-module, \item $N$ is flat over $R$, and \item for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R \kappa(\mathfrak p)$ where $\mathfrak p = R \cap \mathfrak q$ we have $IS + \mathfrak q \not = S$. \end{enumerate} Then the map $N \to N^\wedge$ of $N$ into the $I$-adic completion of $N$ is universally injective as a map of $R$-modules.


flat-lemma-flat-pure-over-complete-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map, and $N$ an $S$-module. Assume \begin{enumerate} \item $R$ is Noetherian and $I$-adically complete, \item $R \to S$ is of finite type, \item $N$ is a finite $S$-module, \item $N$ is flat over $R$, \item $N/IN$ is projective as a $R/I$-module, and \item for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R \kappa(\mathfrak p)$ where $\mathfrak p = R \cap \mathfrak q$ we have $IS + \mathfrak q \not = S$. \end{enumerate} Then $N$ is projective as an $R$-module.


flat-lemma-fibres-irreducible-flat-projective

Let $R$ be a ring. Let $R \to S$ be a ring map. Assume \begin{enumerate} \item $R$ is Noetherian, \item $R \to S$ is of finite type and flat, and \item every fibre ring $S \otimes_R \kappa(\mathfrak p)$ is geometrically integral over $\kappa(\mathfrak p)$. \end{enumerate} Then $S$ is projective as an $R$-module.


flat-lemma-fibres-irreducible-flat-projective-nonnoetherian

Let $R$ be a ring. Let $R \to S$ be a ring map. Assume \begin{enumerate} \item $R \to S$ is of finite presentation and flat, and \item every fibre ring $S \otimes_R \kappa(\mathfrak p)$ is geometrically integral over $\kappa(\mathfrak p)$. \end{enumerate} Then $S$ is projective as an $R$-module.


flat-lemma-induction-step

Let $(R, \mathfrak m)$ be a local ring. Let $R \to S$ be a finitely presented flat ring map with geometrically integral fibres. Write $\mathfrak p = \mathfrak mS$. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak m$. Let $N$ be a finite $S$-module. There exist $r \geq 0$ and an $S$-module map $$ \alpha : S^{\oplus r} \longrightarrow N $$ such that $\alpha : \kappa(\mathfrak p)^{\oplus r} \to N \otimes_S \kappa(\mathfrak p)$ is an isomorphism. For any such $\alpha$ the following are equivalent: \begin{enumerate} \item $N_{\mathfrak q}$ is $R$-flat, \item $\alpha$ is $R$-universally injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat, \item $\alpha$ is injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat, \item $\alpha_{\mathfrak p}$ is an isomorphism and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat, and \item $\alpha_{\mathfrak q}$ is injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat. \end{enumerate}


flat-lemma-complete-devissage-flat-finite-type-module

Let $(R, \mathfrak m)$ be a local ring. Let $R \to S$ be a ring map of finite presentation. Let $N$ be a finite $S$-module. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak m$. Assume that $N_{\mathfrak q}$ is flat over $R$, and assume there exists a complete d\'evissage of $N/S/R$ at $\mathfrak q$. Then $N$ is a finitely presented $S$-module, free as an $R$-module, and there exists an isomorphism $$ N \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_n^{\oplus r_n} $$ as $R$-modules where each $B_i$ is a smooth $R$-algebra with geometrically irreducible fibres.


flat-lemma-open-in-fibre-where-flat

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $s \in S$. Then the set $$ \{x \in X_s \mid \mathcal{F} \text{ flat over }S\text{ at }x\} $$ is open in the fibre $X_s$.


flat-lemma-finite-type-flat-at-point

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat at $x$ over $S$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an open subscheme $$ V \subset X \times_S \Spec(\mathcal{O}_{S', s'}) $$ which contains the unique point of $X \times_S \Spec(\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\mathcal{F}$ to $V$ is flat over $\mathcal{O}_{S', s'}$.


flat-lemma-finite-type-flat-along-fibre

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an open subscheme $$ V \subset X \times_S \Spec(\mathcal{O}_{S', s'}) $$ which contains the fibre $X_s = X \times_S s'$ such that the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.


flat-lemma-finite-type-flat-at-point-local

Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. If $x \in X$ and $\mathcal{F}$ is flat at $x$ over $S$, then $\mathcal{F}_x$ is an $\mathcal{O}_{X, x}$-module of finite presentation.


flat-lemma-flat-finite-type-finitely-presented-over-dense-open

\begin{slogan} $S$-flat and finite type extensions of finitely presented modules on a (good) open are also $X$-finitely presented. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset S$ be open. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite type and flat over $S$, \item $U \subset S$ is retrocompact and scheme theoretically dense, \item $\mathcal{F}|_{f^{-1}U}$ is of finite presentation. \end{enumerate} Then $\mathcal{F}$ is of finite presentation.


flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X

Let $f : X \to S$ be a morphism of schemes. Let $U \subset S$ be open. Assume \begin{enumerate} \item $f$ is locally of finite type and flat, \item $U \subset S$ is retrocompact and scheme theoretically dense, \item $f|_{f^{-1}U} : f^{-1}U \to U$ is locally of finite presentation. \end{enumerate} Then $f$ is of locally of finite presentation.


flat-lemma-flat-finite-presentation-dimension-over-dense-open

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite type. Let $U \subset S$ be a dense open such that $X_U \to U$ has relative dimension $\leq e$, see Morphisms, Definition \ref{morphisms-definition-relative-dimension-d}. If also either \begin{enumerate} \item $f$ is locally of finite presentation, or \item $U \subset S$ is retrocompact, \end{enumerate} then $f$ has relative dimension $\leq e$.


flat-lemma-proper-flat-finite-over-dense-open

Let $f : X \to S$ be a morphism of schemes which is flat and proper. Let $U \subset S$ be a dense open such that $X_U \to U$ is finite. If also either $f$ is locally of finite presentation or $U \subset S$ is retrocompact, then $f$ is finite.


flat-lemma-zariski

Let $f : X \to S$ be a morphism of schemes and $U \subset S$ an open. If \begin{enumerate} \item $f$ is separated, locally of finite type, and flat, \item $f^{-1}(U) \to U$ is an isomorphism, and \item $U \subset S$ is retrocompact and scheme theoretically dense, \end{enumerate} then $f$ is an open immersion.


flat-lemma-induction-step-fp

Let $R$ be a ring. Let $R \to S$ be a finitely presented flat ring map with geometrically integral fibres. Let $\mathfrak q \subset S$ be a prime ideal lying over the prime $\mathfrak r \subset R$. Set $\mathfrak p = \mathfrak r S$. Let $N$ be a finitely presented $S$-module. There exists $r \geq 0$ and an $S$-module map $$ \alpha : S^{\oplus r} \longrightarrow N $$ such that $\alpha : \kappa(\mathfrak p)^{\oplus r} \to N \otimes_S \kappa(\mathfrak p)$ is an isomorphism. For any such $\alpha$ the following are equivalent: \begin{enumerate} \item $N_{\mathfrak q}$ is $R$-flat, \item there exists an $f \in R$, $f \not \in \mathfrak r$ such that $\alpha_f : S_f^{\oplus r} \to N_f$ is $R_f$-universally injective and a $g \in S$, $g \not \in \mathfrak q$ such that $\Coker(\alpha)_g$ is $R$-flat, \item $\alpha_{\mathfrak r}$ is $R_{\mathfrak r}$-universally injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat \item $\alpha_{\mathfrak r}$ is injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat, \item $\alpha_{\mathfrak p}$ is an isomorphism and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat, and \item $\alpha_{\mathfrak q}$ is injective and $\Coker(\alpha)_{\mathfrak q}$ is $R$-flat. \end{enumerate}


flat-lemma-complete-devissage-flat-finitely-presented-module

Let $R \to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module flat over $R$. Let $\mathfrak r \subset R$ be a prime ideal. Assume there exists a complete d\'evissage of $N/S/R$ over $\mathfrak r$. Then there exists an $f \in R$, $f \not \in \mathfrak r$ such that $$ N_f \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_n^{\oplus r_n} $$ as $R$-modules where each $B_i$ is a smooth $R_f$-algebra with geometrically irreducible fibres. Moreover, $N_f$ is projective as an $R_f$-module.


flat-lemma-finite-presentation-flat-along-fibre

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite presentation, and \item $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] } $$ such that $g$ is \'etale, $X_s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma(X', g^*\mathcal{F})$ is a projective $\Gamma(S', \mathcal{O}_{S'})$-module.


flat-lemma-finite-type-flat-at-point-free

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat at $x$ over $S$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of pointed schemes $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S', s'}), s') \ar[l] } $$ such that $X' \to X \times_S \Spec(\mathcal{O}_{S', s'})$ is \'etale, $\kappa(x) = \kappa(x')$, the scheme $X'$ is affine of finite presentation over $\mathcal{O}_{S', s'}$, the sheaf $g^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{X'}$, and such that $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.


flat-lemma-finite-type-flat-at-point-free-variant

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat at $x$ over $S$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of pointed schemes $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (\Spec(\mathcal{O}_{S', s'}), s') \ar[l] } $$ such that $X' \to X \times_S \Spec(\mathcal{O}_{S', s'})$ is \'etale, $\kappa(x) = \kappa(x')$, the scheme $X'$ is affine, and such that $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.


flat-lemma-finite-type-flat-along-fibre-free

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat over $S$ at all points of $X_s$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] } $$ such that $X' \to X \times_S \Spec(\mathcal{O}_{S', s'})$ is \'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine of finite presentation over $\mathcal{O}_{S', s'}$, the sheaf $g^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{X'}$, and such that $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.


flat-lemma-finite-type-flat-along-fibre-free-variant

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat over $S$ at all points of $X_s$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] } $$ such that $X' \to X \times_S \Spec(\mathcal{O}_{S', s'})$ is \'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine, and such that $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.


flat-lemma-weak-bourbaki-pre-pre

Let $R \to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak p \subset R$. Set $\overline{S} = S \otimes_R \kappa(\mathfrak p)$, $\overline{\mathfrak q} = \mathfrak q \overline{S}$, and $\overline{N} = N \otimes_R \kappa(\mathfrak p)$. Then we can find a $g \in S$ with $g \not \in \mathfrak q$ such that $\overline{g} \in \mathfrak r$ for all $\mathfrak r \in \text{Ass}_{\overline{S}}(\overline{N})$ such that $\mathfrak r \not \subset \overline{\mathfrak q}$.


flat-lemma-weak-bourbaki-pre

Let $R \to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module which is flat as an $R$-module. Let $M$ be an $R$-module. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak p \subset R$. Then $$ \mathfrak q \in \text{WeakAss}_S(M \otimes_R N) \Leftrightarrow \Big( \mathfrak p \in \text{WeakAss}_R(M) \text{ and } \overline{\mathfrak q} \in \text{Ass}_{\overline{S}}(\overline{N}) \Big) $$ Here $\overline{S} = S \otimes_R \kappa(\mathfrak p)$, $\overline{\mathfrak q} = \mathfrak q \overline{S}$, and $\overline{N} = N \otimes_R \kappa(\mathfrak p)$.


flat-lemma-bourbaki-finite-type-general-base-at-point

Let $S$ be a scheme. Let $f : X \to S$ be locally of finite type. Let $x \in X$ with image $s \in S$. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$. If $\mathcal{F}$ is flat at $x$ over $S$, then $$ x \in \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}_S(\mathcal{G}) \text{ and } x \in \text{Ass}_{X_s}(\mathcal{F}_s). $$


flat-lemma-bourbaki-finite-type-general-base

Let $f : X \to S$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$ which is flat over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have $$ \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \bigcup\nolimits_{s \in \text{WeakAss}_S(\mathcal{G})} \text{Ass}_{X_s}(\mathcal{F}_s) $$


flat-lemma-finite-type-flat-algebra

Let $R \to S$ be a ring map of finite presentation. Let $M$ be a finite $S$-module. Assume $\text{WeakAss}_S(S)$ is finite. Then $$ U = \{\mathfrak q \subset S \mid M_{\mathfrak q}\text{ flat over }R\} $$ is open in $\Spec(S)$ and for every $g \in S$ such that $D(g) \subset U$ the localization $M_g$ is a finitely presented $S_g$-module flat over $R$.


flat-lemma-finite-type-flat-X

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Assume the set of weakly associated points of $S$ is locally finite in $S$. Then the set of points $x \in X$ where $f$ is flat is an open subscheme $U \subset X$ and $U \to S$ is flat and locally of finite presentation.


flat-lemma-finite-type-flat-over-integral

Let $f : X \to S$ be a morphism of schemes which is locally of finite type and flat. If $S$ is integral, then $f$ is locally of finite presentation.


flat-lemma-explain-why-pure

Let $R$ be a local ring with maximal ideal $\mathfrak m$. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Assume \begin{enumerate} \item $N$ is projective as an $R$-module, and \item $S/\mathfrak mS$ is Noetherian and $N/\mathfrak mN$ is a finite $S/\mathfrak mS$-module. \end{enumerate} Then for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R \kappa(\mathfrak p)$ where $\mathfrak p = R \cap \mathfrak q$ we have $\mathfrak q + \mathfrak m S \not = S$.


flat-lemma-explain-why-pure-direct-sum-finite-modules

Let $R$ be a local ring with maximal ideal $\mathfrak m$. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is isomorphic as an $R$-module to a direct sum of finite $R$-modules. Then for any $R$-module $M$ and for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R M$ we have $\mathfrak q + \mathfrak m S \not = S$.


flat-lemma-impure-finite-presentation

In Situation \ref{situation-pre-pure}. If there exists an impurity of $\mathcal{F}$ above $s$, then there exists an impurity $(g : T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ such that $g$ is locally of finite presentation and $t$ a closed point of the fibre of $g$ above $s$.


flat-lemma-impure-limit

In Situation \ref{situation-pre-pure}. Let $(g : T \to S, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $s$. Assume $T = \lim_{i \in I} T_i$ is a directed limit of affine schemes over $S$. Then for some $i$ the triple $(T_i \to S, t'_i \leadsto t_i, \xi_i)$ is an impurity of $\mathcal{F}$ above $s$.


flat-lemma-quasi-finite-impurity-elementary

In Situation \ref{situation-pre-pure}. If there exists an impurity $(g : T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ with $g$ quasi-finite at $t$, then there exists an impurity $(g : T \to S, t' \leadsto t, \xi)$ such that $(T, t) \to (S, s)$ is an elementary \'etale neighbourhood.


flat-lemma-Noetherian-impurity-quasi-finite

In Situation \ref{situation-pre-pure}. Assume that $S$ is locally Noetherian. If there exists an impurity of $\mathcal{F}$ above $s$, then there exists an impurity $(g : T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$.


flat-lemma-impurity-on-henselization

In Situation \ref{situation-pre-pure}. If there exists an impurity $(S^h \to S, s' \leadsto s, \xi)$ of $\mathcal{F}$ above $s$ then there exists an impurity $(T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ where $(T, t) \to (S, s)$ is an elementary \'etale neighbourhood.


flat-lemma-base-change-universally

Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is universally pure along $X_s$, and \item for every morphism of pointed schemes $(S', s') \to (S, s)$ the pullback $\mathcal{F}_{S'}$ is pure along $X_{s'}$. \end{enumerate} In particular, $\mathcal{F}$ is universally pure relative to $S$ if and only if every base change $\mathcal{F}_{S'}$ of $\mathcal{F}$ is pure relative to $S'$.


flat-lemma-quasi-finite-base-change

Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. If $S' \to S$ is quasi-finite at $s'$ and $\mathcal{F}$ is pure along $X_s$, then $\mathcal{F}_{S'}$ is pure along $X_{s'}$.


flat-lemma-flat-descend-pure

Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. Assume $S' \to S$ is flat at $s'$. \begin{enumerate} \item If $\mathcal{F}_{S'}$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_s$. \item If $\mathcal{F}_{S'}$ is universally pure along $X_{s'}$, then $\mathcal{F}$ is universally pure along $X_s$. \end{enumerate}


flat-lemma-supported-on-closed

Let $i : Z \to X$ be a closed immersion of schemes of finite type over a scheme $S$. Let $s \in S$. Let $\mathcal{F}$ be a finite type, quasi-coherent sheaf on $Z$. Then $\mathcal{F}$ is (universally) pure along $Z_s$ if and only if $i_*\mathcal{F}$ is (universally) pure along $X_s$.


flat-lemma-proper-pure

Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If the support of $\mathcal{F}$ is proper over $S$, then $\mathcal{F}$ is universally pure relative to $S$. \item If $f$ is proper, then $\mathcal{F}$ is universally pure relative to $S$. \item If $f$ is proper, then $X$ is universally pure relative to $S$. \end{enumerate}


flat-lemma-affine-locally-projective-pure

Let $f : X \to S$ be a finite type, affine morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module such that $f_*\mathcal{F}$ is locally projective on $S$, see Properties, Definition \ref{properties-definition-locally-projective}. Then $\mathcal{F}$ is universally pure over $S$.


flat-lemma-associated-point-specializes

Let $f : X \to S$ be a morphism of schemes of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $s \in S$. Assume that $\mathcal{F}$ is flat over $S$ at all points of $X_s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$ with $f(x') = s'$ such that $s' \leadsto s$ is a specialization in $S$. If $x'$ specializes to a point of $X_s$, then $x' \leadsto x$ with $x \in \text{Ass}_{X_s}(\mathcal{F}_s)$.


flat-lemma-criterion

Let $f : X \to S$ be a morphism of schemes of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $s \in S$. Let $(S', s') \to (S, s)$ be an elementary \'etale neighbourhood and let $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] } $$ be a commutative diagram of morphisms of schemes. Assume \begin{enumerate} \item $\mathcal{F}$ is flat over $S$ at all points of $X_s$, \item $X' \to S'$ is of finite type, \item $g^*\mathcal{F}$ is pure along $X'_{s'}$, \item $g : X' \to X$ is \'etale, and \item $g(X')$ contains $\text{Ass}_{X_s}(\mathcal{F}_s)$. \end{enumerate} In this situation $\mathcal{F}$ is pure along $X_s$ if and only if the image of $X' \to X \times_S S'$ contains the points of $\text{Ass}_{X \times_S S'/S'}(\mathcal{F} \times_S S')$ lying over points in $S'$ which specialize to $s'$.


flat-lemma-finite-type-flat-pure-along-fibre-is-universal

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. Assume \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, \item $\mathcal{F}$ is flat over $S$ at all points of $X_s$, and \item $\mathcal{F}$ is pure along $X_s$. \end{enumerate} Then $\mathcal{F}$ is universally pure along $X_s$.


flat-lemma-flat-finite-presentation-purity-open

Let $f : X \to S$ be a morphism of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation flat over $S$. Then the set $$ U = \{s \in S \mid \mathcal{F}\text{ is pure along }X_s\} $$ is open in $S$.


flat-lemma-injectivity-map-source-flat-pure

Let $f : X \to S$ be a morphism of finite type. Let $\mathcal{F}$ be a quasi-coherent sheaf of finite type on $X$. Assume $S$ is local with closed point $s$. Assume $\mathcal{F}$ is pure along $X_s$ and that $\mathcal{F}$ is flat over $S$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ of quasi-coherent $\mathcal{O}_X$-modules. Then the following are equivalent \begin{enumerate} \item the map on stalks $\varphi_x$ is injective for all $x \in \text{Ass}_{X_s}(\mathcal{F}_s)$, and \item $\varphi$ is injective. \end{enumerate}


flat-lemma-flat-finite-presentation-affine-neighbourhood-projective

Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module which is of finite presentation. Let $x \in X$ with $s = f(x) \in S$. If $\mathcal{F}$ is flat at $x$ over $S$ there exists an affine elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an affine open $U' \subset X \times_S S'$ which contains $x' = (x, s')$ such that $\Gamma(U', \mathcal{F}|_{U'})$ is a projective $\Gamma(S', \mathcal{O}_{S'})$-module.


flat-lemma-flat-finite-type-affine-neighbourhood-projective

Let $f : X \to S$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module which is of finite type. Let $x \in X$ with $s = f(x) \in S$. If $\mathcal{F}$ is flat at $x$ over $S$ there exists an affine elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an affine open $U' \subset X \times_S \Spec(\mathcal{O}_{S', s'})$ which contains $x' = (x, s')$ such that $\Gamma(U', \mathcal{F}|_{U'})$ is a free $\mathcal{O}_{S', s'}$-module.


flat-lemma-flat-finite-type-local-colimit-free

Let $A \to B$ be a local ring map of local rings which is essentially of finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module. If $A$ is henselian, then $N$ is a filtered colimit $$ N = \colim_i F_i $$ of free $A$-modules $F_i$ such that all transition maps $u_i : F_i \to F_{i'}$ of the system induce injective maps $\overline{u}_i : F_i/\mathfrak m_AF_i \to F_{i'}/\mathfrak m_AF_{i'}$. Also, $N$ is a Mittag-Leffler $A$-module.


flat-lemma-flat-finite-type-local-valuation-ring-has-content

Let $A \to B$ be a local ring map of local rings which is essentially of finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module. If $A$ is a valuation ring, then any element of $N$ has a content ideal $I \subset A$ (More on Algebra, Definition \ref{more-algebra-definition-content-ideal}).


flat-lemma-iso-sheaf

In Situation \ref{situation-iso}. \begin{enumerate} \item Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology. \item If $f$ is quasi-compact and $\mathcal{G}$ is of finite type, then $F_{surj}$ is limit preserving. \item If $f$ is quasi-compact and $\mathcal{F}$ of finite type, then $F_{zero}$ is limit preserving. \item If $f$ is quasi-compact, $\mathcal{F}$ is of finite type, and $\mathcal{G}$ is of finite presentation, then $F_{iso}$ is limit preserving. \end{enumerate}


flat-lemma-flat-at-point

In Situation \ref{situation-flat-at-point}. \begin{enumerate} \item If $A' \to A''$ is a flat morphism in $\mathcal{C}$ then $F_{lf}(A') = F_{lf}(A'')$. \item If $A \to B$ is essentially of finite presentation and $M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit preserving: If $\{A_i\}_{i \in I}$ is a directed system of objects of $\mathcal{C}$, then $F_{lf}(\colim_i A_i) = \colim_i F_{lf}(A_i)$. \end{enumerate}


flat-lemma-flat-at-point-finite

In Situation \ref{situation-flat-at-point}. Let $B \to C$ is a local map of local $A$-algebras and $N$ a $C$-module. Denote $F'_{lf} : \mathcal{C} \to \textit{Sets}$ the functor associated to the pair $(C, N)$. If $M \cong N$ as $B$-modules and $B \to C$ is finite, then $F_{lf} = F'_{lf}$.


flat-lemma-flat-at-point-go-up

In Situation \ref{situation-flat-at-point} suppose that $B \to C$ is a flat local homomorphism of local rings. Set $N = M \otimes_B C$. Denote $F'_{lf} : \mathcal{C} \to \textit{Sets}$ the functor associated to the pair $(C, N)$. Then $F_{lf} = F'_{lf}$.


flat-lemma-free-at-generic-points

In Situation \ref{situation-free-at-generic-points}. \begin{enumerate} \item The functor $H_p$ satisfies the sheaf property for the fpqc topology. \item If $\mathcal{F}$ is of finite presentation, then functor $H_p$ is limit preserving. \end{enumerate}


flat-lemma-pre-flat-dimension-n

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $n \geq 0$. The following are equivalent \begin{enumerate} \item for $s \in S$ the closed subset $Z \subset X_s$ of points where $\mathcal{F}$ is not flat over $S$ (see Lemma \ref{lemma-open-in-fibre-where-flat}) satisfies $\dim(Z) < n$, and \item for $x \in X$ such that $\mathcal{F}$ is not flat at $x$ over $S$ we have $\text{trdeg}_{\kappa(f(x))}(\kappa(x)) < n$. \end{enumerate} If this is true, then it remains true after any base change.


flat-lemma-flat-dimension-n

In Situation \ref{situation-flat-dimension-n}. \begin{enumerate} \item The functor $F_n$ satisfies the sheaf property for the fpqc topology. \item If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_n$ is limit preserving. \end{enumerate}


flat-lemma-generic-flatness-stratification

Let $f : X \to S$ be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation. Then there exists a $t \geq 0$ and closed subschemes $$ S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset $$ such that $S_i \to S$ is defined by a finite type ideal sheaf, $S_0 \subset S$ is a thickening, and $\mathcal{F}$ pulled back to $X \times_S (S_i \setminus S_{i + 1})$ is flat over $S_i \setminus S_{i + 1}$.


flat-lemma-universally-separating

Let $S$ be a scheme. Let $g : X' \to X$ be a flat morphism of schemes over $S$ with $X$ locally of finite type over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module which is flat over $S$. If $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$ then the canonical map $$ \mathcal{F} \longrightarrow g_*g^*\mathcal{F} $$ is injective, and remains injective after any base change.


flat-lemma-flattening-module-map

Let $A$ be a ring. Let $u : M \to N$ be a surjective map of $A$-modules. If $M$ is projective as an $A$-module, then there exists an ideal $I \subset A$ such that for any ring map $\varphi : A \to B$ the following are equivalent \begin{enumerate} \item $u \otimes 1 : M \otimes_A B \to N \otimes_A B$ is an isomorphism, and \item $\varphi(I) = 0$. \end{enumerate}


flat-lemma-Weil-restriction-closed-subschemes

Let $f:X\to S$ be a morphism of schemes which is of finite presentation, flat, and pure. Let $Y$ be a closed subscheme of $X$. Let $F=f_*Y$ be the Weil restriction functor of $Y$ along $f$, defined by $$ F : (\Sch/S)^{opp} \to \textit{Sets}, \quad T \mapsto \left\{ \begin{matrix} \{*\} & \text{if} & Y_T\to X_T \text{ is an isomorphism, }\\ \emptyset & \text{else.} & \end{matrix} \right. $$ Then $F$ is representable by a closed immersion $Z\to S$. Moreover $Z\to S$ is of finite presentation if $Y\to S$ is.


flat-lemma-freebie

Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $E \subset X_s$ be a subset. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent \begin{enumerate} \item $\mathcal{F}_T$ is flat over $T$ at all points of the fibre $X_t$ which map to a point of $E \subset X_s$, and \item $\Spec(\mathcal{O}_{T, t}) \to S$ factors through $Z$. \end{enumerate} Moreover, if $X \to S$ is locally of finite presentation, $\mathcal{F}$ is of finite presentation, and $E \subset X_s$ is closed and quasi-compact, then $Z \to S$ is of finite presentation.


flat-lemma-reduce-finite-type-injective-into-flat-mod-m

Let $A_0$ be a local ring. If the lemma holds for every Situation \ref{situation-mod-injective} with $A = A_0$, with $B$ a localization of a polynomial algebra over $A$, and $N$ of finite presentation over $B$, then the lemma holds for every Situation \ref{situation-mod-injective} with $A = A_0$.


flat-lemma-henselian-finite-type-injective-into-flat-mod-m

If in Situation \ref{situation-mod-injective} the ring $A$ is henselian then the lemma holds.


flat-lemma-upstairs-finite-type-injective-into-flat-mod-m

Let $A \to B$ be a local ring homomorphism of local rings which is essentially of finite type. Let $u : N \to M$ be a $B$-module map. If $N$ is a finite $B$-module, $M$ is flat over $A$, and $\overline{u} : N/\mathfrak m_A N \to M/\mathfrak m_A M$ is injective, then $u$ is $A$-universally injective, $N$ is of finite presentation over $B$, and $N$ is flat over $A$.


flat-lemma-properties-pure-spreadout

In (\ref{equation-star}) if there exists a pure spreadout, then \begin{enumerate} \item elements of $N$ have content ideals in $A$, and \item if $u : N \to M$ is a morphism to a flat $A$-module $M$ such that $N/\mathfrak m N \to M/\mathfrak m M$ is injective for all maximal ideals $\mathfrak m$ of $A$, then $u$ is $A$-universally injective. \end{enumerate}


flat-lemma-find-pure-spreadout

In (\ref{equation-star}) for every $\mathfrak p \in \Spec(A)$ there is a finitely generated ideal $I \subset \mathfrak pA_\mathfrak p$ such that over $A_\mathfrak p/I$ we have a pure spreadout.


flat-lemma-universally-injective-if-flat

In (\ref{equation-star}) assume $N$ is $A$-flat, $M$ is a flat $A$-module, and $u : N \to M$ is an $A$-module map such that $u \otimes \text{id}_{\kappa(\mathfrak p)}$ is injective for all $\mathfrak p \in \Spec(A)$. Then $u$ is $A$-universally injective.


flat-lemma-big-intersection-is-zero

Let $A$ be a local domain which is not a field. Let $S$ be a set of finitely generated ideals of $A$. Assume that $S$ is closed under products and such that $\bigcup_{I \in S} V(I)$ is the complement of the generic point of $\Spec(A)$. Then $\bigcap_{I \in S} I = (0)$.


flat-lemma-closed-points-complement

Let $A$ be a local ring. Let $I, J \subset A$ be ideals. If $J$ is finitely generated and $I \subset J^n$ for all $n \geq 1$, then $V(I)$ contains the closed points of $\Spec(A) \setminus V(J)$.


flat-lemma-make-smaller-flatness-ideal

Let $A$ be a local ring. Let $I \subset A$ be an ideal. Let $U \subset \Spec(A)$ be quasi-compact open. Let $M$ be an $A$-module. Assume that \begin{enumerate} \item $M/IM$ is flat over $A/I$, \item $M$ is flat over $U$, \end{enumerate} Then $M/I_2M$ is flat over $A/I_2$ where $I_2 = \Ker(I \to \Gamma(U, I/I^2))$.


flat-lemma-free-at-generic-points-representable

In Situation \ref{situation-free-at-generic-points}. For each $p \geq 0$ the functor $H_p$ (\ref{equation-free-at-generic-points}) is representable by a locally closed immersion $S_p \to S$. If $\mathcal{F}$ is of finite presentation, then $S_p \to S$ is of finite presentation.


flat-lemma-localize-flat-dimension-n

In Situation \ref{situation-flat-dimension-n}. Let $h : X' \to X$ be an \'etale morphism. Set $\mathcal{F}' = h^*\mathcal{F}$ and $f' = f \circ h$. Let $F_n'$ be (\ref{equation-flat-dimension-n}) associated to $(f' : X' \to S, \mathcal{F}')$. Then $F_n$ is a subfunctor of $F_n'$ and if $h(X') \supset \text{Ass}_{X/S}(\mathcal{F})$, then $F_n = F'_n$.


flat-lemma-compare-H-F

Assume that $X \to S$ is a smooth morphism of affine schemes with geometrically irreducible fibres of dimension $d$ and that $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module of finite presentation. Then $F_d = \coprod_{p = 0, \ldots, c} H_p$ for some $c \geq 0$ with $F_d$ as in (\ref{equation-flat-dimension-n}) and $H_p$ as in (\ref{equation-free-at-generic-points}).


flat-lemma-flat-dimension-n-representable

In Situation \ref{situation-flat-dimension-n}. Let $s \in S$ let $d \geq 0$. Assume \begin{enumerate} \item there exists a complete d\'evissage of $\mathcal{F}/X/S$ over some point $s \in S$, \item $X$ is of finite presentation over $S$, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and \item $\mathcal{F}$ is flat in dimensions $\geq d + 1$ over $S$. \end{enumerate} Then after possibly replacing $S$ by an open neighbourhood of $s$ the functor $F_d$ (\ref{equation-flat-dimension-n}) is representable by a monomorphism $Z_d \to S$ of finite presentation.


flat-lemma-when-universal-flattening

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $f$ is of finite presentation, $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and $\mathcal{F}$ is pure relative to $S$, then there exists a universal flattening $S' \to S$ of $\mathcal{F}$. Moreover $S' \to S$ is a monomorphism of finite presentation. \item If $f$ is of finite presentation and $X$ is pure relative to $S$, then there exists a universal flattening $S' \to S$ of $X$. Moreover $S' \to S$ is a monomorphism of finite presentation. \item If $f$ is proper and of finite presentation and $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, then there exists a universal flattening $S' \to S$ of $\mathcal{F}$. Moreover $S' \to S$ is a monomorphism of finite presentation. \item If $f$ is proper and of finite presentation then there exists a universal flattening $S' \to S$ of $X$. \end{enumerate}


flat-lemma-compute-what-it-should-be

In Situation \ref{situation-existence} consider $$ K = R\lim_{D_\QCoh(\mathcal{O}_X)}(\mathcal{F}_n) = DQ_X(R\lim_{D(\mathcal{O}_X)}\mathcal{F}_n) $$ Then $K$ is in $D^b_{\QCoh}(\mathcal{O}_X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\geq 0$.


flat-lemma-compute-against-perfect

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. For any perfect object $E$ of $D(\mathcal{O}_X)$ we have \begin{enumerate} \item $M = R\Gamma(X, K \otimes^\mathbf{L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Gamma(X_n, \mathcal{F}_n \otimes^\mathbf{L} E|_{X_n}) = M \otimes_A^\mathbf{L} A_n$ in $D(A_n)$, \item $N = R\Hom_X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Hom_{X_n}(E|_{X_n}, \mathcal{F}_n) = N \otimes_A^\mathbf{L} A_n$ in $D(A_n)$. \end{enumerate} In both statements $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.


flat-lemma-relative-pseudo-coherence

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Then $K$ is pseudo-coherent relative to $A$.


flat-lemma-compute-over-affine

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. For any quasi-compact open $U \subset X$ we have $$ R\Gamma(U, K) \otimes_A^\mathbf{L} A_n = R\Gamma(U_n, \mathcal{F}_n) $$ in $D(A_n)$ where $U_n = U \cap X_n$.


flat-lemma-finitely-presented

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Denote $X_0 \subset X$ the closed subset consisting of points lying over the closed subset $\Spec(A_1) = \Spec(A_2) = \ldots$ of $\Spec(A)$. There exists an open $W \subset X$ containing $X_0$ such that \begin{enumerate} \item $H^i(K)|_W$ is zero unless $i = 0$, \item $\mathcal{F} = H^0(K)|_W$ is of finite presentation, and \item $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$. \end{enumerate}


flat-lemma-proper-support

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Let $W \subset X$ be as in Lemma \ref{lemma-finitely-presented}. Set $\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$, the support of $\mathcal{F}$ is proper over $A$.


flat-lemma-monomorphism-isomorphism

Let $A = \lim A_n$ be a limit of a system of rings whose transition maps are surjective and with locally nilpotent kernels. Let $S = \Spec(A)$. Let $T \to S$ be a monomorphism which is locally of finite type. If $\Spec(A_n) \to S$ factors through $T$ for all $n$, then $T = S$.


flat-lemma-compute-what-it-should-be-derived

In Situation \ref{situation-existence-derived} consider $$ K = R\lim_{D_\QCoh(\mathcal{O}_X)}(K_n) = DQ_X(R\lim_{D(\mathcal{O}_X)} K_n) $$ Then $K$ is in $D^-_{\QCoh}(\mathcal{O}_X)$.


flat-lemma-compute-against-perfect-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. For any perfect object $E$ of $D(\mathcal{O}_X)$ the cohomology $$ M = R\Gamma(X, K \otimes^\mathbf{L} E) $$ is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism $$ R\Gamma(X_n, K_n \otimes^\mathbf{L} E|_{X_n}) = M \otimes_A^\mathbf{L} A_n $$ in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.


flat-lemma-relative-pseudo-coherence-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. Then $K$ is pseudo-coherent on $X$.


flat-lemma-compute-over-affine-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. For any quasi-compact open $U \subset X$ we have $$ R\Gamma(U, K) \otimes_A^\mathbf{L} A_n = R\Gamma(U_n, K_n) $$ in $D(A_n)$ where $U_n = U \cap X_n$.


flat-lemma-helper-blowup-affine-space

Let $R$ be a ring and let $f \in R$. Let $r\geq 0$ be an integer. Let $R \to S$ be a ring map and let $M$ be an $S$-module. Assume \begin{enumerate} \item $R \to S$ is of finite presentation and flat, \item every fibre ring $S \otimes_R \kappa(\mathfrak p)$ is geometrically integral over $R$, \item $M$ is a finite $S$-module, \item $M_f$ is a finitely presented $S_f$-module, \item for all $\mathfrak p \in R$, $f \not \in \mathfrak p$ with $\mathfrak q = \mathfrak pS$ the module $M_{\mathfrak q}$ is free of rank $r$ over $S_\mathfrak q$. \end{enumerate} Then there exists a finitely generated ideal $I \subset R$ with $V(f) = V(I)$ such that for all $a \in I$ with $R' = R[\frac{I}{a}]$ the quotient $$ M' = (M \otimes_R R')/a\text{-power torsion} $$ over $S' = S \otimes_R R'$ satisfies the following: for every prime $\mathfrak p' \subset R'$ there exists a $g \in S'$, $g \not \in \mathfrak p'S'$ such that $M'_g$ is a free $S'_g$-module of rank $r$.


flat-lemma-flatten-module-pre

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume \begin{enumerate} \item $X \to S$ is affine, of finite presentation, flat, geometrically integral fibres, \item $\mathcal{F}$ is a module of finite type, \item $\mathcal{F}_U$ is of finite presentation, \item $\mathcal{F}$ is flat over $S$ at all generic points of fibres lying over points of $U$. \end{enumerate} Then there exists a $U$-admissible blowup $S' \to S$ and an open subscheme $V \subset X_{S'}$ such that (a) the strict transform $\mathcal{F}'$ of $\mathcal{F}$ restricts to a finitely locally free $\mathcal{O}_V$-module and (b) $V \to S'$ is surjective.


flat-lemma-trick-fitting-ideal

Let $A \to C$ be a finite locally free ring map of rank $d$. Let $h \in C$ be an element such that $C_h$ is \'etale over $A$. Let $J \subset C$ be an ideal. Set $I = \text{Fit}_0(C/J)$ where we think of $C/J$ as a finite $A$-module. Then $IC_h = JJ'$ for some ideal $J' \subset C_h$. If $J$ is finitely generated so are $I$ and $J'$.


flat-lemma-push-ideal

Let $A \to B$ be an \'etale ring map. Let $a \in A$ be a nonzerodivisor. Let $J \subset B$ be a finite type ideal with $V(J) \subset V(aB)$. For every $\mathfrak q \subset B$ there exists a finite type ideal $I \subset A$ with $V(I) \subset V(a)$ and $g \in B$, $g \not \in \mathfrak q$ such that $IB_g = JJ'$ for some finite type ideal $J' \subset B_g$.


flat-lemma-flatten-module-etale-localize

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume there exist finitely many commutative diagrams $$ \xymatrix{ & X_i \ar[r]_{j_i} \ar[d] & X \ar[d] \\ S_i^* \ar[r] & S_i \ar[r]^{e_i} & S } $$ where \begin{enumerate} \item $e_i : S_i \to S$ are quasi-compact \'etale morphisms and $S = \bigcup e_i(S_i)$, \item $j_i : X_i \to X$ are \'etale morphisms and $X = \bigcup j_i(X_i)$, \item $S^*_i \to S_i$ is an $e_i^{-1}(U)$-admissible blowup such that the strict transform $\mathcal{F}_i^*$ of $j_i^*\mathcal{F}$ is flat over $S^*_i$. \end{enumerate} Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $\mathcal{F}$ is flat over $S'$.


flat-lemma-flat-after-blowing-up

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume \begin{enumerate} \item $X \to S$ is of finite type and quasi-separated, and \item $X_U \to U$ is flat and locally of finite presentation. \end{enumerate} Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is flat and of finite presentation over $S'$.


flat-lemma-finite-after-blowing-up

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume \begin{enumerate} \item $X \to S$ is proper, and \item $X_U \to U$ is finite locally free. \end{enumerate} Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is finite locally free over $S'$.


flat-lemma-zariski-after-blowup

Let $\varphi : X \to S$ be a separated morphism of finite type with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is isomorphic to an open subscheme of $S'$.


flat-lemma-dominate-modification-by-blowup

Let $\varphi : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$, i.e., such that there exists a factorization $S' \to X \to S$ of the blowup morphism.