more-algebra-lemma-largest-ideal-henselian

Let $A$ be a ring. There exists a largest ideal $I \subset A$ such that $(A, I)$ is a henselian pair.


more-algebra-lemma-irreducible-henselian-pair-connected

Let $(A, I)$ be a henselian pair. Let $\mathfrak p \subset A$ be a prime ideal. Then $V(\mathfrak p + I)$ is connected.


more-algebra-lemma-henselization

The inclusion functor $$ \text{category of henselian pairs} \longrightarrow \text{category of pairs} $$ has a left adjoint $(A, I) \mapsto (A^h, I^h)$.


more-algebra-lemma-henselization-flat

Let $(A, I)$ be a pair. Let $(A^h, I^h)$ be as in  Lemma \ref{lemma-henselization}. Then $A \to A^h$ is flat, $I^h = IA^h$ and $A/I^n \to A^h/I^nA^h$ is an isomorphism for all $n$.


more-algebra-lemma-henselization-Noetherian-pair

\begin{slogan} The henselization of a Noetherian pair is a Noetherian pair with the same completion \end{slogan} Let $(A, I)$ be a pair with $A$ Noetherian.  Let $(A^h, I^h)$ be as in  Lemma \ref{lemma-henselization}. Then the map of $I$-adic completions $$ A^\wedge \to (A^h)^\wedge $$ is an isomorphism. Moreover, $A^h$ is Noetherian, the maps $A \to A^h \to A^\wedge$ are flat, and $A^h \to A^\wedge$ is faithfully flat.


more-algebra-lemma-henselization-colimit

Let $(A, I) = \colim (A_i, I_i)$ be a filtered colimit of pairs. The functor of Lemma \ref{lemma-henselization} gives $A^h = \colim A_i^h$ and $I^h = \colim I_i^h$.


more-algebra-lemma-henselization-change-ideal

\begin{slogan} The henselization of a pair only depends on the radical of the ideal \end{slogan} Let $A$ be a ring with ideals $I$ and $J$. If $V(I) = V(J)$ then the functor of Lemma \ref{lemma-henselization} produces the same ring for the pair $(A, I)$ as for the pair $(A, J)$.


more-algebra-lemma-henselization-integral

\begin{slogan} Henselization commutes with integral base change \end{slogan} Let $(A, I) \to (B, J)$ be a map of pairs such that $V(J) = V(IB)$. Let $(A^h , I^h) \to (B^h, J^h)$ be the induced map on henselizations (Lemma \ref{lemma-henselization}). If $A \to B$ is integral, then the induced map $A^h \otimes_A B \to B^h$ is an isomorphism.


more-algebra-lemma-lift-finite-projective-module

Let $(R, I)$ be a henselian pair. Let $\overline{P}$ be a finite projective $R/I$-module. Then there exists a finite projective $R$-module $P$ such that $P/IP \cong \overline{P}$.


more-algebra-lemma-finite-etale-equivalence

Let $(A, I)$ be a henselian pair. The functor $B \to B/IB$ determines an equivalence between finite \'etale $A$-algebras and finite \'etale $A/I$-algebras.


more-algebra-lemma-lim-finite-projective-gives-finite-projective

Let $A = \lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $A_n$-modules $M_n$ and $A_{n + 1}$-module maps $M_{n + 1} \to M_n$. Assume \begin{enumerate} \item the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels, \item $M_1$ is a finite projective $A_1$-module, \item $M_n$ is a finite flat $A_n$-module, and \item the maps induce isomorphisms $M_{n + 1} \otimes_{A_{n + 1}} A_n \to M_n$. \end{enumerate} Then $M = \lim M_n$ is a finite projective $A$-module and $M \otimes_A A_n \to M_n$ is an isomorphism for all $n$.


more-algebra-lemma-absolutely-integrally-closed

Let $A$ be a ring. The following are equivalent \begin{enumerate} \item $A$ is absolutely integrally closed, and \item any monic $f \in A[T]$ has a root in $A$. \end{enumerate}


more-algebra-lemma-absolutely-integrally-closed-quotient-localization

Let $A$ be absolutely integrally closed. \begin{enumerate} \item Any quotient ring $A/I$ of $A$ is absolutely integrally closed. \item Any localization $S^{-1}A$ is absolutely integrally closed. \end{enumerate}


more-algebra-lemma-integrally-closed-in-absolutely-integrally-closed

Let $A$ be a ring. Let $S \subset A$ be a multiplicative subset. If $S^{-1}A$ is absolutely integrally closed and $A \subset S^{-1}A$ is integrally closed in $S^{-1}A$, then $A$ is absolutely integrally closed.


more-algebra-lemma-absolutely-integrally-closed-henselian-pair

Let $A$ be absolutely integrally closed. Let $I \subset A$ be an ideal. Then $(A, I)$ is a henselian pair if (and only if) the following conditions hold \begin{enumerate} \item $I$ is contained in the Jacobson radical of $A$, \item $A \to A/I$ induces a bijection on idempotents. \end{enumerate}


more-algebra-lemma-auto-ass-implies-P

An auto-associated ring $R$ has the following property: (P) Every proper finitely generated ideal $I \subset R$ has a nonzero annihilator.


more-algebra-lemma-P-universally-injective

Let $R$ be a ring having property (P) of Lemma \ref{lemma-auto-ass-implies-P}. Let $u : N \to M$ be a homomorphism of projective $R$-modules. Then $u$ is universally injective if and only if $u$ is injective.


more-algebra-lemma-P-fPD-zero

Let $R$ be a ring. The following are equivalent \begin{enumerate} \item $R$ has property (P) of Lemma \ref{lemma-auto-ass-implies-P}, \item any injective map of projective $R$-modules is universally injective, \item if $u : N \to M$ is injective and $N$, $M$ are finite projective $R$-modules then $\Coker(u)$ is a finite projective $R$-module, \item if $N \subset M$ and $N$, $M$ are finite projective as $R$-modules, then $N$ is a direct summand of $M$, and \item any injective map $R \to R^{\oplus n}$ is a split injection. \end{enumerate}


more-algebra-lemma-exact-length-1

Let $R$ be a ring. Suppose that $\varphi : R^m \to R^n$ is a map of finite free modules. The following are equivalent \begin{enumerate} \item $\varphi$ is injective, \item the rank of $\varphi$ is $m$ and the annihilator of $I(\varphi)$ in $R$ is zero. \end{enumerate} If $R$ is Noetherian these are also equivalent to \begin{enumerate} \item[(3)] the rank of $\varphi$ is $m$ and either $I(\varphi) = R$ or it contains a nonzerodivisor. \end{enumerate} Here the rank of $\varphi$ and $I(\varphi)$ are defined as in Algebra, Definition \ref{algebra-definition-rank}.


more-algebra-lemma-coker-injective-free

Let $R$ be a ring. Suppose that $\varphi : R^n \to R^n$ be an injective map of finite free modules of the same rank. Then $\Hom_R(\Coker(\varphi), R) = 0$.


more-algebra-lemma-intersection-flat

Let $R$ be a ring. Let $M$ be an $R$-module. Let $I_1$, $I_2$ be ideals of $R$. If $M/I_1M$ is flat over $R/I_1$ and $M/I_2M$ is flat over $R/I_2$, then $M/(I_1 \cap I_2)M$ is flat over $R/(I_1 \cap I_2)$.


more-algebra-lemma-flattening-artinian

Let $R$ be an Artinian ring. Let $M$ be an $R$-module. Then there exists a smallest ideal $I \subset R$ such that $M/IM$ is flat over $R/I$.


more-algebra-lemma-flattening-artinian-universal-property

Let $R$ be an Artinian ring. Let $M$ be an $R$-module. Let $I \subset R$ be the smallest ideal $I \subset R$ such that $M/IM$ is flat over $R/I$. Then $I$ has the following universal property: For every ring map $\varphi : R \to R'$ we have $$ R' \otimes_R M\text{ is flat over }R' \Leftrightarrow \text{we have }\varphi(I) = 0. $$


more-algebra-lemma-base-change-flat-at-primes-over

Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal. Let $M$ be an $S$-module. Let $R \to R'$ be a ring map and $IR' \subset I' \subset R'$ an ideal. If (\ref{equation-flat-at-primes-over}) holds for $(R \to S, I, M)$, then (\ref{equation-flat-at-primes-over}) holds for $(R' \to S \otimes_R R', I', M \otimes_R R')$.


more-algebra-lemma-limit-preserving-flat-at-primes-over

Let $R \to S$ be a ring map of finite presentation. Let $M$ be an $S$-module of finite presentation. Let $R' = \colim_{\lambda \in \Lambda} R_\lambda$ be a directed colimit of $R$-algebras. Let $I_\lambda \subset R_\lambda$ be ideals such that $I_\lambda R_\mu \subset I_\mu$ for all $\mu \geq \lambda$ and set $I' = \colim_\lambda I_\lambda$. If (\ref{equation-flat-at-primes-over}) holds for $(R' \to S \otimes_R R', I', M \otimes_R R')$, then there exists a $\lambda \in \Lambda$ such that (\ref{equation-flat-at-primes-over}) holds for $(R_\lambda \to S \otimes_R R_\lambda, I_\lambda, M \otimes_R R_\lambda)$.


more-algebra-lemma-flat-descent-flat-at-primes

In Situation \ref{situation-flattening-general} let $R' \to R''$ be an $R$-algebra map. Let $I' \subset R'$ and $I'R'' \subset I'' \subset R''$ be ideals. Assume \begin{enumerate} \item the map $V(I'') \to V(I')$ induced by $\Spec(R'') \to \Spec(R')$ is surjective, and \item $R''_{\mathfrak p''}$ is flat over $R'$ for all primes $\mathfrak p'' \in V(I'')$. \end{enumerate} If (\ref{equation-flat-at-primes}) holds for $(R'', I'')$, then (\ref{equation-flat-at-primes}) holds for $(R', I')$.


more-algebra-lemma-limit-preserving-flat-at-primes

In Situation \ref{situation-flattening-general} assume $R \to S$ is essentially of finite presentation and $M$ is an $S$-module of finite presentation. Let $R' = \colim_{\lambda \in \Lambda} R_\lambda$ be a directed colimit of $R$-algebras. Let $I_\lambda \subset R_\lambda$ be ideals such that $I_\lambda R_\mu \subset I_\mu$ for all $\mu \geq \lambda$ and set $I' = \colim_\lambda I_\lambda$. If (\ref{equation-flat-at-primes}) holds for $(R', I')$, then there exists a $\lambda \in \Lambda$ such that (\ref{equation-flat-at-primes}) holds for $(R_\lambda, I_\lambda)$.


more-algebra-lemma-flattening-complete-local-noetherian

Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Assume \begin{enumerate} \item $(R, \mathfrak m)$ is a complete local Noetherian ring, \item $S$ is a Noetherian ring, and \item $M$ is finite over $S$. \end{enumerate} Then there exists an ideal $I \subset \mathfrak m$ such that \begin{enumerate} \item $(M/IM)_{\mathfrak q}$ is flat over $R/I$ for all primes $\mathfrak q$ of $S/IS$ lying over $\mathfrak m$, and \item if $J \subset R$ is an ideal such that $(M/JM)_{\mathfrak q}$ is flat over $R/J$ for all primes $\mathfrak q$ lying over $\mathfrak m$, then $I \subset J$. \end{enumerate} In other words, $I$ is the smallest ideal of $R$ such that (\ref{equation-flat-at-primes-over}) holds for $(\overline{R} \to \overline{S}, \overline{\mathfrak m}, \overline{M})$ where $\overline{R} = R/I$, $\overline{S} = S/IS$, $\overline{\mathfrak m} = \mathfrak m/I$ and $\overline{M} = M/IM$.


more-algebra-lemma-flattening-complete-local-noetherian-property-by-finite-type

With notation $R \to S$, $M$, and $I$ and assumptions as in Lemma \ref{lemma-flattening-complete-local-noetherian}. Consider a local homomorphism of local rings $\varphi : (R, \mathfrak m) \to (R', \mathfrak m')$ such that $R'$ is Noetherian. Then the following are equivalent \begin{enumerate} \item condition (\ref{equation-flat-at-primes-over}) holds for $(R' \to S \otimes_R R', \mathfrak m', M \otimes_R R')$, and \item $\varphi(I) = 0$. \end{enumerate}


more-algebra-lemma-flattening-complete-local-universal-property

With notation $R \to S$, $M$, and $I$ and assumptions as in Lemma \ref{lemma-flattening-complete-local-noetherian}. In addition assume that $R \to S$ is of finite type. Then for any local homomorphism of local rings $\varphi : (R, \mathfrak m) \to (R', \mathfrak m')$ the following are equivalent \begin{enumerate} \item condition (\ref{equation-flat-at-primes-over}) holds for $(R' \to S \otimes_R R', \mathfrak m', M \otimes_R R')$, and \item $\varphi(I) = 0$. \end{enumerate}


more-algebra-lemma-have-one-root

Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. Let $\alpha \in R$ be such that $P(\alpha) = 0$. Then $P(T) = (T - \alpha)Q(T)$ for some monic polynomial $Q(T) \in R[T]$.


more-algebra-lemma-adjoin-one-root

Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. There exists a finite free ring map $R \to R'$ such that $P(T) = (T - \alpha)Q(T)$ for some $\alpha \in R'$ and some monic polynomial $Q(T) \in R'[T]$.


more-algebra-lemma-finite-split

Let $R \to S$ be a finite ring map. There exists a finite free ring extension $R \subset R'$ such that $S \otimes_R R'$ is a quotient of a ring of the form $$ R'[T_1, \ldots, T_n]/(P_1(T_1), \ldots, P_n(T_n)) $$ with $P_i(T) = \prod_{j = 1, \ldots, d_i} (T - \alpha_{ij})$ for some $\alpha_{ij} \in R'$.


more-algebra-lemma-split-image

Let $R$ be a ring. Let $S = R[T_1, \ldots, T_n]/J$. Assume $J$ contains elements of the form $P_i(T_i)$ with $P_i(T) = \prod_{j = 1, \ldots, d_i} (T - \alpha_{ij})$ for some $\alpha_{ij} \in R$. For $\underline{k} = (k_1, \ldots, k_n)$ with $1 \leq k_i \leq d_i$ consider the ring map $$ \Phi_{\underline{k}} : R[T_1, \ldots, T_n] \to R, \quad T_i \longmapsto \alpha_{ik_i} $$ Set $J_{\underline{k}} = \Phi_{\underline{k}}(J)$. Then the image of $\Spec(S) \to \Spec(R)$ is equal to $V(\bigcap J_{\underline{k}})$.


more-algebra-lemma-descent-flatness-injective-finite-Noetherian-rings

Let $R \to S$ be a finite injective homomorphism of Noetherian rings. Let $M$ be an $R$-module. If $M \otimes_R S$ is a flat $S$-module, then $M$ is a flat $R$-module.


more-algebra-lemma-torsion

Let $R$ be a domain. Let $M$ be an $R$-module. The set of torsion elements of $M$ forms a submodule $M_{tors} \subset M$. The quotient module $M/M_{tors}$ is torsion free.


more-algebra-lemma-localize-torsion

Let $R$ be a domain. Let $M$ be a torsion free $R$-module. For any multiplicative set $S \subset R$ the module $S^{-1}M$ is a torsion free $S^{-1}R$-module.


more-algebra-lemma-flat-pullback-torsion

Let $R \to R'$ be a flat homomorphism of domains. If $M$ is a torsion free $R$-module, then $M \otimes_R R'$ is a torsion free $R'$-module.


more-algebra-lemma-extension-torsion-free

Let $R$ be a domain. Let $0 \to M \to M' \to M'' \to 0$ be a short exact sequence of $R$-modules. If $M$ and $M''$ are torsion free, then $M'$ is torsion free.


more-algebra-lemma-check-torsion

Let $R$ be a domain. Let $M$ be an $R$-module. Then $M$ is torsion free if and only if $M_\mathfrak m$ is a torsion free $R_\mathfrak m$-module for all maximal ideals $\mathfrak m$ of $R$.


more-algebra-lemma-finite-torsion-free-submodule-free

Let $R$ be a domain. Let $M$ be a finite $R$-module. Then $M$ is torsion free if and only if $M$ is a submodule of a finite free module.


more-algebra-lemma-torsion-free-finite-noetherian-domain

Let $R$ be a Noetherian domain. Let $M$ be a nonzero finite $R$-module. The following are equivalent \begin{enumerate} \item $M$ is torsion free, \item $M$ is a submodule of a finite free module, \item $(0)$ is the only associated prime of $M$, \item $(0)$ is in the support of $M$ and $M$ has property $(S_1)$, and \item $(0)$ is in the support of $M$ and $M$ has no embedded associated prime. \end{enumerate}


more-algebra-lemma-flat-torsion-free

Let $R$ be a domain. Any flat $R$-module is torsion free.


more-algebra-lemma-valuation-ring-torsion-free-flat

Let $A$ be a valuation ring. An $A$-module $M$ is flat over $A$ if and only if $M$ is torsion free.


more-algebra-lemma-dedekind-torsion-free-flat

Let $A$ be a Dedekind domain (for example a discrete valuation ring or more generally a PID). \begin{enumerate} \item An $A$-module is flat if and only if it is torsion free. \item A finite torsion free $A$-module is finite locally free. \item A finite torsion free $A$-module is finite free if $A$ is a PID. \end{enumerate}


more-algebra-lemma-hom-into-torsion-free

Let $R$ be a domain. Let $M$, $N$ be $R$-modules. If $N$ is torsion free, so is $\Hom_R(M, N)$.


more-algebra-lemma-reflexive-torsion-free

Let $R$ be a domain and let $M$ be an $R$-module. \begin{enumerate} \item If $M$ is reflexive, then $M$ is torsion free. \item If $M$ is finite, then $j : M \to \Hom_R(\Hom_R(M, R), R)$ is injective if and only if $M$ is torsion free \end{enumerate}


more-algebra-lemma-cokernel-map-double-dual-dvr

Let $R$ be a discrete valuation ring and let $M$ be a finite $R$-module. Then the map $j : M \to \Hom_R(\Hom_R(M, R), R)$ is surjective.


more-algebra-lemma-check-reflexive

Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent: \begin{enumerate} \item $M$ is reflexive, \item $M_\mathfrak p$ is a reflexive $R_\mathfrak p$-module for all primes $\mathfrak p \subset R$, and \item $M_\mathfrak m$ is a reflexive $R_\mathfrak m$-module for all maximal ideals $\mathfrak m$ of $R$. \end{enumerate}


more-algebra-lemma-sequence-reflexive

Let $R$ be a Noetherian domain. Let $0 \to M \to M' \to M''$ an exact sequence of finite $R$-modules. If $M'$ is reflexive and $M''$ is torsion free, then $M$ is reflexive.


more-algebra-lemma-characterize-reflexive

Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent \begin{enumerate} \item $M$ is reflexive, \item there exists a short exact sequence $0 \to M \to F \to N \to 0$ with $F$ finite free and $N$ torsion free. \end{enumerate}


more-algebra-lemma-flat-pullback-reflexive

Let $R \to R'$ be a flat homomorphism of Noetherian domains. If $M$ is a finite reflexive $R$-module, then $M \otimes_R R'$ is a finite reflexive $R'$-module.


more-algebra-lemma-dual-reflexive

Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. Let $N$ be a finite reflexive $R$-module. Then $\Hom_R(M, N)$ is reflexive.


more-algebra-lemma-hom-into-depth

Let $R$ be a Noetherian local ring. Let $M$, $N$ be finite $R$-modules. \begin{enumerate} \item If $N$ has depth $\geq 1$, then $\Hom_R(M, N)$ has depth $\geq 1$. \item If $N$ has depth $\geq 2$, then $\Hom_R(M, N)$ has depth $\geq 2$. \end{enumerate}


more-algebra-lemma-hom-into-S2

Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules. \begin{enumerate} \item If $N$ has property $(S_1)$, then $\Hom_R(M, N)$ has property $(S_1)$. \item If $N$ has property $(S_2)$, then $\Hom_R(M, N)$ has property $(S_2)$. \item If $R$ is a domain, $N$ is torsion free and $(S_2)$, then $\Hom_R(M, N)$ is torsion free and has property $(S_2)$. \end{enumerate}


more-algebra-lemma-check-injective-on-ass

Let $R$ be a Noetherian ring. Let $\varphi : M \to N$ be a map of $R$-modules. Assume that for every prime $\mathfrak p$ of $R$ at least one of the following happens \begin{enumerate} \item $M_\mathfrak p \to N_\mathfrak p$ is injective, or \item $\mathfrak p \not \in \text{Ass}(M)$. \end{enumerate} Then $\varphi$ is injective.


more-algebra-lemma-check-isomorphism-via-depth-and-ass

Let $R$ be a Noetherian ring. Let $\varphi : M \to N$ be a map of $R$-modules. Assume $M$ is finite and that for every prime $\mathfrak p$ of $R$ one of the following happens \begin{enumerate} \item $M_\mathfrak p \to N_\mathfrak p$ is an isomorphism, or \item $\text{depth}(M_\mathfrak p) \geq 2$ and $\mathfrak p \not \in \text{Ass}(N)$. \end{enumerate} Then $\varphi$ is an isomorphism.


more-algebra-lemma-isom-depth-2-torsion-free

Let $R$ be a Noetherian domain. Let $\varphi : M \to N$ be a map of $R$-modules. Assume $M$ is finite, $N$ is torsion free, and that for every prime $\mathfrak p$ of $R$ one of the following happens \begin{enumerate} \item $M_\mathfrak p \to N_\mathfrak p$ is an isomorphism, or \item $\text{depth}(M_\mathfrak p) \geq 2$. \end{enumerate} Then $\varphi$ is an isomorphism.


more-algebra-lemma-reflexive-depth-2

Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent \begin{enumerate} \item $M$ is reflexive, \item for every prime $\mathfrak p$ of $R$ one of the following happens \begin{enumerate} \item $M_\mathfrak p$ is a reflexive $R_\mathfrak p$-module, or \item $\text{depth}(M_\mathfrak p) \geq 2$. \end{enumerate} \end{enumerate}


more-algebra-lemma-reflexive-S2

Let $R$ be a Noetherian domain. Let $M$ be a finite reflexive $R$-module. Let $\mathfrak p \subset R$ be a prime ideal. \begin{enumerate} \item If $\text{depth}(R_\mathfrak p) \geq 2$, then $\text{depth}(M_\mathfrak p) \geq 2$. \item If $R$ is $(S_2)$, then $M$ is $(S_2)$. \end{enumerate}


more-algebra-lemma-reflexive-over-normal

Let $R$ be a Noetherian normal domain with fraction field $K$. Let $M$ be a finite $R$-module. The following are equivalent \begin{enumerate} \item $M$ is reflexive, \item $M$ is torsion free and has property $(S_2)$, \item $M$ is torsion free and $M = \bigcap_{\text{height}(\mathfrak p) = 1} M_{\mathfrak p}$ where the intersection happens in $M_K = M \otimes_R K$. \end{enumerate}


more-algebra-lemma-describe-reflexive-hull

Let $R$ be a Noetherian normal domain. Let $M$ be a finite $R$-module. Then the reflexive hull of $M$ is the intersection $$ M^{**} = \bigcap\nolimits_{\text{height}(\mathfrak p) = 1} M_{\mathfrak p}/(M_\mathfrak p)_{tors} = \bigcap\nolimits_{\text{height}(\mathfrak p) = 1} (M/M_{tors})_\mathfrak p $$ taken in $M \otimes_R K$.


more-algebra-lemma-integral-closure-reflexive

Let $A$ be a Noetherian normal domain with fraction field $K$. Let $L$ be a finite extension of $K$. If the integral closure $B$ of $A$ in $L$ is finite over $A$, then $B$ is reflexive as an $A$-module.


more-algebra-lemma-content-finitely-generated

Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \in M$. The content ideal of $x$, if it exists, is finitely generated.


more-algebra-lemma-equal-content

Let $(A, \mathfrak m)$ be a local ring. Let $u : M \to N$ be a map of flat $A$-modules such that $\overline{u} : M/\mathfrak m M \to N/\mathfrak m N$ is injective. If $x \in M$ has content ideal $I$, then $u(x)$ has content ideal $I$ as well.


more-algebra-lemma-content-exists-flat-Mittag-Leffler

Let $A$ be a ring. Let $M$ be a flat Mittag-Leffler module. Then every element of $M$ has a content ideal.


more-algebra-lemma-flat-finite-type-finite-presentation-local-module

Let $R$ be a ring. Let $S = R[x_1, \ldots, x_n]$ be a polynomial ring over $R$. Let $M$ be an $S$-module. Assume \begin{enumerate} \item there exist finitely many primes $\mathfrak p_1, \ldots, \mathfrak p_m$ of $R$ such that the map $R \to \prod R_{\mathfrak p_j}$ is injective, \item $M$ is a finite $S$-module, \item $M$ flat over $R$, and \item for every prime $\mathfrak p$ of $R$ the module $M_{\mathfrak p}$ is of finite presentation over $S_{\mathfrak p}$. \end{enumerate} Then $M$ is of finite presentation over $S$.


more-algebra-lemma-flat-finite-type-finite-presentation-local

Let $R \to S$ be a ring homomorphism. Assume \begin{enumerate} \item there exist finitely many primes $\mathfrak p_1, \ldots, \mathfrak p_m$ of $R$ such that the map $R \to \prod R_{\mathfrak p_j}$ is injective, \item $R \to S$ is of finite type, \item $S$ flat over $R$, and \item for every prime $\mathfrak p$ of $R$ the ring $S_{\mathfrak p}$ is of finite presentation over $R_{\mathfrak p}$. \end{enumerate} Then $S$ is of finite presentation over $R$.


more-algebra-lemma-flat-graded-finite-type-finite-presentation-module

Let $R$ be a ring. Let $S = R[x_1, \ldots, x_n]$ be a graded polynomial algebra over $R$, i.e., $\deg(x_i) > 0$ but not necessarily equal to $1$. Let $M$ be a graded $S$-module. Assume \begin{enumerate} \item $R$ is a local ring, \item $M$ is a finite $S$-module, and \item $M$ is flat over $R$. \end{enumerate} Then $M$ is finitely presented as an $S$-module.


more-algebra-lemma-flat-graded-finite-type-finite-presentation

Let $R$ be a ring. Let $S = \bigoplus_{n \geq 0} S_n$ be a graded $R$-algebra. Let $M = \bigoplus_{d \in \mathbf{Z}} M_d$ be a graded $S$-module. Assume $S$ is finitely generated as an $R$-algebra, assume $S_0$ is a finite $R$-algebra, and assume there exist finitely many primes $\mathfrak p_j$, $i = 1, \ldots, m$ such that $R \to \prod R_{\mathfrak p_j}$ is injective. \begin{enumerate} \item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra. \item If $M$ is flat as an $R$-module and finite as an $S$-module, then $M$ is finitely presented as an $S$-module. \end{enumerate}


more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation

\begin{reference} \cite[Theorem 3]{Nagata-Finitely} \end{reference} Let $A$ be a valuation ring. Let $A \to B$ be a ring map of finite type. Let $M$ be a finite $B$-module. \begin{enumerate} \item If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra. \item If $M$ is flat as an $A$-module, then $M$ is finitely presented as a $B$-module. \end{enumerate}


more-algebra-lemma-blowup-fitting-ideal

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$ and $I = \text{Fit}_k(M)$. For every $a \in I$ with $R' = R[\frac{I}{a}]$ the strict transform $$ M' = (M \otimes_R R')/a\text{-power torsion} $$ has $\text{Fit}_k(M') = R'$.


more-algebra-lemma-blowup-fitting-ideal-locally-free

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$ and $I = \text{Fit}_k(M)$. Asssume that $M_\mathfrak p$ is free of rank $k$ for every $\mathfrak p \not \in V(I)$. Then for every $a \in I$ with $R' = R[\frac{I}{a}]$ the strict transform $$ M' = (M \otimes_R R')/a\text{-power torsion} $$ is locally free of rank $k$.


more-algebra-lemma-blowup-module

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $f \in R$ be an element such that $M_f$ is finite locally free of rank $r$. 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 strict transform $$ M' = (M \otimes_R R')/a\text{-power torsion} $$ is locally free of rank $r$.


more-algebra-lemma-ui-completion-direct-sum-into-product

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 $I$. There is a canonical map $$ \left(\bigoplus\nolimits_{\alpha \in A} R\right)^\wedge \longrightarrow \prod\nolimits_{\alpha \in A} R $$ from the $I$-adic completion of the direct sum into the product which is universally injective.


more-algebra-lemma-completed-direct-sum-flat

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian. The completion $(\bigoplus\nolimits_{\alpha \in A} R)^\wedge$ is a flat $R$-module.


more-algebra-lemma-tor-strictly-pro-zero

\begin{reference} This is \cite[Lemma 9.9]{quillenhomology}; note that the author forgot the word ``strict'' in the statement although it was clearly intended. \end{reference} Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $M$ be a finite $A$-module. For every $p > 0$ there exists a $c > 0$ such that $\text{Tor}_p^A(M, A/I^n) \to \text{Tor}_p^A(M, A/I^{n - c})$ is zero for all $n \geq c$.


more-algebra-lemma-limit-flat

Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $(M_n)$ be an inverse system of $A$-modules such that \begin{enumerate} \item $M_n$ is a flat $A/I^n$-module, \item $M_{n + 1} \to M_n$ is surjective. \end{enumerate} Then $M = \lim M_n$ is a flat $A$-module and $Q \otimes_A M = \lim Q \otimes_A M_n$ for every finite $A$-module $Q$.


more-algebra-lemma-flat-after-completion

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. Assume \begin{enumerate} \item $I$ is finitely generated, \item $R/I$ is Noetherian, \item $M/IM$ is flat over $R/I$, \item $\text{Tor}_1^R(M, R/I) = 0$. \end{enumerate} Then the $I$-adic completion $R^\wedge$ is a Noetherian ring and $M^\wedge$ is flat over $R^\wedge$.


more-algebra-lemma-functorial

Let $\varphi : E \to R$ and $\varphi' : E' \to R$ be $R$-module maps. Let $\psi : E \to E'$ be an $R$-module map such that $\varphi' \circ \psi = \varphi$. Then $\psi$ induces a homomorphism of differential graded algebras $K_\bullet(\varphi) \to K_\bullet(\varphi')$.


more-algebra-lemma-change-basis

Let $f_1, \ldots, f_r \in R$ be a sequence. Let $(x_{ij})$ be an invertible $r \times r$-matrix with coefficients in $R$. Then the complexes $K_\bullet(f_\bullet)$ and $$ K_\bullet(\sum x_{1j}f_j, \sum x_{2j}f_j, \ldots, \sum x_{rj}f_j) $$ are isomorphic.


more-algebra-lemma-homotopy-koszul-abstract

Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. Let $e \in E$ with image $f = \varphi(e)$ in $R$. Then $$ f = de + ed $$ as endomorphisms of $K_\bullet(\varphi)$.


more-algebra-lemma-homotopy-koszul

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$ be a sequence. Multiplication by $f_i$ on $K_\bullet(f_\bullet)$ is homotopic to zero, and in particular the cohomology modules $H_i(K_\bullet(f_\bullet))$ are annihilated by the ideal $(f_1, \ldots, f_r)$.


more-algebra-lemma-cone-koszul-abstract

Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. Let $f \in R$. Set $E' = E \oplus R$ and define $\varphi' : E' \to R$ by $\varphi$ on $E$ and multiplication by $f$ on $R$. The complex $K_\bullet(\varphi')$ is isomorphic to the cone of the map of complexes $$ f : K_\bullet(\varphi) \longrightarrow K_\bullet(\varphi). $$


more-algebra-lemma-cone-koszul

Let $R$ be a ring. Let $f_1, \ldots, f_r$ be a sequence of elements of $R$. The complex $K_\bullet(f_1, \ldots, f_r)$ is isomorphic to the cone of the map of complexes $$ f_r : K_\bullet(f_1, \ldots, f_{r - 1}) \longrightarrow K_\bullet(f_1, \ldots, f_{r - 1}). $$


more-algebra-lemma-cone-squared

Let $R$ be a ring. Let $A_\bullet$ be a complex of $R$-modules. Let $f, g \in R$. Let $C(f)_\bullet$ be the cone of $f : A_\bullet \to A_\bullet$. Define similarly $C(g)_\bullet$ and $C(fg)_\bullet$. Then $C(fg)_\bullet$ is homotopy equivalent to the cone of a map $$ C(f)_\bullet[1] \longrightarrow C(g)_\bullet $$


more-algebra-lemma-koszul-mult-abstract

Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. Let $f, g \in R$. Set $E' = E \oplus R$ and define $\varphi'_f, \varphi'_g, \varphi'_{fg} : E' \to R$ by $\varphi$ on $E$ and multiplication by $f, g, fg$ on $R$. The complex $K_\bullet(\varphi'_{fg})$ is isomorphic to the cone of a map of complexes $$ K_\bullet(\varphi'_f)[1] \longrightarrow K_\bullet(\varphi'_g). $$


more-algebra-lemma-koszul-mult

Let $R$ be a ring. Let $f_1, \ldots, f_{r - 1}$ be a sequence of elements of $R$. Let $f, g \in R$. The complex $K_\bullet(f_1, \ldots, f_{r - 1}, fg)$ is homotopy equivalent to the cone of a map of complexes $$ K_\bullet(f_1, \ldots, f_{r - 1}, f)[1] \longrightarrow K_\bullet(f_1, \ldots, f_{r - 1}, g) $$


more-algebra-lemma-join-sequences-koszul-complex

Let $R$ be a ring. Let $f_1, \ldots, f_r$, $g_1, \ldots, g_s$ be elements of $R$. Then there is an isomorphism of Koszul complexes $$ K_\bullet(R, f_1, \ldots, f_r, g_1, \ldots, g_s) = \text{Tot}(K_\bullet(R, f_1, \ldots, f_r) \otimes_R K_\bullet(R, g_1, \ldots, g_s)). $$


more-algebra-lemma-extended-alternating-is-complex

The extended alternating {\v C}ech complexes defined above are complexes of $R$-modules.


more-algebra-lemma-extended-alternating-form-module

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$. Let $M$ be an $R$-module. The extended alternating {\v C}ech complex of $M$ is the tensor product over $R$ of $M$ with the extended alternating {\v C}ech complex of $R$.


more-algebra-lemma-extended-alternating-base-change

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$. Let $M$ be an $R$-module. Let $R \to S$ be a ring map, denote $g_1, \ldots, g_r \in S$ the images of $f_1, \ldots, f_r$, and set $N = M \otimes_R S$. The extended alternating {\v C}ech complex constructed using $S$, $g_1, \ldots, g_r$, and $N$ is the tensor product of the extended alternating {\v C}ech complex of $M$ with $S$ over $R$.


more-algebra-lemma-extended-alternating-homotopy-zero

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$. Let $M$ be an $R$-module. If there exists an $i \in \{1, \ldots, r\}$ such that $f_i$ is a unit, then the extended alternating {\v C}ech complex of $M$ is homotopy equivalent to $0$.


more-algebra-lemma-extended-alternating-torsion

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$. Let $M$ be an $R$-module. Let $H^q$ be the $q$th cohomology module of the extended alternation {\v C}ech complex of $M$. Then \begin{enumerate} \item $H^q = 0$ if $q \not \in [0, r]$, \item for $x \in H^i$ there exists an $n \geq 1$ such that $f_i^n x = 0$ for $i = 1, \ldots, r$, \item the support of $H^q$ is contained in $V(f_1, \ldots, f_r)$, \item if there is an $f \in (f_1, \ldots, f_r)$ which acts invertibly on $M$, then $H^q = 0$. \end{enumerate}


more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$. The extended alternating {\v C}ech complex $$ R \to \bigoplus\nolimits_{i_0} R_{f_{i_0}} \to \bigoplus\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_r} $$ is a colimit of the Koszul complexes $K(R, f_1^n, \ldots, f_r^n)$; see proof for a precise statement.


more-algebra-lemma-regular-koszul-regular

An $M$-regular sequence is $M$-Koszul-regular. A regular sequence is Koszul-regular.


more-algebra-lemma-koszul-regular-H1-regular

A $M$-Koszul-regular sequence is $M$-$H_1$-regular. A Koszul-regular sequence is $H_1$-regular.


more-algebra-lemma-mult-koszul-regular

Let $f_1, \ldots, f_{r - 1} \in R$ be a sequence and $f, g \in R$. Let $M$ be an $R$-module. \begin{enumerate} \item If $f_1, \ldots, f_{r - 1}, f$ and $f_1, \ldots, f_{r - 1}, g$ are $M$-$H_1$-regular then $f_1, \ldots, f_{r - 1}, fg$ is $M$-$H_1$-regular too. \item If $f_1, \ldots, f_{r - 1}, f$ and $f_1, \ldots, f_{r - 1}, g$ are $M$-Koszul-regular then $f_1, \ldots, f_{r - 1}, fg$ is $M$-Koszul-regular too. \end{enumerate}


more-algebra-lemma-koszul-regular-flat-base-change

Let $\varphi : R \to S$ be a flat ring map. Let $f_1, \ldots, f_r \in R$. Let $M$ be an $R$-module and set $N = M \otimes_R S$. \begin{enumerate} \item If $f_1, \ldots, f_r$ in $R$ is an $M$-$H_1$-regular sequence, then $\varphi(f_1), \ldots, \varphi(f_r)$ is an $N$-$H_1$-regular sequence in $S$. \item If $f_1, \ldots, f_r$ is an $M$-Koszul-regular sequence in $R$, then $\varphi(f_1), \ldots, \varphi(f_r)$ is an $N$-Koszul-regular sequence in $S$. \end{enumerate}


more-algebra-lemma-H1-regular-quasi-regular

An $M$-$H_1$-regular sequence is $M$-quasi-regular.


more-algebra-lemma-noetherian-finite-all-equivalent

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \ldots, f_r \in \mathfrak m$. The following are equivalent \begin{enumerate} \item $f_1, \ldots, f_r$ is an $M$-regular sequence, \item $f_1, \ldots, f_r$ is a $M$-Koszul-regular sequence, \item $f_1, \ldots, f_r$ is an $M$-$H_1$-regular sequence, \item $f_1, \ldots, f_r$ is an $M$-quasi-regular sequence. \end{enumerate} In particular the sequence $f_1, \ldots, f_r$ is a regular sequence in $R$ if and only if it is a Koszul regular sequence, if and only if it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence.


more-algebra-lemma-H1-regular-in-quotient

Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $g_1, \ldots, g_m$ be a sequence in $A$ whose image in $A/I$ is $H_1$-regular. Then $I \cap (g_1, \ldots, g_m) = I(g_1, \ldots, g_m)$.


more-algebra-lemma-conormal-sequence-H1-regular

Let $A$ be a ring. Let $I \subset J \subset A$ be ideals. Assume that $J/I \subset A/I$ is generated by an $H_1$-regular sequence. Then $I \cap J^2 = IJ$.


more-algebra-lemma-join-quasi-regular-H1-regular

Let $A$ be a ring. Let $I$ be an ideal generated by a quasi-regular sequence $f_1, \ldots, f_n$ in $A$. Let $g_1, \ldots, g_m \in A$ be elements whose images $\overline{g}_1, \ldots, \overline{g}_m$ form an $H_1$-regular sequence in $A/I$. Then $f_1, \ldots, f_n, g_1, \ldots, g_m$ is a quasi-regular sequence in $A$.


more-algebra-lemma-join-H1-regular-sequences

Let $A$ be a ring. Let $I$ be an ideal generated by an $H_1$-regular sequence $f_1, \ldots, f_n$ in $A$. Let $g_1, \ldots, g_m \in A$ be elements whose images $\overline{g}_1, \ldots, \overline{g}_m$ form an $H_1$-regular sequence in $A/I$. Then $f_1, \ldots, f_n, g_1, \ldots, g_m$ is an $H_1$-regular sequence in $A$.


more-algebra-lemma-truncate-H1-regular

Let $A$ be a ring. Let $f_1, \ldots, f_n, g_1, \ldots, g_m \in A$ be an $H_1$-regular sequence. Then the images $\overline{g}_1, \ldots, \overline{g}_m$ in $A/(f_1, \ldots, f_n)$ form an $H_1$-regular sequence.


more-algebra-lemma-join-koszul-regular-sequences

Let $A$ be a ring. Let $I$ be an ideal generated by a Koszul-regular sequence $f_1, \ldots, f_n$ in $A$. Let $g_1, \ldots, g_m \in A$ be elements whose images $\overline{g}_1, \ldots, \overline{g}_m$ form a Koszul-regular sequence in $A/I$. Then $f_1, \ldots, f_n, g_1, \ldots, g_m$ is a Koszul-regular sequence in $A$.


more-algebra-lemma-truncate-koszul-regular

Let $A$ be a ring. Let $f_1, \ldots, f_n, g_1, \ldots, g_m \in A$. If both $f_1, \ldots, f_n$ and $f_1, \ldots, f_n, g_1, \ldots, g_m$ are Koszul-regular sequences in $A$, then $\overline{g}_1, \ldots, \overline{g}_m$ in $A/(f_1, \ldots, f_n)$ form a Koszul-regular sequence.


more-algebra-lemma-independence-of-generators

Let $R$ be a ring. Let $I$ be an ideal generated by $f_1, \ldots, f_r \in R$. \begin{enumerate} \item If $I$ can be generated by a quasi-regular sequence of length $r$, then $f_1, \ldots, f_r$ is a quasi-regular sequence. \item If $I$ can be generated by an $H_1$-regular sequence of length $r$, then $f_1, \ldots, f_r$ is an $H_1$-regular sequence. \item If $I$ can be generated by a Koszul-regular sequence of length $r$, then $f_1, \ldots, f_r$ is a Koszul-regular sequence. \end{enumerate}


more-algebra-lemma-make-nonzero-divisor

\begin{reference} This is a particular case of \cite[Corollary]{McCoy} \end{reference} Let $R$ be a ring. Let $a_1, \ldots, a_n \in R$ be elements such that $R \to R^{\oplus n}$, $x \mapsto (xa_1, \ldots, xa_n)$ is injective. Then the element $\sum a_i t_i$ of the polynomial ring $R[t_1, \ldots, t_n]$ is a nonzerodivisor.


more-algebra-lemma-Koszul-regular-flat-locally-regular

Let $R$ be a ring. Let $f_1, \ldots, f_n$ be a Koszul-regular sequence in $R$ such that $(f_1, \ldots, f_n) \not = R$. Consider the faithfully flat, smooth ring map $$ R \longrightarrow S = R[\{t_{ij}\}_{i \leq j}, t_{11}^{-1}, t_{22}^{-1}, \ldots, t_{nn}^{-1}] $$ For $1 \leq i \leq n$ set $$ g_i = \sum\nolimits_{i \leq j} t_{ij} f_j \in S. $$ Then $g_1, \ldots, g_n$ is a regular sequence in $S$ and $(f_1, \ldots, f_n)S = (g_1, \ldots, g_n)$.


more-algebra-lemma-vanishing-extended-alternating-koszul

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$ be an Koszul-regular sequence. Then the extended alternating {\v C}ech complex $R \to \bigoplus\nolimits_{i_0} R_{f_{i_0}} \to \bigoplus\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_r}$ from Section \ref{section-alternating-cech} only has cohomology in degree $r$.


more-algebra-lemma-blowup-regular-sequence

Let $a, a_2, \ldots, a_r$ be an $H_1$-regular sequence in a ring $R$ (for example a Koszul regular sequence or a regular sequence, see Lemmas \ref{lemma-regular-koszul-regular} and \ref{lemma-koszul-regular-H1-regular}). With $I = (a, a_2, \ldots, a_r)$ the blowup algebra $R' = R[\frac{I}{a}]$ is isomorphic to $R'' = R[y_2, \ldots, y_r]/(a y_i - a_i)$.


more-algebra-lemma-base-change-H1-regular

Let $A \to B$ be a ring map. Let $f_1, \ldots, f_r$ be a sequence in $B$ such that $B/(f_1, \ldots, f_r)$ is $A$-flat. Let $A \to A'$ be a ring map. Then the canonical map $$ H_1(K_\bullet(B, f_1, \ldots, f_r)) \otimes_A A' \longrightarrow H_1(K_\bullet(B', f'_1, \ldots, f'_r)) $$ is an isomorphism. Here $B' = B \otimes_A A'$ and $f_i' \in B'$ is the image of $f_i$.


more-algebra-lemma-relative-regular-immersion-algebra

Let $A \to B$ and $A \to A'$ be ring maps. Set $B' = B \otimes_A A'$. Let $f_1, \ldots, f_r \in B$. Assume $B/(f_1, \ldots, f_r)B$ is flat over $A$ \begin{enumerate} \item If $f_1, \ldots, f_r$ is a quasi-regular sequence, then the image in $B'$ is a quasi-regular sequence. \item If $f_1, \ldots, f_r$ is a $H_1$-regular sequence, then the image in $B'$ is a $H_1$-regular sequence. \end{enumerate}


more-algebra-lemma-cut-by-koszul-flat

Let $A' \to B'$ be a ring map. Let $I \subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \ldots, f'_r \in B'$. Assume \begin{enumerate} \item $A' \to B'$ is flat and of finite presentation, \item $I$ is locally nilpotent, \item the images $f_1, \ldots, f_r \in B$ form a quasi-regular sequence, \item $B/(f_1, \ldots, f_r)$ is flat over $A$. \end{enumerate} Then $B'/(f'_1, \ldots, f'_r)$ is flat over $A'$.


more-algebra-lemma-cut-by-koszul

Let $A' \to B'$ be a ring map. Let $I \subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \ldots, f'_r \in B'$. Assume \begin{enumerate} \item $A' \to B'$ is flat and of finite presentation (for example smooth), \item $I$ is locally nilpotent, \item the images $f_1, \ldots, f_r \in B$ form a quasi-regular sequence, \item $B/(f_1, \ldots, f_r)$ is smooth over $A$. \end{enumerate} Then $B'/(f'_1, \ldots, f'_r)$ is smooth over $A'$.


more-algebra-lemma-quasi-regular-ideal-finite

A quasi-regular ideal is finitely generated.


more-algebra-lemma-quasi-regular-ideal-finite-projective

Let $I \subset R$ be a quasi-regular ideal of a ring. Then $I/I^2$ is a finite projective $R/I$-module.


more-algebra-lemma-flat-descent-regular-ideal

Let $A \to B$ be a faithfully flat ring map. Let $I \subset A$ be an ideal. If $IB$ is a Koszul-regular (resp.\ $H_1$-regular, resp.\ quasi-regular) ideal in $B$, then $I$ is a Koszul-regular (resp.\ $H_1$-regular, resp.\ quasi-regular) ideal in $A$.


more-algebra-lemma-conormal-sequence-H1-regular-ideal

Let $A$ be a ring. Let $I \subset J \subset A$ be ideals. Assume that $J/I \subset A/I$ is a $H_1$-regular ideal. Then $I \cap J^2 = IJ$.


more-algebra-lemma-lci-local

Let $R \to S$ be a ring map. Let $g_1, \ldots, g_m \in S$ generate the unit ideal. If each $R \to S_{g_j}$ is a local complete intersection so is $R \to S$.


more-algebra-lemma-relative-global-complete-intersection-koszul

Let $R$ be a ring. If $R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ is a relative global complete intersection, then $f_1, \ldots, f_c$ is a Koszul regular sequence.


more-algebra-lemma-syntomic-lci

Let $R \to S$ be a ring map. The following are equivalent \begin{enumerate} \item $R \to S$ is syntomic (Algebra, Definition \ref{algebra-definition-lci}), and \item $R \to S$ is flat and a local complete intersection. \end{enumerate}


more-algebra-lemma-transitive-lci-at-end

Let $A \to B \to C$ be ring maps. Assume $B \to C$ is a local complete intersection homomorphism. Choose a presentation $\alpha : A[x_s, s \in S] \to B$ with kernel $I$. Choose a presentation $\beta : B[y_1, \ldots, y_m] \to C$ with kernel $J$. Let $\gamma : A[x_s, y_t] \to C$ be the induced presentation of $C$ with kernel $K$. Then we get a canonical commutative diagram $$ \xymatrix{ 0 \ar[r] & \Omega_{A[x_s]/A} \otimes C \ar[r] & \Omega_{A[x_s, y_t]/A} \otimes C \ar[r] & \Omega_{B[y_t]/B} \otimes C \ar[r] & 0 \\ 0 \ar[r] & I/I^2 \otimes C \ar[r] \ar[u] & K/K^2 \ar[r] \ar[u] & J/J^2 \ar[r] \ar[u] & 0 } $$ with exact rows. In particular, the six term exact sequence of Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL} can be completed with a zero on the left, i.e., the sequence $$ 0 \to H_1(\NL_{B/A} \otimes_B C) \to H_1(L_{C/A}) \to H_1(L_{C/B}) \to \Omega_{B/A} \otimes_B C \to \Omega_{C/A} \to \Omega_{C/B} \to 0 $$ is exact.


more-algebra-lemma-transitive-colimit-lci-at-end

Let $A \to B \to C$ be ring maps. If $B \to C$ is a filtered colimit of local complete intersection homomorphisms then the conclusion of Lemma \ref{lemma-transitive-lci-at-end} remains valid.


more-algebra-lemma-henselization-NL

Let $A \to B$ be a local homomorphism of local rings. Let $A^h \to B^h$, resp.\ $A^{sh} \to B^{sh}$ be the induced map on henselizations, resp.\ strict henselizations (Algebra, Lemma \ref{algebra-lemma-henselian-functorial}, resp.\ Lemma \ref{algebra-lemma-strictly-henselian-functorial}). Then $\NL_{B/A} \otimes_B B^h \to \NL_{B^h/A^h}$ and $\NL_{B/A} \otimes_B B^{sh} \to \NL_{B^{sh}/A^{sh}}$ induce isomorphisms on cohomology groups.


more-algebra-lemma-cartier-equality

Let $K/k$ be a finitely generated field extension. Then $\Omega_{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and $\text{trdeg}_k(K) = \dim_K \Omega_{K/k} - \dim_K H_1(L_{K/k})$.


more-algebra-lemma-transitivity-gamma

Let $K \subset L \subset M$ be field extensions. Then the Jacobi-Zariski sequence $$ 0 \to H_1(L_{L/K}) \otimes_L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega_{L/K} \otimes_L M \to \Omega_{M/K} \to \Omega_{M/L} \to 0 $$ is exact.


more-algebra-lemma-gamma-commutative-diagram

Given a commutative diagram of fields $$ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } $$ with $k \subset k'$ and $K \subset K'$ finitely generated field extensions the kernel and cokernel of the maps $$ \alpha : \Omega_{K/k} \otimes_K K' \to \Omega_{K'/k'} \quad\text{and}\quad \beta : H_1(L_{K/k}) \otimes_K K' \to H_1(L_{K'/k'}) $$ are finite dimensional and $$ \dim \Ker(\alpha) - \dim \Coker(\alpha) -\dim \Ker(\beta) + \dim \Coker(\beta) = \text{trdeg}_k(k') - \text{trdeg}_K(K') $$


more-algebra-lemma-geometrically-regular-over-field

Let $k$ be a field of characteristic $p > 0$. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. Assume $A$ is geometrically regular over $k$. Let $k \subset F \subset K$ be a finitely generated subextension. Let $\varphi : k[y_1, \ldots, y_m] \to A$ be a $k$-algebra map such that $y_i$ maps to an element of $F$ in $K$ and such that $\text{d}y_1, \ldots, \text{d}y_m$ map to a basis of $\Omega_{F/k}$. Set $\mathfrak p = \varphi^{-1}(\mathfrak m)$. Then $$ k[y_1, \ldots, y_m]_\mathfrak p \to A $$ is flat and $A/\mathfrak pA$ is regular.


more-algebra-lemma-continuous

Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ and $J \subset S$ be ideals and endow $R$ with the $I$-adic topology and $S$ with the $J$-adic topology. Then $\varphi$ is a homomorphism of topological rings if and only if $\varphi(I^n) \subset J$ for some $n \geq 1$.


more-algebra-lemma-baire-category-complete-module

Let $M$ be a topological abelian group. Assume $M$ is linearly topologized, complete, and has a countable fundamental system of neighbourhoods of $0$. If $U_n \subset M$, $n \geq 1$ are open dense subsets, then $\bigcap_{n \geq 1} U_n$ is dense.


more-algebra-lemma-consequence-baire-complete-module

With same assumptions as Lemma \ref{lemma-baire-category-complete-module} if $M = \bigcup_{n \geq 1} N_n$ for some closed subgroups $N_n$, then $N_n$ is open for some $n$.


more-algebra-lemma-open-mapping

Let $u : N \to M$ be a continuous map of linearly topologized abelian groups. Assume that $N$ is complete, $M$ separated, and $N$ has a countable fundamental system of neighbourhoods of $0$. Then exactly one of the following holds \begin{enumerate} \item $u$ is open, or \item for some open subgroup $N' \subset N$ the image $u(N')$ is nowhere dense in $M$. \end{enumerate}


more-algebra-lemma-formally-smooth

Let $\varphi : R \to S$ be a ring map. \begin{enumerate} \item If $R \to S$ is formally smooth in the sense of Algebra, Definition \ref{algebra-definition-formally-smooth}, then $R \to S$ is formally smooth for any linear topology on $R$ and any pre-adic topology on $S$ such that $R \to S$ is continuous. \item Let $\mathfrak n \subset S$ and $\mathfrak m \subset R$ ideals such that $\varphi$ is continuous for the $\mathfrak m$-adic topology on $R$ and the $\mathfrak n$-adic topology on $S$. Then the following are equivalent \begin{enumerate} \item $\varphi$ is formally smooth for the $\mathfrak m$-adic topology on $R$ and the $\mathfrak n$-adic topology on $S$, and \item $\varphi$ is formally smooth for the discrete topology on $R$ and the $\mathfrak n$-adic topology on $S$. \end{enumerate} \end{enumerate}


more-algebra-lemma-formally-smooth-completion

Let $(R, \mathfrak m)$ and $(S, \mathfrak n)$ be rings endowed with finitely generated ideals. Endow $R$ and $S$ with the $\mathfrak m$-adic and $\mathfrak n$-adic topologies. Let $R \to S$ be a homomorphism of topological rings. The following are equivalent \begin{enumerate} \item $R \to S$ is formally smooth for the $\mathfrak n$-adic topology, \item $R \to S^\wedge$ is formally smooth for the $\mathfrak n^\wedge$-adic topology, \item $R^\wedge \to S^\wedge$ is formally smooth for the $\mathfrak n^\wedge$-adic topology. \end{enumerate} Here $R^\wedge$ and $S^\wedge$ are the $\mathfrak m$-adic and $\mathfrak n$-adic completions of $R$ and $S$.


more-algebra-lemma-lift-continuous

Let $R \to S$ be a ring map. Let $\mathfrak n$ be an ideal of $S$. Assume that $R \to S$ is formally smooth in the $\mathfrak n$-adic topology. Consider a solid commutative diagram $$ \xymatrix{ S \ar[r]_\psi \ar@{-->}[rd] & A/J \\ R \ar[r] \ar[u] & A \ar[u] } $$ of homomorphisms of topological rings where $A$ is adic and $A/J$ is the quotient (as topological ring) of $A$ by a closed ideal $J \subset A$ such that $J^t$ is contained in an ideal of definition of $A$ for some $t \geq 1$. Then there exists a dotted arrow in the category of topological rings which makes the diagram commute.


more-algebra-lemma-increase-ideal

Let $R \to S$ be a ring map. Let $\mathfrak n \subset \mathfrak n' \subset S$ be ideals. If $R \to S$ is formally smooth for the $\mathfrak n$-adic topology, then $R \to S$ is formally smooth for the $\mathfrak n'$-adic topology.


more-algebra-lemma-compose-formally-smooth

A composition of formally smooth continuous homomorphisms of linearly topologized rings is formally smooth.


more-algebra-lemma-base-change-fs

Let $R$, $S$ be rings. Let $\mathfrak n \subset S$ be an ideal. Let $R \to S$ be formally smooth for the $\mathfrak n$-adic topology. Let $R \to R'$ be any ring map. Then $R' \to S' = S \otimes_R R'$ is formally smooth in the $\mathfrak n' = \mathfrak nS'$-adic topology.


more-algebra-lemma-descent-fs

Let $R$, $S$ be rings. Let $\mathfrak n \subset S$ be an ideal. Let $R \to R'$ be a ring map. Set $S' = S \otimes_R R'$ and $\mathfrak n' = \mathfrak nS$. If \begin{enumerate} \item the map $R \to R'$ embeds $R$ as a direct summand of $R'$ as an $R$-module, and \item $R' \to S'$ is formally smooth for the $\mathfrak n'$-adic topology, \end{enumerate} then $R \to S$ is formally smooth in the $\mathfrak n$-adic topology.


more-algebra-lemma-fs-local

Let $(R, \mathfrak m) \to (S, \mathfrak n)$ be a local homomorphism of local rings. The following are equivalent \begin{enumerate} \item $R \to S$ is formally smooth in the $\mathfrak n$-adic topology, \item for every solid commutative diagram $$ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/J \\ R \ar[r] \ar[u] & A \ar[u] } $$ of local homomorphisms of local rings where $J \subset A$ is an ideal of square zero, $\mathfrak m_A^n = 0$ for some $n > 0$, and $S \to A/J$ induces an isomorphism on residue fields, a dotted arrow exists which makes the diagram commute. \end{enumerate} If $S$ is Noetherian these conditions are also equivalent to \begin{enumerate} \item[(3)] same as in (2) but only for diagrams where in addition $A \to A/J$ is a small extension (Algebra, Definition \ref{algebra-definition-small-extension}). \end{enumerate}


more-algebra-lemma-fs-implies-regular

Let $k$ be a field and let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. If $k \to A$ is formally smooth for the $\mathfrak m$-adic topology, then $A$ is a regular local ring.


more-algebra-lemma-lift-residue-field

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be a complete local $k$-algebra. If $\kappa/k$ is separable, then there exists a $k$-algebra map $\kappa \to A$ such that $\kappa \to A \to \kappa$ is $\text{id}_\kappa$.


more-algebra-lemma-power-series-over-residue-field

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be a complete local $k$-algebra. If $\kappa/k$ is separable and $A$ regular, then there exists an isomorphism of $A \cong \kappa[[t_1, \ldots, t_d]]$ as $k$-algebras.


more-algebra-lemma-regular-implies-fs

Let $k$ be a field. Let $(A, \mathfrak m, K)$ be a regular local $k$-algebra such that $K/k$ is separable. Then $k \to A$ is formally smooth in the $\mathfrak m$-adic topology.


more-algebra-lemma-formally-smooth-finite-type

Let $A \to B$ be a finite type ring map with $A$ Noetherian. Let $\mathfrak q \subset B$ be a prime ideal lying over $\mathfrak p \subset A$. The following are equivalent \begin{enumerate} \item $A \to B$ is smooth at $\mathfrak q$, and \item $A_\mathfrak p \to B_\mathfrak q$ is formally smooth in the $\mathfrak q$-adic topology. \end{enumerate}


more-algebra-lemma-power-series-ring-over-Cohen-fs

Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \ldots, x_n]]$. Let $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \ldots, x_n]]$. Let $\Lambda$ be a Cohen ring. Let $C = \Lambda[[x_1, \ldots, x_n]]$. \begin{enumerate} \item $\mathbf{Q} \to A$ is formally smooth in the  $\mathfrak m$-adic topology. \item $\mathbf{F}_p \to B$ is formally smooth in the  $\mathfrak m$-adic topology. \item $\mathbf{Z} \to C$ is formally smooth in the $\mathfrak m$-adic topology. \end{enumerate}


more-algebra-lemma-quotient-power-series-ring-over-Cohen

Let $K$ be a field and $A = K[[x_1, \ldots, x_n]]$. Let $\Lambda$ be a Cohen ring and let $B = \Lambda[[x_1, \ldots, x_n]]$. \begin{enumerate} \item If $y_1, \ldots, y_n \in A$ is a regular system of parameters then $K[[y_1, \ldots, y_n]] \to A$ is an isomorphism. \item If $z_1, \ldots, z_r \in A$ form part of a regular system of parameters for $A$, then $r \leq n$ and $A/(z_1, \ldots, z_r) \cong K[[y_1, \ldots, y_{n - r}]]$. \item If $p, y_1, \ldots, y_n \in B$ is a regular system of parameters then $\Lambda[[y_1, \ldots, y_n]] \to B$ is an isomorphism. \item If $p, z_1, \ldots, z_r \in B$ form part of a regular system of parameters for $B$, then $r \leq n$ and $B/(z_1, \ldots, z_r) \cong \Lambda[[y_1, \ldots, y_{n - r}]]$. \end{enumerate}


more-algebra-lemma-embed-map-Noetherian-complete-local-rings

Let $A \to B$ be a local homomorphism of Noetherian complete local rings. Then there exists a commutative diagram $$ \xymatrix{ S \ar[r] & B \\ R \ar[u] \ar[r] & A \ar[u] } $$ with the following properties: \begin{enumerate} \item the horizontal arrows are surjective, \item if the characteristic of $A/\mathfrak m_A$ is zero, then $S$ and $R$ are power series rings over fields, \item if the characteristic of $A/\mathfrak m_A$ is $p > 0$, then $S$ and $R$ are power series rings over Cohen rings, and \item $R \to S$ maps a regular system of parameters of $R$ to part of a regular system of parameters of $S$. \end{enumerate} In particular $R \to S$ is flat (see Algebra, Lemma \ref{algebra-lemma-flat-over-regular}) with regular fibre $S/\mathfrak m_R S$ (see Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}).


more-algebra-lemma-dominate-two-surjections

Let $S \to R$ and $S' \to R$ be surjective maps of complete Noetherian local rings. Then $S \times_R S'$ is a complete Noetherian local ring.


more-algebra-lemma-formally-smooth-flat

Let $A \to B$ be a local homomorphism of Noetherian local rings. Assume $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. Then $A \to B$ is flat.


more-algebra-lemma-lift-fs

Let $A$ be a Noetherian complete local ring with residue field $k$. Let $B$ be a Noetherian complete local $k$-algebra. Assume $k \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. Then there exists a Noetherian complete local ring $C$ and a local homomorphism $A \to C$ which is formally smooth in the $\mathfrak m_C$-adic topology such that $C \otimes_A k \cong B$.


more-algebra-lemma-regular-local

Let $R \to \Lambda$ be a ring map with $\Lambda$ Noetherian. The following are equivalent \begin{enumerate} \item $R \to \Lambda$ is regular, \item $R_\mathfrak p \to \Lambda_\mathfrak q$ is regular for all $\mathfrak q \subset \Lambda$ lying over $\mathfrak p \subset R$, and \item $R_\mathfrak m \to \Lambda_{\mathfrak m'}$ is regular for all maximal ideals $\mathfrak m' \subset \Lambda$ lying over $\mathfrak m$ in $R$. \end{enumerate}


more-algebra-lemma-regular-base-change

Let $R \to \Lambda$ be a regular ring map. For any finite type ring map $R \to R'$ the base change $R' \to \Lambda \otimes_R R'$ is regular too.


more-algebra-lemma-regular-composition

Let $A \to B \to C$ be regular ring maps. If the fibre rings of $A \to C$ are Noetherian, then $A \to C$ is regular.


more-algebra-lemma-colimit-smooth-regular

Let $R$ be a ring. Let $(A_i, \varphi_{ii'})$ be a directed system of smooth $R$-algebras. Set $\Lambda = \colim A_i$. If the fibre rings $\Lambda \otimes_R \kappa(\mathfrak p)$ are Noetherian for all $\mathfrak p \subset R$, then $R \to \Lambda$ is regular.


more-algebra-lemma-regular-permanence

Let $A \to B \to C$ be ring maps. If $A \to C$ is regular and $B \to C$ is flat and surjective on spectra, then $A \to B$ is regular.


more-algebra-lemma-reduced-goes-up

Let $\varphi : R \to S$ be a ring map. Assume \begin{enumerate} \item $\varphi$ is regular, \item $S$ is Noetherian, and \item $R$ is Noetherian and reduced. \end{enumerate} Then $S$ is reduced.


more-algebra-lemma-normal-goes-up

Let $\varphi : R \to S$ be a ring map. Assume \begin{enumerate} \item $\varphi$ is regular, \item $S$ is Noetherian, and \item $R$ is Noetherian and normal. \end{enumerate} Then $S$ is normal.


more-algebra-lemma-completion-dimension

Let $A$ be a Noetherian local ring. Then $\dim(A) = \dim(A^\wedge)$.


more-algebra-lemma-completion-depth

Let $A$ be a Noetherian local ring. Then $\text{depth}(A) = \text{depth}(A^\wedge)$.


more-algebra-lemma-completion-CM

Let $A$ be a Noetherian local ring. Then $A$ is Cohen-Macaulay if and only if $A^\wedge$ is so.


more-algebra-lemma-completion-regular

Let $A$ be a Noetherian local ring. Then $A$ is regular if and only if $A^\wedge$ is so.


more-algebra-lemma-completion-dvr

Let $A$ be a Noetherian local ring. Then $A$ is a discrete valuation ring if and only if $A^\wedge$ is so.


more-algebra-lemma-completion-reduced

Let $A$ be a Noetherian local ring. \begin{enumerate} \item If $A^\wedge$ is reduced, then so is $A$. \item In general $A$ reduced does not imply $A^\wedge$ is reduced. \item If $A$ is Nagata, then $A$ is reduced if and only if $A^\wedge$ is reduced. \end{enumerate}


more-algebra-lemma-flat-completion

Let $A \to B$ be a local homomorphism of Noetherian local rings. Then the induced map of completions $A^\wedge \to B^\wedge$ is flat if and only if $A \to B$ is flat.


more-algebra-lemma-flat-unramified

Let $A \to B$ be a flat local homomorphism of Noetherian local rings such that $\mathfrak m_A B = \mathfrak m_B$ and $\kappa(\mathfrak m_A) = \kappa(\mathfrak m_B)$. Then $A \to B$ induces an isomorphism $A^\wedge \to B^\wedge$ of completions.


more-algebra-lemma-Noetherian-etale-extension

If $A \to B$ is an \'etale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $A_{\mathfrak p}$ is Noetherian if and only if $B_{\mathfrak q}$ is Noetherian.


more-algebra-lemma-dimension-etale-extension

If $A \to B$ is an \'etale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $\dim(A_{\mathfrak p}) = \dim(B_{\mathfrak q})$.


more-algebra-lemma-regular-etale-extension

If $A \to B$ is an \'etale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $A_{\mathfrak p}$ is regular if and only if $B_{\mathfrak q}$ is regular.


more-algebra-lemma-Dedekind-etale-extension

If $A \to B$ is an \'etale ring map and $A$ is a Dedekind domain, then $B$ is a finite product of Dedekind domains. In particular, the localizations $B_\mathfrak q$ for $\mathfrak q \subset B$ maximal are discrete valuation rings.


more-algebra-lemma-dumb-properties-henselization

Let $(R, \mathfrak m, \kappa)$ be a local ring. Then we have the following \begin{enumerate} \item $R \to R^h \to R^{sh}$ are faithfully flat ring maps, \item $\mathfrak m R^h = \mathfrak m^h$ and $\mathfrak m R^{sh} = \mathfrak m^h R^{sh} = \mathfrak m^{sh}$, \item $R/\mathfrak m^n = R^h/\mathfrak m^nR^h$ for all $n$, \item there exist elements $x_i \in R^{sh}$ such that $R^{sh}/\mathfrak m^nR^{sh}$ is a free $R/\mathfrak m^n$-module on $x_i \bmod \mathfrak m^nR^{sh}$. \end{enumerate}


more-algebra-lemma-henselization-formally-smooth

Let $(R, \mathfrak m, \kappa)$ be a local ring. Then \begin{enumerate} \item $R \to R^h$, $R^h \to R^{sh}$, and $R \to R^{sh}$ are formally \'etale, \item $R \to R^h$, $R^h \to R^{sh}$, resp.\ $R \to R^{sh}$ are formally smooth in the $\mathfrak m^h$, $\mathfrak m^{sh}$, resp.\ $\mathfrak m^{sh}$-topology. \end{enumerate}


more-algebra-lemma-henselization-noetherian

\begin{reference} \cite[IV, Theorem 18.6.6 and Proposition 18.8.8]{EGA} \end{reference} Let $R$ be a local ring. The following are equivalent \begin{enumerate} \item $R$ is Noetherian, \item $R^h$ is Noetherian, and \item $R^{sh}$ is Noetherian. \end{enumerate} In this case we have \begin{enumerate} \item[(a)] $(R^h)^\wedge$ and $(R^{sh})^\wedge$ are Noetherian complete local rings, \item[(b)] $R^\wedge \to (R^h)^\wedge$ is an isomorphism, \item[(c)] $R^h \to (R^h)^\wedge$ and $R^{sh} \to (R^{sh})^\wedge$ are flat, \item[(d)] $R^\wedge \to (R^{sh})^\wedge$ is formally smooth in the $\mathfrak m_{(R^{sh})^\wedge}$-adic topology, \item[(e)] $(R^\wedge)^{sh} = R^\wedge \otimes_{R^h} R^{sh}$, and \item[(f)] $((R^\wedge)^{sh})^\wedge = (R^{sh})^\wedge$. \end{enumerate}


more-algebra-lemma-henselization-reduced

\begin{slogan} Reducedness passes to the (strict) henselization. \end{slogan} Let $R$ be a local ring. The following are equivalent: $R$ is reduced, the henselization $R^h$ of $R$ is reduced, and the strict henselization $R^{sh}$ of $R$ is reduced.


more-algebra-lemma-henselization-nil

Let $R$ be a local ring. Let $nil(R)$ denote the ideal of nilpotent elements of $R$. Then $nil(R)R^h = nil(R^h)$ and $nil(R)R^{sh} = nil(R^{sh})$.


more-algebra-lemma-henselization-normal

Let $R$ be a local ring. The following are equivalent: $R$ is a normal domain, the henselization $R^h$ of $R$ is a normal domain, and the strict henselization $R^{sh}$ of $R$ is a normal domain.


more-algebra-lemma-henselization-dimension

Given any local ring $R$ we have $\dim(R) = \dim(R^h) = \dim(R^{sh})$.


more-algebra-lemma-henselization-depth

Given a Noetherian local ring $R$ we have $\text{depth}(R) = \text{depth}(R^h) = \text{depth}(R^{sh})$.


more-algebra-lemma-henselization-CM

Let $R$ be a Noetherian local ring. The following are equivalent: $R$ is Cohen-Macaulay, the henselization $R^h$ of $R$ is Cohen-Macaulay, and the strict henselization $R^{sh}$ of $R$ is Cohen-Macaulay.


more-algebra-lemma-henselization-regular

Let $R$ be a Noetherian local ring. The following are equivalent: $R$ is a regular local ring, the henselization $R^h$ of $R$ is a regular local ring, and the strict henselization $R^{sh}$ of $R$ is a regular local ring.


more-algebra-lemma-henselization-dvr

Let $R$ be a Noetherian local ring. Then $R$ is a discrete valuation ring if and only if $R^h$ is a discrete valuation ring if and only if $R^{sh}$ is a discrete valuation ring.


more-algebra-lemma-filtered-colimit-etale-noetherian-fibres

Let $A$ be a ring. Let $B$ be a filtered colimit of \'etale $A$-algebras. Let $\mathfrak p$ be a prime of $A$. If $B$ is Noetherian, then there are finitely many primes $\mathfrak q_1, \ldots, \mathfrak q_r$ lying over $\mathfrak p$, we have $B \otimes_A \kappa(\mathfrak p) = \prod \kappa(\mathfrak q_i)$, and each of the field extensions $\kappa(\mathfrak p) \subset \kappa(\mathfrak q_i)$ is separable algebraic.


more-algebra-lemma-fibres-henselization

Let $R$ be a Noetherian local ring. Let $\mathfrak p \subset R$ be a prime. Then $$ R^h \otimes_R \kappa(\mathfrak p) = \prod\nolimits_{i = 1, \ldots, t} \kappa(\mathfrak q_i) \quad\text{resp.}\quad R^{sh} \otimes_R \kappa(\mathfrak p) = \prod\nolimits_{i = 1, \ldots, s} \kappa(\mathfrak r_i) $$ where $\mathfrak q_1, \ldots, \mathfrak q_t$, resp.\ $\mathfrak r_1, \ldots, \mathfrak r_s$ are the prime of $R^h$, resp.\ $R^{sh}$ lying over $\mathfrak p$. Moreover, the field extensions $\kappa(\mathfrak p) \subset \kappa(\mathfrak q_i)$ resp.\ $\kappa(\mathfrak p) \subset \kappa(\mathfrak r_i)$ are separable algebraic.


more-algebra-lemma-p-basis

Let $k \subset K$ be a field extension. Assume $k$ has characteristic $p > 0$. Let $\{x_i\}$ be a subset of $K$. The following are equivalent \begin{enumerate} \item the elements $\{x_i\}$ are $p$-independent over $k$, and \item the elements $\text{d}x_i$ are $K$-linearly independent in $\Omega_{K/k}$. \end{enumerate} Any $p$-independent collection can be extended to a $p$-basis of $K$ over $k$. In particular, the field $K$ has a $p$-basis over $k$. Moreover, the following are equivalent: \begin{enumerate} \item[(a)] $\{x_i\}$ is a $p$-basis of $K$ over $k$, and \item[(b)] $\text{d}x_i$ is a basis of the $K$-vector space $\Omega_{K/k}$. \end{enumerate}


more-algebra-lemma-intersection-subfields-subspace

Let $k \subset K$ be a field extension. Let $\{K_\alpha\}_{\alpha \in A}$ be a collection of subfields of $K$ with the following properties \begin{enumerate} \item $k \subset K_\alpha$ for all $\alpha \in A$, \item $k = \bigcap_{\alpha \in A} K_\alpha$, \item for $\alpha, \alpha' \in A$ there exists an $\alpha'' \in A$ such that $K_{\alpha''} \subset K_\alpha \cap K_{\alpha'}$. \end{enumerate} Then for $n \geq 1$ and $V \subset K^{\oplus n}$ a $K$-vector space we have $V \cap k^{\oplus n} \not = 0$ if and only if $V \cap K_\alpha^{\oplus n} \not = 0$ for all $\alpha \in A$.


more-algebra-lemma-intersection-subfields

Let $K$ be a field of characteristic $p$. Let $\{K_\alpha\}_{\alpha \in A}$ be a collection of subfields of $K$ with the following properties \begin{enumerate} \item $K^p \subset K_\alpha$ for all $\alpha \in A$, \item $K^p = \bigcap_{\alpha \in A} K_\alpha$, \item for $\alpha, \alpha' \in A$ there exists an $\alpha'' \in A$ such that $K_{\alpha''} \subset K_\alpha \cap K_{\alpha'}$. \end{enumerate} Then \begin{enumerate} \item the intersection of the kernels of the maps $\Omega_{K/\mathbf{F}_p} \to \Omega_{K/K_\alpha}$ is zero, \item for any finite extension $K \subset L$ we have $L^p = \bigcap_{\alpha \in A} L^pK_\alpha$. \end{enumerate}


more-algebra-lemma-power-series-ring-subfields

Let $k$ be a field of characteristic $p > 0$. Let $n, m \geq 0$. Let $K$ be the fraction field of $k[[x_1, \ldots, x_n]][y_1, \ldots, y_m]$. As $k'$ ranges through all subfields $k/k'/k^p$ with $[k : k'] < \infty$ the subfields $$ \text{fraction field of } k'[[x_1^p, \ldots, x_n^p]][y_1^p, \ldots, y_m^p] \subset K $$ form a family of subfields as in Lemma \ref{lemma-intersection-subfields}. Moreover, each of the ring extensions $k'[[x_1^p, \ldots, x_n^p]][y_1^p, \ldots, y_m^p] \subset k[[x_1, \ldots, x_n]][y_1, \ldots, y_m]$ is finite.


more-algebra-lemma-J-1

Let $R$ be a Noetherian ring. Let $X = \Spec(R)$. The ring $R$ is J-1 if and only if $V(\mathfrak p) \cap \text{Reg}(X)$ contains a nonempty open subset of $V(\mathfrak p)$ for all $\mathfrak p \in \text{Reg}(X)$.


more-algebra-lemma-intersection-regular-with-closed

Let $R$ be a Noetherian ring. Let $X = \Spec(R)$. Assume that for all primes $\mathfrak p \subset R$ the ring $R/\mathfrak p$ is J-0. Then $R$ is J-1.


more-algebra-lemma-J-0-goes-down

Let $R \to S$ be a ring map. Assume that \begin{enumerate} \item $R$ is a Noetherian domain, \item $R \to S$ is injective and of finite type, and \item $S$ is a domain and J-0. \end{enumerate} Then $R$ is J-0.


more-algebra-lemma-J-0-goes-up

Let $R \to S$ be a ring map. Assume that \begin{enumerate} \item $R$ is a Noetherian domain and J-0, \item $R \to S$ is injective and of finite type, and \item $S$ is a domain, and \item the induced extension of fraction fields is separable. \end{enumerate} Then $S$ is J-0.


more-algebra-lemma-J-2

Let $R$ be a Noetherian ring. The following are equivalent \begin{enumerate} \item $R$ is J-2, \item every finite type $R$-algebra which is a domain is J-0, \item every finite $R$-algebra is J-1, \item for every prime $\mathfrak p$ and every finite purely inseparable extension $\kappa(\mathfrak p) \subset L$ there exists a finite $R$-algebra $R'$ which is a domain, which is J-0, and whose field of fractions is $L$. \end{enumerate}


more-algebra-lemma-derivation-extends

Let $R$ be a ring. Let $D : R \to R$ be a derivation. \begin{enumerate} \item For any ideal $I \subset R$ the derivation $D$ extends canonically to a derivation $D^\wedge : R^\wedge \to R^\wedge$ on the $I$-adic completion. \item For any multiplicative subset $S \subset R$ the derivation $D$ extends uniquely to the localization $S^{-1}R$ of $R$. \end{enumerate} If $R \subset R'$ is a finite type extension of rings such that $R_g \cong R'_g$ for some $g \in R$ which is a nonzerodivisor in $R'$, then $g^ND$ extends to $R'$ for some $N \geq 0$.


more-algebra-lemma-quotient-regular

\begin{slogan} The Jacobian criterion for hypersurfaces, done right. \end{slogan} Let $R$ be a regular ring. Let $f \in R$. Assume there exists a derivation $D : R \to R$ such that $D(f)$ is a unit of $R/(f)$. Then $R/(f)$ is regular.


more-algebra-lemma-quotient-sequence-regular

Let $(R, \mathfrak m, \kappa)$ be a regular local ring. Let $m \geq 1$. Let $f_1, \ldots, f_m \in \mathfrak m$. Assume there exist derivations $D_1, \ldots, D_m : R \to R$ such that $\det_{1 \leq i, j \leq m}(D_i(f_j))$ is a unit of $R$. Then $R/(f_1, \ldots, f_m)$ is regular and $f_1, \ldots, f_m$ is a regular sequence.


more-algebra-lemma-degree-p-extension-regular

Let $R$ be a regular ring. Let $f \in R$. Assume there exists a derivation $D : R \to R$ such that $D(f)$ is a unit of $R$. Then $R[z]/(z^n - f)$ is regular for any integer $n \geq 1$. More generally, $R[z]/(p(z) - f)$ is regular for any $p \in \mathbf{Z}[z]$.


more-algebra-lemma-find-D

Let $p$ be a prime number. Let $B$ be a domain with $p = 0$ in $B$. Let $f \in B$ be an element which is not a $p$th power in the fraction field of $B$. If $B$ is of finite type over a Noetherian complete local ring, then there exists a derivation $D : B \to B$ such that $D(f)$ is not zero.


more-algebra-lemma-complete-Noetherian-domain-J-0

Let $A$ be a Noetherian complete local domain. Then $A$ is J-0.


more-algebra-lemma-lift-derivation-through-fs

Let $A \to B$ be a local homomorphism of Noetherian local rings. Let $D : A \to A$ be a derivation. Assume that $B$ is complete and $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. Then there exists an extension $D' : B \to B$ of $D$.


more-algebra-lemma-check-G-ring-easy

Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every pair of primes $\mathfrak q \subset \mathfrak p \subset R$ the algebra $$ (R/\mathfrak q)_\mathfrak p^\wedge \otimes_{R/\mathfrak q} \kappa(\mathfrak q) $$ is geometrically regular over $\kappa(\mathfrak q)$.


more-algebra-lemma-G-ring-goes-up-quasi-finite

Let $R \to R'$ be a finite type map of Noetherian rings and let $$ \xymatrix{ \mathfrak q' \ar[r] & \mathfrak p' \ar[r] & R' \\ \mathfrak q \ar[r] \ar@{-}[u] & \mathfrak p \ar[r] \ar@{-}[u] & R \ar[u] } $$ be primes. Assume $R \to R'$ is quasi-finite at $\mathfrak p'$. \begin{enumerate} \item If the formal fibre $R_\mathfrak p^\wedge \otimes_R \kappa(\mathfrak q)$ is geometrically regular over $\kappa(\mathfrak q)$, then the formal fibre $R'_{\mathfrak p'} \otimes_{R'} \kappa(\mathfrak q')$ is geometrically regular over $\kappa(\mathfrak q')$. \item If the formal fibres of $R_\mathfrak p$ are geometrically regular, then the formal fibres of $R'_{\mathfrak p'}$ are geometrically regular. \item If $R \to R'$ is quasi-finite and $R$ is a G-ring, then $R'$ is a G-ring. \end{enumerate}


more-algebra-lemma-helper-G-ring

Let $k$ be a field of characteristic $p$. Let $A = k[[x_1, \ldots, x_n]][y_1, \ldots, y_n]$ and denote $K$ the fraction field of $A$. Let $\mathfrak p \subset A$ be a prime. Then $A_\mathfrak p^\wedge \otimes_A K$ is geometrically regular over $K$.


more-algebra-lemma-check-G-ring-maximal-ideals

Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if $R_\mathfrak m$ has geometrically regular formal fibres for every maximal ideal $\mathfrak m$ of $R$.


more-algebra-lemma-henselization-G-ring

Let $R$ be a Noetherian local ring which is a G-ring. Then the henselization $R^h$ and the strict henselization $R^{sh}$ are G-rings.


more-algebra-lemma-another-helper-G-ring

Let $p$ be a prime number. Let $A$ be a Noetherian complete local domain with fraction field $K$ of characteristic $p$. Let $\mathfrak q \subset A[x]$ be a maximal ideal lying over the maximal ideal of $A$ and let $(0) \not = \mathfrak r \subset \mathfrak q$ be a prime lying over $(0) \subset A$. Then $A[x]_\mathfrak q^\wedge \otimes_{A[x]} \kappa(\mathfrak r)$ is geometrically regular over $\kappa(\mathfrak r)$.


more-algebra-lemma-map-G-ring-to-completion-regular

\begin{reference} \cite[Theorem 79]{MatCA} \end{reference} Let $A$ be a G-ring. Let $I \subset A$ be an ideal and let $A^\wedge$ be the completion of $A$ with respect to $I$. Then $A \to A^\wedge$ is regular.


more-algebra-lemma-henselization-pair-G-ring

\begin{slogan} Being a G-ring is stable under Henselizations along ideals \end{slogan} \begin{reference} \cite[Theorem 5.3 i)]{Greco} \end{reference} Let $A$ be a G-ring. Let $I \subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see Lemma \ref{lemma-henselization}. Then $A^h$ is a G-ring.


more-algebra-lemma-check-P-ring-easy

Let $R$ be a Noetherian ring. Let $P$ be a property as above. Then $R$ is a $P$-ring if and only if for every pair of primes $\mathfrak q \subset \mathfrak p \subset R$ the $\kappa(\mathfrak q)$-algebra $$ (R/\mathfrak q)_\mathfrak p^\wedge \otimes_{R/\mathfrak q} \kappa(\mathfrak q) $$ has property $P$.


more-algebra-lemma-P-local

Let $R \to \Lambda$ be a homomorphism of Noetherian rings. Assume $P$ has property (B). The following are equivalent \begin{enumerate} \item the fibres of $R \to \Lambda$ have $P$, \item the fibres of $R_\mathfrak p \to \Lambda_\mathfrak q$ have $P$ for all $\mathfrak q \subset \Lambda$ lying over $\mathfrak p \subset R$, and \item the fibres of $R_\mathfrak m \to \Lambda_{\mathfrak m'}$ have $P$ for all maximal ideals $\mathfrak m' \subset \Lambda$ lying over $\mathfrak m$ in $R$. \end{enumerate}


more-algebra-lemma-P-ring-goes-up-quasi-finite

Let $R \to R'$ be a finite type map of Noetherian rings and let $$ \xymatrix{ \mathfrak q' \ar[r] & \mathfrak p' \ar[r] & R' \\ \mathfrak q \ar[r] \ar@{-}[u] & \mathfrak p \ar[r] \ar@{-}[u] & R \ar[u] } $$ be primes. Assume $R \to R'$ is quasi-finite at $\mathfrak p'$. Assume $P$ satisfies (A) and (B). \begin{enumerate} \item If $\kappa(\mathfrak q) \to R_\mathfrak p^\wedge \otimes_R \kappa(\mathfrak q)$ has $P$, then $\kappa(\mathfrak q') \to R'_{\mathfrak p'} \otimes_{R'} \kappa(\mathfrak q')$ has $P$. \item If the formal fibres of $R_\mathfrak p$ have $P$, then the formal fibres of $R'_{\mathfrak p'}$ have $P$. \item If $R \to R'$ is quasi-finite and $R$ is a $P$-ring, then $R'$ is a $P$-ring. \end{enumerate}


more-algebra-lemma-check-P-ring-maximal-ideals

Let $R$ be a Noetherian ring. Assume $P$ satisfies (C) and (D). Then $R$ is a $P$-ring if and only if the formal fibres of $R_\mathfrak m$ have $P$ for every maximal ideal $\mathfrak m$ of $R$.


more-algebra-lemma-henselization-pair-P-ring

\begin{slogan} Henselization of a ring inherits good properties of formal fibers \end{slogan} Let $A$ be a $P$-ring where $P$ satisfies (B), (C), (D), and (E). Let $I \subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see Lemma \ref{lemma-henselization}. Then $A^h$ is a $P$-ring.


more-algebra-lemma-henselization-P-ring

Let $R$ be a Noetherian local ring which is a $P$-ring where $P$ satisfies (B), (C), (D), and (E). Then the henselization $R^h$ and the strict henselization $R^{sh}$ are $P$-rings.


more-algebra-lemma-formal-fibres-reduced

Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$``$R$ is geometrically reduced over $k$''.


more-algebra-lemma-formal-fibres-normal

Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$``$R$ is geometrically normal over $k$''.


more-algebra-lemma-formal-fibres-Sk

Fix $n \geq 1$. Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$``$R$ has $(S_n)$''.


more-algebra-lemma-finite-type-over-excellent

Any localization of a finite type ring over a (quasi-)excellent ring is (quasi-)excellent.


more-algebra-lemma-Nagata-local-ring

Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent \begin{enumerate} \item $A$ is Nagata, and \item the formal fibres of $A$ are geometrically reduced. \end{enumerate}


more-algebra-lemma-completion-normal-local-ring

Let $(A, \mathfrak m)$ be a Noetherian local ring. If $A$ is normal and the formal fibres of $A$ are normal (for example if $A$ is excellent or quasi-excellent), then $A^\wedge$ is normal.


more-algebra-lemma-injective-abelian

An abelian group $J$ is an injective object in the category of abelian groups if and only if $J$ is divisible.


more-algebra-lemma-relation-ext-ext

Let $R$ be a ring. Let $\mathcal{A}$ be the abelian category of $R$-modules. There is a canonical isomorphism $\Ext_\mathcal{A}(M, N) = \Ext^1_R(M, N)$ compatible with the long exact sequences of Algebra, Lemmas \ref{algebra-lemma-long-exact-seq-ext} and \ref{algebra-lemma-reverse-long-exact-seq-ext} and the $6$-term exact sequences of Homology, Lemma \ref{homology-lemma-six-term-sequence-ext}.


more-algebra-lemma-characterize-injective-bis

Let $R$ be a ring. Let $J$ be an $R$-module. The following are equivalent \begin{enumerate} \item $J$ is injective, \item $\Ext^1_R(R/I, J) = 0$ for every ideal $I \subset R$, and \item for an ideal $I \subset R$ and module map $I \to J$ there exists an extension $R \to J$. \end{enumerate}


more-algebra-lemma-vee-exact

Let $R$ be a ring. The functor $M \mapsto M^\vee$ is exact.


more-algebra-lemma-ev-injective

For any $R$-module $M$ the evaluation map $ev : M \to (M^\vee)^\vee$ is injective.


more-algebra-lemma-injectives-modules

Let $R$ be a ring. The construction above defines a covariant functor $M \mapsto (M \to J(M))$ from the category of $R$-modules to the category of arrows of $R$-modules such that for every module $M$ the output $M \to J(M)$ is an injective map of $M$ into an injective $R$-module $J(M)$.


more-algebra-lemma-K-injective-flat

Let $R \to S$ be a flat ring map. If $I^\bullet$ is a K-injective complex of $S$-modules, then $I^\bullet$ is K-injective as a complex of $R$-modules.


more-algebra-lemma-hom-K-injective

Let $A \to B$ be a ring map. If $I^\bullet$ is a K-injective complex of $A$-modules, then $\Hom_A(B, I^\bullet)$ is a K-injective complex of $B$-modules.


more-algebra-lemma-derived-tor-homotopy

Let $R$ be a ring. Let $P^\bullet$ be a complex of $R$-modules. Let $\alpha, \beta : L^\bullet \to M^\bullet$ be homotopy equivalent maps of complexes. Then $\alpha$ and $\beta$ induce homotopy equivalent maps $$ \text{Tot}(\alpha \otimes \text{id}_P), \text{Tot}(\beta \otimes \text{id}_P) : \text{Tot}(L^\bullet \otimes_R P^\bullet) \longrightarrow \text{Tot}(M^\bullet \otimes_R P^\bullet). $$ In particular the construction $L^\bullet \mapsto \text{Tot}(L^\bullet \otimes_R P^\bullet)$ defines an endo-functor of the homotopy category of complexes.


more-algebra-lemma-derived-tor-exact

Let $R$ be a ring. Let $P^\bullet$ be a complex of $R$-modules. The functors $$ K(\text{Mod}_R) \longrightarrow K(\text{Mod}_R), \quad L^\bullet \longmapsto \text{Tot}(P^\bullet \otimes_R L^\bullet) $$ and $$ K(\text{Mod}_R) \longrightarrow K(\text{Mod}_R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes_R P^\bullet) $$ are exact functors of triangulated categories.


more-algebra-lemma-K-flat-quasi-isomorphism

Let $R$ be a ring. Let $K^\bullet$ be a K-flat complex. Then the functor $$ K(\text{Mod}_R) \longrightarrow K(\text{Mod}_R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes_R K^\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms.


more-algebra-lemma-base-change-K-flat

Let $R \to R'$ be a ring map. If $K^\bullet$ is a K-flat complex of $R$-modules, then $K^\bullet \otimes_R R'$ is a K-flat complex of $R'$-modules.


more-algebra-lemma-tensor-product-K-flat

Let $R$ be a ring. If $K^\bullet$, $L^\bullet$ are K-flat complexes of $R$-modules, then $\text{Tot}(K^\bullet \otimes_R L^\bullet)$ is a K-flat complex of $R$-modules.


more-algebra-lemma-K-flat-two-out-of-three

Let $R$ be a ring. Let $(K_1^\bullet, K_2^\bullet, K_3^\bullet)$ be a distinguished triangle in $K(\text{Mod}_R)$. If two out of three of $K_i^\bullet$ are K-flat, so is the third.


more-algebra-lemma-K-flat-two-out-of-three-ses

Let $R$ be a ring. Let $0 \to K_1^\bullet \to K_2^\bullet \to K_3^\bullet \to 0$ be a short exact sequence of complexes. If $K_3^n$ is flat for all $n \in \mathbf{Z}$ and two out of three of $K_i^\bullet$ are K-flat, so is the third.


more-algebra-lemma-derived-tor-quasi-isomorphism

Let $R$ be a ring. Let $P^\bullet$ be a bounded above complex of flat $R$-modules. Then $P^\bullet$ is K-flat.


more-algebra-lemma-colimit-K-flat

Let $R$ be a ring. Let $K_1^\bullet \to K_2^\bullet \to \ldots$ be a system of K-flat complexes. Then $\colim_i K_i^\bullet$ is K-flat. More generally any filtered colimit of K-flat complexes is K-flat.


more-algebra-lemma-universally-acyclic-K-flat

Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. If $K^\bullet \otimes_R M$ is acyclic for all finitely presented $R$-modules $M$, then $K^\bullet$ is K-flat.


more-algebra-lemma-K-flat-resolution

Let $R$ be a ring. For any complex $M^\bullet$ there exists a K-flat complex $K^\bullet$ whose terms are flat $R$-modules and a quasi-isomorphism $K^\bullet \to M^\bullet$ which is termwise surjective.


more-algebra-lemma-derived-tor-quasi-isomorphism-other-side

Let $R$ be a ring. Let $\alpha : P^\bullet \to Q^\bullet$ be a quasi-isomorphism of K-flat complexes of $R$-modules. For every complex $L^\bullet$ of $R$-modules the induced map $$ \text{Tot}(\text{id}_L \otimes \alpha) : \text{Tot}(L^\bullet \otimes_R P^\bullet) \longrightarrow \text{Tot}(L^\bullet \otimes_R Q^\bullet) $$ is a quasi-isomorphism.


more-algebra-lemma-flip-douoble-tensor-product

Let $R$ be a ring. Let $K^\bullet, L^\bullet$ be complexes of $R$-modules. There is a canonical isomorphism $$ K^\bullet \otimes_R^\mathbf{L} L^\bullet \longrightarrow L^\bullet \otimes_R^\mathbf{L} K^\bullet $$ functorial in both complexes which uses a sign of $(-1)^{pq}$ for the map $K^p \otimes_R L^q \to L^q \otimes_R K^p$ (see proof for explanation).


more-algebra-lemma-triple-tensor-product

Let $R$ be a ring. Let $K^\bullet, L^\bullet, M^\bullet$ be complexes of $R$-modules. There is a canonical isomorphism $$ (K^\bullet \otimes_R^\mathbf{L} L^\bullet) \otimes_R^\mathbf{L} M^\bullet = K^\bullet \otimes_R^\mathbf{L} (L^\bullet \otimes_R^\mathbf{L} M^\bullet) $$ functorial in all three complexes.


more-algebra-lemma-derived-base-change

The construction above is independent of choices and defines an exact functor of triangulated categories $- \otimes_R^\mathbf{L} N^\bullet : D(R) \to D(A)$. There is a functorial isomorphism $$ E^\bullet \otimes_R^\mathbf{L} N^\bullet = (E^\bullet \otimes_R^\mathbf{L} A) \otimes_A^\mathbf{L} N^\bullet $$ for $E^\bullet$ in $D(R)$.


more-algebra-lemma-functoriality-derived-base-change

Let $R \to A$ be a ring map. Let $f : L^\bullet \to N^\bullet$ be a map of complexes of $A$-modules. Then $f$ induces a transformation of functors $$ 1 \otimes f : - \otimes_A^\mathbf{L} L^\bullet \longrightarrow - \otimes_A^\mathbf{L} N^\bullet $$ If $f$ is a quasi-isomorphism, then $1 \otimes f$ is an isomorphism of functors.


more-algebra-lemma-double-base-change

Let $A \to B \to C$ be ring maps. Let $N^\bullet$ be a complex of $B$-modules and $K^\bullet$ a complex of $C$-modules. The compositions of the functors $$ D(A) \xrightarrow{- \otimes_A^\mathbf{L} N^\bullet} D(B) \xrightarrow{- \otimes_B^\mathbf{L} K^\bullet} D(C) $$ is the functor $- \otimes_A^\mathbf{L} (N^\bullet \otimes_B^\mathbf{L} K^\bullet) : D(A) \to D(C)$. If $M$, $N$, $K$ are modules over $A$, $B$, $C$, then we have $$ (M \otimes_A^\mathbf{L} N) \otimes_B^\mathbf{L} K = M \otimes_A^\mathbf{L} (N \otimes_B^\mathbf{L} K) = (M \otimes_A^\mathbf{L} C) \otimes_C^\mathbf{L} (N \otimes_B^\mathbf{L} K) $$ in $D(C)$. We also have a canonical isomorphism $$ (M \otimes_A^\mathbf{L} N) \otimes_B^\mathbf{L} K \longrightarrow (M \otimes_A^\mathbf{L} K) \otimes_C^\mathbf{L} (N \otimes_B^\mathbf{L} C) $$ using signs. Similar results holds for complexes.


more-algebra-lemma-base-change-comparison

The comparison map (\ref{equation-comparison-map}) is an isomorphism if $A' = A \otimes_R R'$ and $A$ and $R'$ are Tor independent over $R$.


more-algebra-lemma-tor-independent-flat

Consider a commutative diagram of rings $$ \xymatrix{ A' & R' \ar[r] \ar[l] & B' \\ A \ar[u] & R \ar[l] \ar[u] \ar[r] & B \ar[u] } $$ Assume that $R'$ is flat over $R$ and $A'$ is flat over $A \otimes_R R'$ and $B'$ is flat over $R' \otimes_R B$. Then $$ \text{Tor}_i^R(A, B) \otimes_{(A \otimes_R B)} (A' \otimes_{R'} B') = \text{Tor}_i^{R'}(A', B') $$


more-algebra-lemma-flat-base-change-tor-independent

Let $R \to A$ and $R \to B$ be ring maps. Let $R \to R'$ be a ring map and set $A' = A \otimes_R R'$ and $B' = B \otimes_R R'$. If $A$ and $B$ are tor independent over $R$ and $R \to R'$ is flat, then $A'$ and $B'$ are tor independent over $R'$.


more-algebra-lemma-lemma-tor-independent-flat-compare

Assumptions as in Lemma \ref{lemma-tor-independent-flat}. For $M \in D(A)$ there are canonical isomorphisms $$ H^i((M \otimes_A^\mathbf{L} A') \otimes_{R'}^\mathbf{L} B') = H^i(M \otimes_R^\mathbf{L} B) \otimes_{(A \otimes_R B)} (A' \otimes_{R'} B') $$ of $A' \otimes_{R'} B'$-modules.


more-algebra-lemma-tor-independent

Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. The following are equivalent \begin{enumerate} \item $A$ and $B$ are Tor independent over $R$, \item for every pair of primes $\mathfrak p \subset A$ and $\mathfrak q \subset B$ lying over the same prime $\mathfrak r \subset R$ the rings $A_\mathfrak p$ and $B_\mathfrak q$ are Tor independent over $R_\mathfrak r$, and \item For every prime $\mathfrak s$ of $A \otimes_R B$ the module $$ \text{Tor}_i^R(A, B)_\mathfrak s = \text{Tor}_i^{R_\mathfrak r}(A_\mathfrak p, B_\mathfrak q)_\mathfrak s $$ (where $\mathfrak p = A \cap \mathfrak s$, $\mathfrak q = B \cap \mathfrak s$ and $\mathfrak r = R \cap \mathfrak s$) is zero. \end{enumerate}


more-algebra-lemma-functoriality-product-tor

Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \to C$ be an $R$-algebra map. Then the induced map $$ \text{Tor}^R_{\star}(B, A) \longrightarrow \text{Tor}^R_{\star}(C, A) $$ is an $A$-algebra homomorphism.


more-algebra-lemma-cone-pseudo-coherent

Let $R$ be a ring and $m \in \mathbf{Z}$. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. \begin{enumerate} \item If $K^\bullet$ is $(m + 1)$-pseudo-coherent and $L^\bullet$ is $m$-pseudo-coherent then $M^\bullet$ is $m$-pseudo-coherent. \item If $K^\bullet, M^\bullet$ are $m$-pseudo-coherent, then $L^\bullet$ is $m$-pseudo-coherent. \item If $L^\bullet$ is $(m + 1)$-pseudo-coherent and $M^\bullet$ is $m$-pseudo-coherent, then $K^\bullet$ is $(m + 1)$-pseudo-coherent. \end{enumerate}


more-algebra-lemma-finite-cohomology

Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. Let $m \in \mathbf{Z}$. \begin{enumerate} \item If $K^\bullet$ is $m$-pseudo-coherent and $H^i(K^\bullet) = 0$ for $i > m$, then $H^m(K^\bullet)$ is a finite type $R$-module. \item If $K^\bullet$ is $m$-pseudo-coherent and $H^i(K^\bullet) = 0$ for $i > m + 1$, then $H^{m + 1}(K^\bullet)$ is a finitely presented $R$-module. \end{enumerate}


more-algebra-lemma-n-pseudo-module

Let $R$ be a ring. Let $M$ be an $R$-module. Then \begin{enumerate} \item $M$ is $0$-pseudo-coherent if and only if $M$ is a finite $R$-module, \item $M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely presented $R$-module, \item $M$ is $(-d)$-pseudo-coherent if and only if there exists a resolution $$ R^{\oplus a_d} \to R^{\oplus a_{d - 1}} \to \ldots \to R^{\oplus a_0} \to M \to 0 $$ of length $d$, and \item $M$ is pseudo-coherent if and only if there exists an infinite resolution $$ \ldots \to R^{\oplus a_1} \to R^{\oplus a_0} \to M \to 0 $$ by finite free $R$-modules. \end{enumerate}


more-algebra-lemma-pseudo-coherent

Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. The following are equivalent \begin{enumerate} \item $K^\bullet$ is pseudo-coherent, \item $K^\bullet$ is $m$-pseudo-coherent for every $m \in \mathbf{Z}$, and \item $K^\bullet$ is quasi-isomorphic to a bounded above complex of finite projective $R$-modules. \end{enumerate} If (1), (2), and (3) hold and $H^i(K^\bullet) = 0$ for $i > b$, then we can find a quasi-isomorphism $F^\bullet \to K^\bullet$ with $F^i$ finite free $R$-modules and $F^i = 0$ for $i > b$.


more-algebra-lemma-two-out-of-three-pseudo-coherent

Let $R$ be a ring. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. If two out of three of $K^\bullet, L^\bullet, M^\bullet$ are pseudo-coherent then the third is also pseudo-coherent.


more-algebra-lemma-recognize-pseudo-coherent

Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. Let $m \in \mathbf{Z}$. \begin{enumerate} \item If $H^i(K^\bullet) = 0$ for all $i \geq m$, then $K^\bullet$ is $m$-pseudo-coherent. \item If $H^i(K^\bullet) = 0$ for $i > m$ and $H^m(K^\bullet)$ is a finite $R$-module, then $K^\bullet$ is $m$-pseudo-coherent. \item If $H^i(K^\bullet) = 0$ for $i > m + 1$, the module $H^{m + 1}(K^\bullet)$ is of finite presentation, and $H^m(K^\bullet)$ is of finite type, then $K^\bullet$ is $m$-pseudo-coherent. \end{enumerate}


more-algebra-lemma-summands-pseudo-coherent

Let $R$ be a ring. Let $m \in \mathbf{Z}$. If $K^\bullet \oplus L^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) so are $K^\bullet$ and $L^\bullet$.


more-algebra-lemma-complex-pseudo-coherent-modules

Let $R$ be a ring. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a bounded above complex of $R$-modules such that $K^i$ is $(m - i)$-pseudo-coherent for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent. In particular, if $K^\bullet$ is a bounded above complex of pseudo-coherent $R$-modules, then $K^\bullet$ is pseudo-coherent.


more-algebra-lemma-cohomology-pseudo-coherent

Let $R$ be a ring. Let $m \in \mathbf{Z}$. Let $K^\bullet \in D^{-}(R)$ such that $H^i(K^\bullet)$ is $(m - i)$-pseudo-coherent (resp.\ pseudo-coherent) for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent).


more-algebra-lemma-finite-push-pseudo-coherent

Let $A \to B$ be a ring map. Assume that $B$ is pseudo-coherent as an $A$-module. Let $K^\bullet$ be a complex of $B$-modules. The following are equivalent \begin{enumerate} \item $K^\bullet$ is $m$-pseudo-coherent as a complex of $B$-modules, and \item $K^\bullet$ is $m$-pseudo-coherent as a complex of $A$-modules. \end{enumerate} The same equivalence holds for pseudo-coherence.


more-algebra-lemma-pull-pseudo-coherent

Let $A \to B$ be a ring map. Let $K^\bullet$ be an $m$-pseudo-coherent (resp.\ pseudo-coherent) complex of $A$-modules. Then $K^\bullet \otimes_A^{\mathbf{L}} B$ is an $m$-pseudo-coherent (resp.\ pseudo-coherent) complex of $B$-modules.


more-algebra-lemma-flat-base-change-pseudo-coherent

Let $A \to B$ be a flat ring map. Let $M$ be an $m$-pseudo-coherent (resp.\ pseudo-coherent) $A$-module. Then $M \otimes_A B$ is an $m$-pseudo-coherent (resp.\ pseudo-coherent) $B$-module.


more-algebra-lemma-glue-pseudo-coherent

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$ be elements which generate the unit ideal. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \otimes_R R_{f_i}$ is $m$-pseudo-coherent (resp.\ pseudo-coherent), then $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent).


more-algebra-lemma-flat-descent-pseudo-coherent

Let $R$ be a ring. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $R$-modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes_R R'$ is $m$-pseudo-coherent (resp.\ pseudo-coherent), then $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent).


more-algebra-lemma-tensor-pseudo-coherent

Let $R$ be a ring. Let $K, L$ be objects of $D(R)$. \begin{enumerate} \item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \otimes_R^\mathbf{L} L$ is $t$-pseudo-coherent with $t = \max(m + a, n + b)$. \item If $K$ and $L$ are pseudo-coherent, then $K \otimes_R^\mathbf{L} L$ is pseudo-coherent. \end{enumerate}


more-algebra-lemma-Noetherian-pseudo-coherent

Let $R$ be a Noetherian ring. Then \begin{enumerate} \item A complex of $R$-modules $K^\bullet$ is $m$-pseudo-coherent if and only if $K^\bullet \in D^{-}(R)$ and $H^i(K^\bullet)$ is a finite $R$-module for $i \geq m$. \item A complex of $R$-modules $K^\bullet$ is pseudo-coherent if and only if $K^\bullet \in D^{-}(R)$ and $H^i(K^\bullet)$ is a finite $R$-module for all $i$. \item An $R$-module is pseudo-coherent if and only if it is finite. \end{enumerate}


more-algebra-lemma-coherent-pseudo-coherent

Let $R$ be a coherent ring (Algebra, Definition \ref{algebra-definition-coherent}). Let $K \in D^-(R)$. The following are equivalent \begin{enumerate} \item $K$ is $m$-pseudo-coherent, \item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is coherent for $i > m$, and \item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is finitely presented for $i > m$. \end{enumerate} Thus $K$ is pseudo-coherent if and only if $H^i(K)$ is a coherent module for all $i$.


more-algebra-lemma-pseudo-coherence-colimit-ext

Let $R$ be a ring. Let $M = \colim M_i$ be a filtered colimit of $R$-modules. Let $K \in D(R)$ be $m$-pseudo-coherent. Then $\colim \Ext^n_R(K, M_i) = \Ext^n_R(K, M)$ for $n < -m$ and $\colim \Ext^{-m}_R(K, M_i) \to \Ext^{-m}_R(K, M)$ is injective.


more-algebra-lemma-pseudo-coherence-and-ext

Let $R$ be a ring. Let $L$, $M$, $N$ be $R$-modules. \begin{enumerate} \item If $M$ is finitely presented and $L$ is flat, then the canonical map $\Hom_R(M, N) \otimes_R L \to \Hom_R(M, N \otimes_R L)$ is an isomorphism. \item If $M$ is $(-m)$-pseudo-coherent and $L$ is flat, then the canonical map $\Ext^i_R(M, N) \otimes_R L \to \Ext^i_R(M, N \otimes_R L)$ is an isomorphism for $i < m$. \end{enumerate}


more-algebra-lemma-pseudo-coherence-and-base-change-ext

Let $R \to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules. \begin{enumerate} \item If $M$ is a finitely presented $R$-module, then $\Hom_R(M, N) \otimes_R R' = \Hom_{R'}(M \otimes_R R', N \otimes_R R')$. \item If $M$ is $(-m)$-pseudo-coherent, then $\Ext^i_R(M, N) \otimes_R R' = \Ext^i_{R'}(M \otimes_R R', N \otimes_R R')$ for $i < m$. \end{enumerate} In particular if $R$ is Noetherian and $M$ is a finite module this holds for all $i$.


more-algebra-lemma-pseudo-coherent-tensor

Let $R$ be a ring. Let $K \in D^-(R)$. The following are equivalent: \begin{enumerate} \item $K$ is pseudo-coherent, \item for every family $(Q_{\alpha})_{\alpha \in A}$ of $R$-modules, the canonical map $$ \alpha : K \otimes_R^\mathbf{L} \left( \prod\nolimits_\alpha Q_{\alpha} \right) \longrightarrow \prod\nolimits_\alpha (K \otimes_R^\mathbf{L} Q_{\alpha}) $$ is an isomorphism in $D(A)$, \item for every $R$-module $Q$ and every set $A$, the canonical map $$ \beta : K \otimes_R^\mathbf{L} Q^A \longrightarrow (K \otimes_R^\mathbf{L} Q)^A $$ is an isomorphism in $D(A)$, and \item for every set $A$, the canonical map $$ \gamma : K \otimes_R^\mathbf{L} R^A \longrightarrow K^A $$ is an isomorphism in $D(A)$. \end{enumerate} Given $m \in \mathbf{Z}$ the following are equivalent \begin{enumerate} \item[(a)] $K$ is $m$-pseudo-coherent, \item[(b)] for every family $(Q_{\alpha})_{\alpha \in A}$ of $R$-modules, with $\alpha$ as above $H^i(\alpha)$ is an isomorphism for $i > m$ and surjective for $i = m$, \item[(c)] for every $R$-module $Q$ and every set $A$, with $\beta$ as above $H^i(\beta)$ is an isomorphism for $i > m$ and surjective for $i = m$, \item[(d)] for every set $A$, with $\gamma$ as above $H^i(\gamma)$ is an isomorphism for $i > m$ and surjective for $i = m$. \end{enumerate}


more-algebra-lemma-detect-cohomology-pseudo-coherent

Let $R$ be a ring. Let $K \in D(R)$ be pseudo-coherent. Let $i \in \mathbf{Z}$. There exists a finitely presented $R$-module $M$ and a map $K \to M[-i]$ in $D(R)$ which induces an injection $H^i(K) \to M$.


more-algebra-lemma-detect-cohomology

Let $A$ be a Noetherian ring. Let $K \in D(A)$ be pseudo-coherent, i.e., $K \in D^-(A)$ with finite cohomology modules. Let $\mathfrak m$ be a maximal ideal of $A$. If $H^i(K)/\mathfrak m H^i(K) \not = 0$, then there exists a finite $A$-module $E$ annihilated by a power of $\mathfrak m$ and a map $K \to E[-i]$ which is nonzero on $H^i(K)$.


more-algebra-lemma-last-one-flat

Let $R$ be a ring. Let $K^\bullet$ be a bounded above complex of flat $R$-modules with tor-amplitude in $[a, b]$. Then $\Coker(d_K^{a - 1})$ is a flat $R$-module.


more-algebra-lemma-tor-amplitude

Let $R$ be a ring. Let $K^\bullet$ be an object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $K^\bullet$ has tor-amplitude in $[a, b]$. \item $K^\bullet$ is quasi-isomorphic to a complex $E^\bullet$ of flat $R$-modules with $E^i = 0$ for $i \not \in [a, b]$. \end{enumerate}


more-algebra-lemma-bounded-below-tor-amplitude

Let $R$ be a ring. Let $a \in \mathbf{Z}$ and let $K$ be an object of $D(R)$. The following are equivalent \begin{enumerate} \item $K$ has tor-amplitude in $[a, \infty]$, and \item $K$ is quasi-isomorphic to a K-flat complex $E^\bullet$ whose terms are flat $R$-modules with $E^i = 0$ for $i \not \in [a, \infty]$. \end{enumerate}


more-algebra-lemma-cone-tor-amplitude

Let $R$ be a ring. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. Let $a, b \in \mathbf{Z}$. \begin{enumerate} \item If $K^\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and $L^\bullet$ has tor-amplitude in $[a, b]$ then $M^\bullet$ has tor-amplitude in $[a, b]$. \item If $K^\bullet, M^\bullet$ have tor-amplitude in $[a, b]$, then $L^\bullet$ has tor-amplitude in $[a, b]$. \item If $L^\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and $M^\bullet$ has tor-amplitude in $[a, b]$, then $K^\bullet$ has tor-amplitude in $[a + 1, b + 1]$. \end{enumerate}


more-algebra-lemma-tor-dimension

Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \geq 0$. The following are equivalent \begin{enumerate} \item $M$ has tor dimension $\leq d$, and \item there exists a resolution $$ 0 \to F_d \to \ldots \to F_1 \to F_0 \to M \to 0 $$ with $F_i$ a flat $R$-module. \end{enumerate} In particular an $R$-module has tor dimension $0$ if and only if it is a flat $R$-module.


more-algebra-lemma-summands-tor-amplitude

Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. If $K^\bullet \oplus L^\bullet$ has tor amplitude in $[a, b]$ so do $K^\bullet$ and $L^\bullet$.


more-algebra-lemma-complex-finite-tor-dimension-modules

Let $R$ be a ring. Let $K^\bullet$ be a bounded complex of $R$-modules such that $K^i$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet$ has tor amplitude in $[a, b]$. In particular if $K^\bullet$ is a finite complex of $R$-modules of finite tor dimension, then $K^\bullet$ has finite tor dimension.