more-algebra-lemma-cohomology-tor-amplitude

Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet \in D^b(R)$ such that $H^i(K^\bullet)$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet$ has tor amplitude in $[a, b]$. In particular if $K^\bullet \in D^b(R)$ and all its cohomology groups have finite tor dimension then $K^\bullet$ has finite tor dimension.


more-algebra-lemma-push-tor-amplitude

Let $A \to B$ be a ring map. Let $K^\bullet$ and $L^\bullet$ be complexes of $B$-modules. Let $a, b, c, d \in \mathbf{Z}$. If \begin{enumerate} \item $K^\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, \item $L^\bullet$ as a complex of $A$-modules has tor amplitude in $[c, d]$, \end{enumerate} then $K^\bullet \otimes^\mathbf{L}_B L^\bullet$ as a complex of $A$-modules has tor amplitude in $[a + c, b + d]$.


more-algebra-lemma-flat-push-tor-amplitude

Let $A \to B$ be a ring map. Assume that $B$ is flat as an $A$-module. Let $K^\bullet$ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. If $K^\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\bullet$ as a complex of $A$-modules has tor amplitude in $[a, b]$.


more-algebra-lemma-finite-tor-dimension-push-tor-amplitude

Let $A \to B$ be a ring map. Assume that $B$ has tor dimension $\leq d$ as an $A$-module. Let $K^\bullet$ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. If $K^\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\bullet$ as a complex of $A$-modules has tor amplitude in $[a - d, b]$.


more-algebra-lemma-pull-tor-amplitude

Let $A \to B$ be a ring map. Let $a, b \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $A$-modules with tor amplitude in $[a, b]$. Then $K^\bullet \otimes_A^{\mathbf{L}} B$ as a complex of $B$-modules has tor amplitude in $[a, b]$.


more-algebra-lemma-flat-base-change-finite-tor-dimension

Let $A \to B$ be a flat ring map. Let $d \geq 0$. Let $M$ be an $A$-module of tor dimension $\leq d$. Then $M \otimes_A B$ is a $B$-module of tor dimension $\leq d$.


more-algebra-lemma-tor-amplitude-localization

Let $A  \to B$ be a ring map. Let $K^\bullet$ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $K^\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules, \item $K^\bullet_\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every prime $\mathfrak q \subset B$ with $\mathfrak p = A \cap \mathfrak q$, \item $K^\bullet_\mathfrak m$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every maximal ideal $\mathfrak m \subset B$ with $\mathfrak p = A \cap \mathfrak m$. \end{enumerate}


more-algebra-lemma-glue-tor-amplitude

Let $R$ be a ring. Let $f_1, \ldots, f_r \in R$ be elements which generate the unit ideal. Let $a, b \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \otimes_R R_{f_i}$ has tor amplitude in $[a, b]$, then $K^\bullet$ has tor amplitude in $[a, b]$.


more-algebra-lemma-flat-descent-tor-amplitude

Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $R$-modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes_R R'$ has tor amplitude in $[a, b]$, then $K^\bullet$ has tor amplitude in $[a, b]$.


more-algebra-lemma-no-change-tor-amplitude

Given ring maps $R \to A \to B$ with $A \to B$ faithfully flat and $K \in D(A)$ the tor amplitude of $K$ over $R$ is the same as the tor amplitude of $K \otimes_A^\mathbf{L} B$ over $R$.


more-algebra-lemma-finite-gl-dim-tor-dimension

Let $R$ be a ring of finite global dimension $d$. Then \begin{enumerate} \item every module has tor dimension $\leq d$, \item a complex of $R$-modules $K^\bullet$ with $H^i(K^\bullet) \not = 0$ only if $i \in [a, b]$ has tor amplitude in $[a - d, b]$, and \item a complex of $R$-modules $K^\bullet$ has finite tor dimension if and only if $K^\bullet \in D^b(R)$. \end{enumerate}


more-algebra-lemma-projective-amplitude

Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $K$ has projective-amplitude in $[a, b]$, \item $\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i \not \in [-b, -a]$, \item $H^n(K) = 0$ for $n > b$ and $\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i > -a$, and \item $H^n(K) = 0$ for $n \not \in [a - 1, b]$ and $\Ext^{-a + 1}_R(K, N) = 0$ for all $R$-modules $N$. \end{enumerate}


more-algebra-lemma-injective-amplitude

Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $K$ has injective-amplitude in $[a, b]$, \item $\Ext^i_R(N, K) = 0$ for all $R$-modules $N$ and all $i \not \in [a, b]$, \item $\Ext^i(R/I, K) = 0$ for all ideals $I \subset R$ and all $i \not \in [a, b]$. \end{enumerate}


more-algebra-lemma-finite-injective-dimension

Let $R$ be a ring. Let $K \in D(R)$. \begin{enumerate} \item If $K$ is in $D^b(R)$ and $H^i(K)$ has finite injective dimension for all $i$, then $K$ has finite injective dimension. \item If $K^\bullet$ represents $K$, is a bounded complex of $R$-modules, and $K^i$ has finite injective dimension for all $i$, then $K$ has finite injective dimension. \end{enumerate}


more-algebra-lemma-finite-injective-dimension-Noetherian-radical

Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal contained in the Jacobson radical of $R$. Let $K \in D^+(R)$ have finite cohomology modules. Then the following are equivalent \begin{enumerate} \item $K$ has finite injective dimension, and \item there exists a $b$ such that $\Ext^i_R(R/J, K) = 0$ for $i > b$ and any ideal $J \supset I$. \end{enumerate}


more-algebra-lemma-finite-injective-dimension-Noetherian-local

Let $(R, \mathfrak m, \kappa)$ be a local Noetherian ring. Let $K \in D^+(R)$ have finite cohomology modules. Then the following are equivalent \begin{enumerate} \item $K$ has finite injective dimension, and \item $\Ext^i_R(\kappa, K) = 0$ for $i \gg 0$. \end{enumerate}


more-algebra-lemma-ideal-factor-through-projective

Let $R$ be a ring. Let $M$, $N$ be $R$-modules. \begin{enumerate} \item Given an $R$-module map $\varphi : M \to N$ the following are equivalent: (a) $\varphi$ factors through a projective $R$-module, and (b) $\varphi$ factors through a free $R$-module. \item The set of $\varphi : M \to N$ satisfying the equivalent conditions of (1) is an $R$-submodule of $\Hom_R(M, N)$. \item Given maps $\psi : M' \to M$ and $\xi : N \to N'$, if $\varphi : M \to N$ satisfies the equivalent conditions of (1), then $\xi \circ \varphi \circ \psi : M' \to N'$ does too. \end{enumerate}


more-algebra-lemma-near-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. The following conditions are equivalent \begin{enumerate} \item for every $a \in I$ the map $a : M \to M$ factors through a projective $R$-module, \item for every $a \in I$ the map $a : M \to M$ factors through a free $R$-module, and \item $\Ext^1_R(M, N)$ is annihilated by $I$ for every $R$-module $N$. \end{enumerate}


more-algebra-lemma-torsion-near-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. If $M$ is annihilated by $I$, then $M$ is $I$-projective.


more-algebra-lemma-ses-near-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $$ 0 \to K \to P \to M \to 0 $$ be a short exact sequence of $R$-modules. If $M$ is $I$-projective and $P$ is projective, then $K$ is $I$-projective.


more-algebra-lemma-dual-near-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. If $M$ is a finite, $I$-projective $R$-module, then $M^\vee = \Hom_R(M, R)$ is $I$-projective.


more-algebra-lemma-compose

Let $R$ be a ring. Given complexes $K^\bullet, L^\bullet, M^\bullet$ of $R$-modules there is a canonical isomorphism $$ \Hom^\bullet(K^\bullet, \Hom^\bullet(L^\bullet, M^\bullet)) = \Hom^\bullet(\text{Tot}(K^\bullet \otimes_R L^\bullet), M^\bullet) $$ of complexes of $R$-modules.


more-algebra-lemma-composition

Let $R$ be a ring. Given complexes $K^\bullet, L^\bullet, M^\bullet$ of $R$-modules there is a canonical morphism $$ \text{Tot}\left( \Hom^\bullet(L^\bullet, M^\bullet) \otimes_R \Hom^\bullet(K^\bullet, L^\bullet) \right) \longrightarrow \Hom^\bullet(K^\bullet, M^\bullet) $$ of complexes of $R$-modules.


more-algebra-lemma-diagonal-better

Let $R$ be a ring. Given complexes $K^\bullet, L^\bullet, M^\bullet$ of $R$-modules there is a canonical morphism $$ \text{Tot}(K^\bullet \otimes_R \Hom^\bullet(M^\bullet, L^\bullet)) \longrightarrow \Hom^\bullet(M^\bullet, \text{Tot}(K^\bullet \otimes_R L^\bullet)) $$ of complexes of $R$-modules functorial in all three complexes.


more-algebra-lemma-diagonal

Let $R$ be a ring. Given complexes $K^\bullet, L^\bullet$ of $R$-modules there is a canonical morphism $$ K^\bullet \longrightarrow \Hom^\bullet(L^\bullet, \text{Tot}(K^\bullet \otimes_R L^\bullet)) $$ of complexes of $R$-modules functorial in both complexes.


more-algebra-lemma-evaluate-and-more

Let $R$ be a ring. Given complexes $K^\bullet, L^\bullet, M^\bullet$ of $R$-modules there is a canonical morphism $$ \text{Tot}(\Hom^\bullet(L^\bullet, M^\bullet) \otimes_R K^\bullet) \longrightarrow \Hom^\bullet(\Hom^\bullet(K^\bullet, L^\bullet), M^\bullet) $$ of complexes of $R$-modules functorial in all three complexes.


more-algebra-lemma-symmetric-monoidal-cat-complexes

Let $R$ be a ring. The category of complexes of $R$-modules with tensor product defined by $K^\bullet \otimes M^\bullet = \text{Tot}(K^\bullet \otimes_R M^\bullet)$ is a symmetric monoidal category with the associativity and commutativity constraints given above.


more-algebra-lemma-left-dual-module

Let $R$ be a ring. Let $M$ be an $R$-module. Let  $N, \eta, \epsilon$ be a left dual of $M$ in the monoidal category of $R$-modules, see Categories, Definition \ref{categories-definition-dual}. Then \begin{enumerate} \item $M$ and $N$ are finite projective $R$-modules, \item the map $e : \Hom_R(M, R) \to N$, $\lambda \mapsto (\lambda \otimes 1)(\eta)$ is an isomorphism, \item we have $\epsilon(n, m) = e^{-1}(n)(m)$ for $n \in N$ and $m \in M$. \end{enumerate}


more-algebra-lemma-left-dual-complex

Let $R$ be a ring. Let $M^\bullet$ be a complex of $R$-modules. Let $N^\bullet, \eta, \epsilon$ be a left dual of $M^\bullet$ in the monoidal category of complexes of $R$-modules. Then \begin{enumerate} \item $M^\bullet$ and $N^\bullet$ are bounded, \item $M^n$ and $N^n$ are finite projective $R$-modules, \item writing $\epsilon = \sum \epsilon_n$ with $\epsilon_n : N^{-n} \otimes_R M^n \to R$ and $\eta = \sum \eta_n$ with $\eta_n : R \to  M^n \otimes_R N^{-n}$ then $(N^{-n}, \eta_n, \epsilon_n)$ is the left dual of $M^n$ as in Lemma \ref{lemma-left-dual-module}, \item the differential $d_N^n : N^n \to N^{n + 1}$ is equal to $-(-1)^n$ times the map $$ N^n = \Hom_R(M^{-n}, R) \xrightarrow{d_M^{-n - 1}} \Hom_R(M^{-n - 1}, R) = N^{n + 1} $$ where the equality signs are the identifications from Lemma \ref{lemma-left-dual-module} part (2). \end{enumerate} Conversely, given a bounded complex $M^\bullet$ of finite projective $R$-modules, setting $N^n = \Hom_R(M^{-n}, R)$ with differentials as above, setting $\epsilon = \sum \epsilon_n$ with $\epsilon_n : N^{-n} \otimes_R M^n \to R$ given by evaluation, and setting $\eta = \sum \eta_n$ with $\eta_n : R \to M^n \otimes_R N^{-n}$ mapping $1$ to $\text{id}_{M_n}$ we obtain a left dual of $M^\bullet$ in the monoidal category of complexes of $R$-modules.


more-algebra-lemma-internal-hom

Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism $$ R\Hom_R(K, R\Hom_R(L, M)) = R\Hom_R(K \otimes_R^\mathbf{L} L, M) $$ in $D(R)$ functorial in $K, L, M$ which recovers (\ref{equation-internal-hom}) by taking $H^0$.


more-algebra-lemma-RHom-out-of-projective

Let $R$ be a ring. Let $P^\bullet$ be a bounded above complex of projective $R$-modules. Let $L^\bullet$ be a complex of $R$-modules. Then $R\Hom_R(P^\bullet, L^\bullet)$ is represented by the complex $\Hom^\bullet(P^\bullet, L^\bullet)$.


more-algebra-lemma-internal-hom-evaluate

Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism $$ R\Hom_R(L, M) \otimes_R^\mathbf{L} K \longrightarrow R\Hom_R(R\Hom_R(K, L), M) $$ in $D(R)$ functorial in $K, L, M$.


more-algebra-lemma-internal-hom-diagonal-better

Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism $$ K \otimes_R^\mathbf{L} R\Hom_R(M, L) \longrightarrow R\Hom_R(M, K \otimes_R^\mathbf{L} L) $$ in $D(R)$ functorial in $K$, $L$, $M$.


more-algebra-lemma-internal-hom-diagonal

Let $R$ be a ring. Given complexes $K, L$ in $D(R)$ there is a canonical morphism $$ K \longrightarrow R\Hom_R(L, K \otimes_R^\mathbf{L} L) $$ in $D(R)$ functorial in both $K$ and $L$.


more-algebra-lemma-perfect

Let $K^\bullet$ be an object of $D(R)$. The following are equivalent \begin{enumerate} \item $K^\bullet$ is perfect, and \item $K^\bullet$ is pseudo-coherent and has finite tor dimension. \end{enumerate} If (1) and (2) hold and $K^\bullet$ has tor-amplitude in $[a, b]$, then $K^\bullet$ is quasi-isomorphic to a complex $E^\bullet$ of finite projective $R$-modules with $E^i = 0$ for $i \not \in [a, b]$.


more-algebra-lemma-perfect-module

Let $M$ be a module over a ring $R$. The following are equivalent \begin{enumerate} \item $M$ is a perfect module, and \item there exists a resolution $$ 0 \to F_d \to \ldots \to F_1 \to F_0 \to M \to 0 $$ with each $F_i$ a finite projective $R$-module. \end{enumerate}


more-algebra-lemma-two-out-of-three-perfect

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 perfect then the third is also perfect.


more-algebra-lemma-summands-perfect

Let $R$ be a ring. If $K^\bullet \oplus L^\bullet$ is perfect, then so are $K^\bullet$ and $L^\bullet$.


more-algebra-lemma-cohomology-perfect

Let $R$ be a ring. If $K^\bullet \in D^b(R)$ and all its cohomology modules are perfect, then $K^\bullet$ is perfect.


more-algebra-lemma-pull-perfect

Let $A \to B$ be a ring map. Let $K^\bullet$ be a perfect complex of $A$-modules. Then $K^\bullet \otimes_A^{\mathbf{L}} B$ is a perfect complex of $B$-modules.


more-algebra-lemma-glue-perfect

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


more-algebra-lemma-flat-descent-perfect

Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. 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 perfect, then $K^\bullet$ is perfect.


more-algebra-lemma-regular-perfect

Let $R$ be a regular ring of finite dimension. Then \begin{enumerate} \item an $R$-module is perfect if and only if it is a finite $R$-module, and \item a complex of $R$-modules $K^\bullet$ is perfect if and only if $K^\bullet \in D^b(R)$ and each $H^i(K^\bullet)$ is a finite $R$-module. \end{enumerate}


more-algebra-lemma-dual-perfect-complex

Let $A$ be a ring. Let $K \in D(A)$ be perfect. Then $K^\vee = R\Hom_A(K, A)$ is a perfect complex and $K \cong (K^\vee)^\vee$. There are functorial isomorphisms $$ L \otimes_A^\mathbf{L} K^\vee = R\Hom_A(K, L) \quad\text{and}\quad H^0(L \otimes_A^\mathbf{L} K^\vee) = \Ext_A^0(K, L) $$ for $L \in D(A)$.


more-algebra-lemma-colimit-perfect-complexes

Let $R = \colim_{i \in I} R_i$ be a filtered colimit of rings. \begin{enumerate} \item Given a perfect $K$ in $D(R)$ there exists an $i \in I$ and a perfect $K_i$ in $D(R_i)$ such that $K \cong K_i \otimes_{R_i}^\mathbf{L} R$ in $D(R)$. \item Given $0 \in I$ and $K_0, L_0 \in D(R_0)$ with $K_0$ perfect, we have $$ \Hom_{D(R)}(K_0 \otimes_{R_0}^\mathbf{L} R, L_0 \otimes_{R_0}^\mathbf{L} R) = \colim_{i \geq 0} \Hom_{D(R_i)}(K_0 \otimes_{R_0}^\mathbf{L} R_i, L_0 \otimes_{R_0}^\mathbf{L} R_i) $$ \end{enumerate} In other words, the triangulated category of perfect complexes over $R$ is the colimit of the triangulated categories of perfect complexes over $R_i$.


more-algebra-lemma-lift-acyclic-complex

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $\mathcal{P}$ be a class of $R$-modules. Assume \begin{enumerate} \item each $P \in \mathcal{P}$ is a projective $R$-module, \item if $P_1 \in \mathcal{P}$ and $P_1 \oplus P_2 \in \mathcal{P}$, then $P_2 \in \mathcal{P}$, and \item if $f : P_1 \to P_2$, $P_1, P_2 \in \mathcal{P}$ is surjective modulo $I$, then $f$ is surjective. \end{enumerate} Then given any bounded above acyclic complex $E^\bullet$ whose terms are of the form $P/IP$ for $P \in \mathcal{P}$ there exists a bounded above acyclic complex $P^\bullet$ whose terms are in $\mathcal{P}$ lifting $E^\bullet$.


more-algebra-lemma-lift-complex

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $\mathcal{P}$ be a class of $R$-modules. Let $K \in D(R)$ and let $E^\bullet$ be a complex of $R/I$-modules representing $K \otimes_R^\mathbf{L} R/I$. Assume \begin{enumerate} \item each $P \in \mathcal{P}$ is a projective $R$-module, \item $P_1 \in \mathcal{P}$ and $P_1 \oplus P_2 \in \mathcal{P}$ if and only if $P_1, P_2 \in \mathcal{P}$, \item if $f : P_1 \to P_2$, $P_1, P_2 \in \mathcal{P}$ is surjective modulo $I$, then $f$ is surjective, \item $E^\bullet$ is bounded above and $E^i$ is of the form $P/IP$ for $P \in \mathcal{P}$, and \item $K$ can be represented by a bounded above complex whose terms are in $\mathcal{P}$. \end{enumerate} Then there exists a bounded above complex $P^\bullet$ whose terms are in $\mathcal{P}$ with $P^\bullet/IP^\bullet$ isomorphic to $E^\bullet$ and representing $K$ in $D(R)$.


more-algebra-lemma-lift-complex-projectives

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \begin{enumerate} \item $E^\bullet$ is a bounded above complex of projective $R/I$-modules, \item $K \otimes_R^\mathbf{L} R/I$ is represented by $E^\bullet$ in $D(R/I)$, and \item $I$ is a nilpotent ideal. \end{enumerate} Then there exists a bounded above complex $P^\bullet$ of projective $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes_R R/I$ is isomorphic to $E^\bullet$.


more-algebra-lemma-lift-complex-stably-frees

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \begin{enumerate} \item $E^\bullet$ is a bounded above complex of finite stably free $R/I$-modules, \item $K \otimes_R^\mathbf{L} R/I$ is represented by $E^\bullet$ in $D(R/I)$, \item $K^\bullet$ is pseudo-coherent, and \item every element of $1 + I$ is invertible. \end{enumerate} Then there exists a bounded above complex $P^\bullet$ of finite stably free $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes_R R/I$ is isomorphic to $E^\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free.


more-algebra-lemma-lift-pseudo-coherent-from-residue-field

Let $(R, \mathfrak m, \kappa)$ be a local ring. Let $K \in D(R)$ be pseudo-coherent. Set $d_i = \dim_\kappa H^i(K \otimes_R^\mathbf{L} \kappa)$. Then $d_i < \infty$ and for some $b \in \mathbf{Z}$ we have $d_i = 0$ for $i > b$. Then there exists a complex $$ \ldots \to R^{\oplus d_{b - 2}} \to R^{\oplus d_{b - 1}} \to R^{\oplus d_b} \to 0 \to \ldots $$ representing $K$ in $D(R)$. Moreover, this complex is unique up to isomorphism(!).


more-algebra-lemma-lift-perfect-from-residue-field

Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime. Let $K \in D(R)$ be perfect. Set $d_i = \dim_{\kappa(\mathfrak p)} H^i(K \otimes_R^\mathbf{L} \kappa(\mathfrak p))$. Then $d_i < \infty$ and only a finite number are nonzero. Then there exists an $f \in R$, $f \not \in \mathfrak p$ and a complex $$ \ldots \to 0 \to R_f^{\oplus d_a} \to R_f^{\oplus d_{a + 1}} \to \ldots \to R_f^{\oplus d_{b - 1}} \to R_f^{\oplus d_b} \to 0 \to \ldots $$ representing $K \otimes_R^\mathbf{L} R_f$ in $D(R_f)$.


more-algebra-lemma-compare-representatives-perfect

Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime. Let $M^\bullet$ and $N^\bullet$ be bounded complexes of finite projective $R$-modules representing the same object of $D(R)$. Then there exists an $f \in R$, $f \not \in \mathfrak p$ such that there is an isomorphism (!) of complexes $$ M^\bullet_f \oplus P^\bullet \cong N^\bullet_f \oplus Q^\bullet $$ where $P^\bullet$ and $Q^\bullet$ are finite direct sums of trivial complexes, i.e., complexes of the form the form $\ldots \to 0 \to R_f \xrightarrow{1} R_f \to 0 \to \ldots$ (placed in arbitrary degrees).


more-algebra-lemma-lift-complex-finite-projectives

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \begin{enumerate} \item $E^\bullet$ is a bounded above complex of finite projective $R/I$-modules, \item $K \otimes_R^\mathbf{L} R/I$ is represented by $E^\bullet$ in $D(R/I)$, \item $K$ is pseudo-coherent, and \item $(R, I)$ is a henselian pair. \end{enumerate} Then there exists a bounded above complex $P^\bullet$ of finite projective $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes_R R/I$ is isomorphic to $E^\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free.


more-algebra-lemma-splitting-unique

Let $R$ be a ring. Let $K$ and $L$ be objects of $D(R)$. Assume $L$ has projective-amplitude in $[a, b]$, for example if $L$ is perfect of tor-amplitude in $[a, b]$. \begin{enumerate} \item If $H^i(K) = 0$ for $i \geq a$, then $\Hom_{D(R)}(L, K) = 0$. \item If $H^i(K) = 0$ for $i \geq a + 1$, then given any distinguished triangle $K \to M \to L \to K[1]$ there is an isomorphism $M \cong K \oplus L$ in $D(R)$ compatible with the maps in the distinguished triangle. \item If $H^i(K) = 0$ for $i \geq a$, then the isomorphism in (2) exists and is unique. \end{enumerate}


more-algebra-lemma-better-cut-complex-in-two

Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $K^\bullet$ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \in \mathbf{Z}$ the map $$ H^i(K^\bullet) \otimes_R \kappa(\mathfrak p) \longrightarrow H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) $$ is surjective. Then there exists an $f \in R$, $f \not \in \mathfrak p$ such that $\tau_{\geq i + 1}(K^\bullet \otimes_R R_f)$ is a perfect object of $D(R_f)$ with tor amplitude in $[i + 1, \infty]$ and a canonical isomorphism $$ K^\bullet \otimes_R R_f \cong \tau_{\leq i}(K^\bullet \otimes_R R_f) \oplus \tau_{\geq i + 1}(K^\bullet \otimes_R R_f) $$ in $D(R_f)$.


more-algebra-lemma-cut-complex-in-two

Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $i \in \mathbf{Z}$. Let $K^\bullet$ be a pseudo-coherent complex of $R$-modules such that $H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) = 0$. Then there exists an $f \in R$, $f \not \in \mathfrak p$ and a canonical direct sum decomposition $$ K^\bullet \otimes_R R_f = \tau_{\geq i + 1}(K^\bullet \otimes_R R_f) \oplus \tau_{\leq i - 1}(K^\bullet \otimes_R R_f) $$ in $D(R_f)$ with $\tau_{\geq i + 1}(K^\bullet \otimes_R R_f)$ a perfect complex with tor-amplitude in $[i + 1, \infty]$.


more-algebra-lemma-split-using-ext-injective

Let $R$ be a ring. Let $K \in D^-(R)$. Let $a \in \mathbf{Z}$. Assume that for any injective $R$-module map $M \to M'$ the map $\Ext^{-a}_R(K, M) \to \Ext^{-a}_R(K, M')$ is injective. Then there is a unique direct sum decomposition $K \cong \tau_{\leq a}K \oplus \tau_{\geq a + 1}K$ and $\tau_{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$.


more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect

Let $R$ be a ring and let $\mathfrak p \subset R$ be a prime. Let $K$ be pseudo-coherent and bounded below. Set $d_i = \dim_{\kappa(\mathfrak p)} H^i(K \otimes_R^\mathbf{L} \kappa(\mathfrak p))$. If there exists an $a \in \mathbf{Z}$ such that $d_i = 0$ for $i < a$, then there exists an $f \in R$, $f \not \in \mathfrak p$ and a complex $$ \ldots \to 0 \to R_f^{\oplus d_a} \to R_f^{\oplus d_{a + 1}} \to \ldots \to R_f^{\oplus d_{b - 1}} \to R_f^{\oplus d_b} \to 0 \to \ldots $$ representing $K \otimes_R^\mathbf{L} R_f$ in $D(R_f)$. In particular $K \otimes_R^\mathbf{L} R_f$ is perfect.


more-algebra-lemma-check-perfect-pointwise

Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet$ be a pseudo-coherent complex of $R$-modules. The following are equivalent \begin{enumerate} \item $K^\bullet$ is perfect with tor amplitude in $[a, b]$, \item for every prime $\mathfrak p$ we have $H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) = 0$ for all $i \not \in [a, b]$, and \item for every maximal ideal $\mathfrak m$ we have $H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak m)) = 0$ for all $i \not \in [a, b]$. \end{enumerate}


more-algebra-lemma-check-perfect-stalks

Let $R$ be a ring. Let $K^\bullet$ be a pseudo-coherent complex of $R$-modules. Consider the following conditions \begin{enumerate} \item $K^\bullet$ is perfect, \item for every prime ideal $\mathfrak p$ the complex $K^\bullet \otimes_R R_{\mathfrak p}$ is perfect, \item for every maximal ideal $\mathfrak m$ the complex $K^\bullet \otimes_R R_{\mathfrak m}$ is perfect, \item for every prime $\mathfrak p$ we have $H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) = 0$ for all $i \ll 0$, \item for every maximal ideal $\mathfrak m$ we have $H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak m)) = 0$ for all $i \ll 0$. \end{enumerate} We always have the implications $$ (1) \Rightarrow (2) \Leftrightarrow (3) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5) $$ If $K^\bullet$ is bounded below, then all conditions are equivalent.


more-algebra-lemma-projective-amplitude-pseudo-coherent

Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $K$ has projective-amplitude in $[a, b]$, \item $K$ is perfect of tor-amplitude in $[a, b]$, \item $\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i \not \in [-b, -a]$, \item $H^n(K) = 0$ for $n > b$ and $\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i > -a$, and \item $H^n(K) = 0$ for $n \not \in [a - 1, b]$ and $\Ext^{-a + 1}_R(K, N) = 0$ for all finitely presented $R$-modules $N$. \end{enumerate}


more-algebra-lemma-perfect-over-polynomial-ring

Let $A \to B$ be a ring map. Let $a, b \in \mathbf{Z}$. Let $d \geq 0$. Let $K^\bullet$ be a complex of $B$-modules. Assume \begin{enumerate} \item the ring map $A \to B$ is flat, \item for every prime $\mathfrak p \subset A$ the ring $B \otimes_A \kappa(\mathfrak p)$ has finite global dimension $\leq d$, \item $K^\bullet$ is pseudo-coherent as a complex of $B$-modules, and \item $K^\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules. \end{enumerate} Then $K^\bullet$ is perfect as a complex of $B$-modules with tor amplitude in $[a - d, b]$.


more-algebra-lemma-perfect-over-regular-local-ring

Let $A \to B$ be a local ring homomorphism. Let $a, b \in \mathbf{Z}$. Let $d \geq 0$. Let $K^\bullet$ be a complex of $B$-modules. Assume \begin{enumerate} \item the ring map $A \to B$ is flat, \item the ring $B/\mathfrak m_AB$ is regular of dimension $d$, \item $K^\bullet$ is pseudo-coherent as a complex of $B$-modules, and \item $K^\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules, in fact it suffices if $H^i(K^\bullet \otimes_A^\mathbf{L} \kappa(\mathfrak m_A))$ is nonzero only for $i \in [a, b]$. \end{enumerate} Then $K^\bullet$ is perfect as a complex of $B$-modules with tor amplitude in $[a - d, b]$.


more-algebra-lemma-perfect-ring-classical-generator

Let $R$ be a ring. The full subcategory $D_{perf}(R) \subset D(R)$ of perfect objects is the smallest strictly full, saturated, triangulated subcategory containing $R = R[0]$. In other words $D_{perf}(R) = \langle R \rangle$. In particular, $R$ is a classical generator for $D_{perf}(R)$.


more-algebra-lemma-commutes-with-countable-sums

Let $R$ be a ring. Let $K \in D(R)$ be an object such that for every countable set of objects $E_n \in D(R)$ the canonical map $$ \bigoplus \Hom_{D(R)}(K, E_n) \longrightarrow \Hom_{D(R)}(K, \bigoplus E_n) $$ is a bijection. Then, given any system $L_n^\bullet$ of complexes over $\mathbf{N}$ we have that $$ \colim \Hom_{D(R)}(K, L^\bullet_n) \longrightarrow \Hom_{D(R)}(K, L^\bullet) $$ is a bijection, where $L^\bullet$ is the termwise colimit, i.e., $L^m = \colim L_n^m$ for all $m \in \mathbf{Z}$.


more-algebra-lemma-perfect-modulo-nilpotent-ideal

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that \begin{enumerate} \item $K \otimes_R^\mathbf{L} R/I$ is perfect in $D(R/I)$, and \item $I$ is a nilpotent ideal. \end{enumerate} Then $K$ is perfect in $D(R)$.


more-algebra-lemma-ghost-lemma

\begin{reference} \cite{Kelly} \end{reference} Let $R$ be a ring. Let $n \geq 1$. Let $K \in \langle R \rangle_n$ with notation as in Derived Categories, Section \ref{derived-section-generators}. Consider maps $$ K \xrightarrow{f_1} K_1 \xrightarrow{f_2} K_2 \xrightarrow{f_3} \ldots \xrightarrow{f_n} K_n $$ in $D(R)$. If $H^i(f_j) = 0$ for all $i, j$, then $f_n \circ \ldots \circ f_1 = 0$.


more-algebra-lemma-not-regular-not-strong

Let $R$ be a Noetherian ring. If $R$ is a strong generator for $D_{perf}(R)$, then $R$ is regular of finite dimension.


more-algebra-lemma-ext-regular

Let $R$ be a Noetherian regular ring of dimension $d < \infty$. Let $K, L \in D^-(R)$. Assume there exists an $k$ such that $H^i(K) = 0$ for $i \leq k$ and $H^i(L) = 0$ for $i \geq k - d + 1$. Then $\Hom_{D(R)}(K, L) = 0$.


more-algebra-lemma-split-complex-regular

Let $R$ be a Noetherian regular ring of dimension $1 \leq d < \infty$. Let $K \in D(R)$ be perfect and let $k \in \mathbf{Z}$ such that $H^i(K) = 0$ for $i = k - d + 2, \ldots, k$ (empty condition if $d = 1$). Then $K = \tau_{\leq k - d + 1}K \oplus \tau_{\geq k + 1}K$.


more-algebra-lemma-regular-strong-generator

Let $R$ be a Noetherian regular ring of finite dimension. Then $R$ is a strong generator for the full subcategory $D_{perf}(R) \subset D(R)$ of perfect objects.


more-algebra-lemma-relatively-finitely-presented

Let $R \to A$ be a ring map of finite type. Let $M$ be an $A$-module. The following are equivalent \begin{enumerate} \item for some presentation $\alpha : R[x_1, \ldots, x_n] \to A$ the module $M$ is a finitely presented $R[x_1, \ldots, x_n]$-module, \item for all presentations $\alpha : R[x_1, \ldots, x_n] \to A$ the module $M$ is a finitely presented $R[x_1, \ldots, x_n]$-module, and \item for any surjection $A' \to A$ where $A'$ is a finitely presented $R$-algebra, the module $M$ is finitely presented as $A'$-module. \end{enumerate} In this case $M$ is a finitely presented $A$-module.


more-algebra-lemma-finite-extension

Let $R$ be a ring. Let $A \to B$ be a finite map of finite type $R$-algebras. Let $M$ be a $B$-module. Then $M$ is an $A$-module finitely presented relative to $R$ if and only if $M$ is a $B$-module finitely presented relative to $R$.


more-algebra-lemma-localize-relative-finite-presentation

Let $R$ be a ring, $f \in R$ an element, $R_f \to A$ is a finite type ring map, $g \in A$, and $M$ an $A$-module. If $M$ of finite presentation relative to $R_f$, then $M_g$ is an $A_g$-module of finite presentation relative to $R$.


more-algebra-lemma-base-change-relative-finite-presentation

Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. For any ring map $R \to R'$ the $A \otimes_R R'$-module $$ M \otimes_A A' = M \otimes_R R' $$ is finitely presented relative to $R'$.


more-algebra-lemma-pull-relative-finite-presentation

Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. Let $A \to A'$ be a ring map of finite presentation. The $A'$-module $M \otimes_A A'$ is finitely presented relative to $R$.


more-algebra-lemma-composition-relative-finite-presentation

Let $R \to A \to B$ be finite type ring maps. Let $M$ be a $B$-module. If $M$ is finitely presented relative to $A$ and $A$ is of finite presentation over $R$, then $M$ is finitely presented relative to $R$.


more-algebra-lemma-glue-relative-finite-presentation

Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. Let $f_1, \ldots, f_r \in A$ generate the unit ideal. The following are equivalent \begin{enumerate} \item each $M_{f_i}$ is finitely presented relative to $R$, and \item $M$ is finitely presented relative to $R$. \end{enumerate}


more-algebra-lemma-ses-relatively-finite-presentation

Let $R \to A$ be a finite type ring map. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of $A$-modules. \begin{enumerate} \item If $M', M''$ are finitely presented relative to $R$, then so is $M$. \item If $M'$ is a finite type $A$-module and $M$ is finitely presented relative to $R$, then $M''$ is finitely presented relative to $R$. \end{enumerate}


more-algebra-lemma-sum-relatively-finite-presentation

Let $R \to A$ be a finite type ring map. Let $M, M'$ be $A$-modules. If $M \oplus M'$ is finitely presented relative to $R$, then so are $M$ and $M'$.


more-algebra-lemma-pull-push

Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. Consider the $R$-algebra map $R[x] \to R$ which maps $x$ to zero. Then $$ K^\bullet \otimes_{R[x]}^{\mathbf{L}} R \cong K^\bullet \oplus K^\bullet[1] $$ in $D(R)$.


more-algebra-lemma-add-variable-pseudo-coherent

Let $R$ be a ring and $K^\bullet$ a complex of $R$-modules. Let $m \in \mathbf{Z}$. Consider the $R$-algebra map $R[x] \to R$ which maps $x$ to zero. Then $K^\bullet$ is $m$-pseudo-coherent as a complex of $R$-modules if and only if $K^\bullet$ is $m$-pseudo-coherent as a complex of $R[x]$-modules.


more-algebra-lemma-relatively-pseudo-coherent

Let $R \to A$ be a ring map of finite type. Let $K^\bullet$ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item for some presentation $\alpha : R[x_1, \ldots, x_n] \to A$ the complex $K^\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \ldots, x_n]$-modules, \item for all presentations $\alpha : R[x_1, \ldots, x_n] \to A$ the complex $K^\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \ldots, x_n]$-modules. \end{enumerate} In particular the same equivalence holds for pseudo-coherence.


more-algebra-lemma-finite-extension-pseudo-coherent

Let $R$ be a ring. Let $A \to B$ be a finite map of finite type $R$-algebras. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $B$-modules. Then $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$ if and only if $K^\bullet$ seen as a complex of $A$-modules is $m$-pseudo-coherent (pseudo-coherent) relative to $R$.


more-algebra-lemma-cone-relatively-pseudo-coherent

Let $R$ be a ring. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$ be a distinguished triangle in $D(A)$. \begin{enumerate} \item If $K^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $L^\bullet$ is $m$-pseudo-coherent relative to $R$ then $M^\bullet$ is $m$-pseudo-coherent relative to $R$. \item If $K^\bullet, M^\bullet$ are $m$-pseudo-coherent relative to $R$, then $L^\bullet$ is $m$-pseudo-coherent relative to $R$. \item If $L^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $M^\bullet$ is $m$-pseudo-coherent relative to $R$, then $K^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$. \end{enumerate} Moreover, if two out of three of $K^\bullet, L^\bullet, M^\bullet$ are pseudo-coherent relative to $R$, the so is the third.


more-algebra-lemma-rel-n-pseudo-module

Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. Then \begin{enumerate} \item $M$ is $0$-pseudo-coherent relative to $R$ if and only if $M$ is a finite type $A$-module, \item $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if $M$ is a finitely presented relative to $R$, \item $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if for every surjection $R[x_1, \ldots, x_n] \to A$ there exists a resolution $$ R[x_1, \ldots, x_n]^{\oplus a_d} \to R[x_1, \ldots, x_n]^{\oplus a_{d - 1}} \to \ldots \to R[x_1, \ldots, x_n]^{\oplus a_0} \to M \to 0 $$ of length $d$, and \item $M$ is pseudo-coherent relative to $R$ if and only if for every presentation $R[x_1, \ldots, x_n] \to A$ there exists an infinite resolution $$ \ldots \to R[x_1, \ldots, x_n]^{\oplus a_1} \to R[x_1, \ldots, x_n]^{\oplus a_0} \to M \to 0 $$ by finite free $R[x_1, \ldots, x_n]$-modules. \end{enumerate}


more-algebra-lemma-summands-relative-pseudo-coherent

Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet, L^\bullet \in D(A)$. If $K^\bullet \oplus L^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$ so are $K^\bullet$ and $L^\bullet$.


more-algebra-lemma-localize-relative-pseudo-coherent

Let $R$ be a ring, $f \in R$ an element, $R_f \to A$ is a finite type ring map, $g \in A$, and $K^\bullet$ a complex of $A$-modules. If $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R_f$, then $K^\bullet \otimes_A A_g$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$.


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

Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $A$-modules which is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$. Let $R \to R'$ be a ring map such that $A$ and $R'$ are Tor independent over $R$. Set $A' = A \otimes_R R'$. Then $K^\bullet \otimes_A^{\mathbf{L}} A'$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R'$.


more-algebra-lemma-pull-relative-pseudo-coherent

Let $R \to A \to B$ be finite type ring maps. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $A$-modules. Assume $B$ as a $B$-module is pseudo-coherent relative to $A$. If $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$, then $K^\bullet \otimes_A^{\mathbf{L}} B$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$.


more-algebra-lemma-composition-relative-pseudo-coherent

Let $R$ be a ring. Let $A \to B$ be a map of finite type $R$-algebras. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $B$-modules. Assume $A$ is pseudo-coherent relative to $R$. Then the following are equivalent \begin{enumerate} \item $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $A$, and \item $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$. \end{enumerate}


more-algebra-lemma-glue-relative-pseudo-coherent

Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. Let $f_1, \ldots, f_r \in A$ generate the unit ideal. The following are equivalent \begin{enumerate} \item each $K^\bullet \otimes_A A_{f_i}$ is $m$-pseudo-coherent relative to $R$, and \item $K^\bullet$ is $m$-pseudo-coherent relative to $R$. \end{enumerate} The same equivalence holds for pseudo-coherence relative to $R$.


more-algebra-lemma-Noetherian-relative-pseudo-coherent

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


more-algebra-lemma-perfect-ring-map

A ring map $A \to B$ is perfect if and only if $B = A[x_1, \ldots, x_n]/I$ and $B$ as an $A[x_1, \ldots, x_n]$-module has a finite resolution by finite projective $A[x_1, \ldots, x_n]$-modules.


more-algebra-lemma-Noetherian-pseudo-coherent-ring-map

A finite type ring map of Noetherian rings is pseudo-coherent.


more-algebra-lemma-flat-finite-presentation-perfect

A ring map which is flat and of finite presentation is perfect.


more-algebra-lemma-regular-perfect-ring-map

Let $A \to B$ be a finite type ring map with $A$ a regular ring of finite dimension. Then $A \to B$ is perfect.


more-algebra-lemma-lci-perfect

A local complete intersection homomorphism is perfect.


more-algebra-lemma-relative-pseudo-coherent-is-moot

Let $R \to A$ be a pseudo-coherent ring map. Let $K \in D(A)$. The following are equivalent \begin{enumerate} \item $K$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$, and \item $K$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) in $D(A)$. \end{enumerate}


more-algebra-lemma-more-relative-pseudo-coherent-is-moot

Let $R \to B \to A$ be ring maps with $\varphi : B \to A$ surjective and $R \to B$ and $R \to A$ flat and of finite presentation. For $K \in D(A)$ denote $\varphi_*K \in D(B)$ the restriction. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, \item $K$ is pseudo-coherent relative to $R$, \item $K$ is pseudo-coherent relative to $A$, \item $\varphi_*K$ is pseudo-coherent, \item $\varphi_*K$ is pseudo-coherent relative to $R$. \end{enumerate} Similar holds for $m$-pseudo-coherence.


more-algebra-lemma-cone-relatively-perfect

Let $R \to A$ be a flat ring map of finite presentation. The $R$-perfect objects of $D(A)$ form a saturated\footnote{Derived Categories, Definition \ref{derived-definition-saturated}.} triangulated strictly full subcategory.


more-algebra-lemma-perfect-relatively-perfect

Let $R \to A$ be a flat ring map of finite presentation. A perfect object of $D(A)$ is $R$-perfect. If $K, M \in D(A)$ then $K \otimes_A^\mathbf{L} M$ is $R$-perfect if $K$ is perfect and $M$ is $R$-perfect.


more-algebra-lemma-structure-relatively-perfect

Let $R \to A$ be a flat ring map of finite presentation. Let $K \in D(A)$. The following are equivalent \begin{enumerate} \item $K$ is $R$-perfect, and \item $K$ is isomorphic to a finite complex of $R$-flat, finitely presented $A$-modules. \end{enumerate}


more-algebra-lemma-base-change-relatively-perfect

Let $R \to A$ be a flat ring map of finite presentation. Let $R \to R'$ be a ring map and set $A' = A \otimes_R R'$. If $K \in D(A)$ is $R$-perfect, then $K \otimes_A^\mathbf{L} A'$ is $R'$-perfect.


more-algebra-lemma-compute-RHom-relatively-perfect

Let $R \to A$ be a flat ring map. Let $K, L \in D(A)$ with $K$ pseudo-coherent and $L$ finite tor dimension over $R$. We may choose \begin{enumerate} \item a bounded above complex $P^\bullet$ of finite free $A$-modules representing $K$, and \item a bounded complex of $R$-flat $A$-modules $F^\bullet$ representing $L$. \end{enumerate} Given these choices we have \begin{enumerate} \item[(a)] $E^\bullet = \Hom^\bullet(P^\bullet, F^\bullet)$ is a bounded below complex of $R$-flat $A$-modules representing $R\Hom_A(K, L)$, \item[(b)] for any ring map $R \to R'$ with $A' = A \otimes_R R'$ the complex $E^\bullet \otimes_R R'$ represents $R\Hom_{A'}(K \otimes_A^\mathbf{L} A', L \otimes_A^\mathbf{L} A')$. \end{enumerate} If in addition $R \to A$ is of finite presentation and $L$ is $R$-perfect, then we may choose $F^p$ to be finitely presented $A$-modules and consequently $E^n$ will be finitely presented $A$-modules as well.


more-algebra-lemma-colimit-relatively-perfect

Let $R = \colim_{i \in I} R_i$ be a filtered colimit of rings. Let $0 \in I$ and $R_0 \to A_0$ be a flat ring map of finite presentation. For $i \geq 0$ set $A_i = R_i \otimes_{R_0} A_0$ and set $A = R \otimes_{R_0} A_0$. \begin{enumerate} \item Given an $R$-perfect $K$ in $D(A)$ there exists an $i \in I$ and an $R_i$-perfect $K_i$ in $D(A_i)$ such that $K \cong K_i \otimes_{A_i}^\mathbf{L} A$ in $D(A)$. \item Given $K_0, L_0 \in D(A_0)$ with $K_0$ pseudo-coherent and $L_0$ finite tor dimension over $R_0$, then we have $$ \Hom_{D(A)}(K_0 \otimes_{A_0}^\mathbf{L} A, L_0 \otimes_{A_0}^\mathbf{L} A) = \colim_{i \geq 0} \Hom_{D(A_i)}(K_0 \otimes_{A_0}^\mathbf{L} A_i, L_0 \otimes_{A_0}^\mathbf{L} A_i) $$ \end{enumerate} In particular, the triangulated category of $R$-perfect complexes over $A$ is the colimit of the triangulated categories of $R_i$-perfect complexes over $A_i$.


more-algebra-lemma-thickening-relatively-perfect

Let $R' \to A'$ be a flat ring map of finite presentation. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Set $A = A' \otimes_{R'} R$. Let $K' \in D(A')$ and set $K = K' \otimes_{A'}^\mathbf{L} A$ in $D(A)$. If $K$ is $R$-perfect, then $K'$ is $R'$-perfect.


more-algebra-lemma-lift-from-fibre-relatively-perfect

Let $R$ be a ring. Let $A = R[x_1, \ldots, x_d]/I$ be flat and of finite presentation over $R$. Let $\mathfrak q \subset A$ be a prime ideal lying over $\mathfrak p \subset R$. Let $K \in D(A)$ be pseudo-coherent. Let $a, b \in \mathbf{Z}$. If $H^i(K_\mathfrak q \otimes_{R_\mathfrak p}^\mathbf{L} \kappa(\mathfrak p))$ is nonzero only for $i \in [a, b]$, then $K_\mathfrak q$ has tor amplitude in $[a - d, b]$ over $R$.


more-algebra-lemma-bounded-on-fibres-relatively-perfect

Let $R \to A$ be a ring map which is flat and of finite presentation. Let $K \in D(A)$ be pseudo-coherent. The following are equivalent \begin{enumerate} \item $K$ is $R$-perfect, and \item $K$ is bounded below and for every prime ideal $\mathfrak p \subset R$ the object $K \otimes_R^\mathbf{L} \kappa(\mathfrak p)$ is bounded below. \end{enumerate}


more-algebra-lemma-ext-1-zero

Let $R$ be a ring. Let $K \in D(R)$ with $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. The following are equivalent \begin{enumerate} \item $H^{-1}(K) = 0$ and $H^0(K)$ is a projective module and \item $\Ext^1_R(K, M) = 0$ for every $R$-module $M$. \end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \begin{enumerate} \item[(3)] $\Ext^1_R(K, M) = 0$ for every finite $R$-module $M$. \end{enumerate}


more-algebra-lemma-represent-two-term-complex

Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. Then \begin{enumerate} \item $K$ can be represented by a two term complex $K^{-1} \to K^0$ with $K^0$ a free module, and \item if $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then $K$ can be represented by a two term complex $K^{-1} \to K^0$ with $K^0$ a finite free module and $K^{-1}$ finite. \end{enumerate}


more-algebra-lemma-map-out-of-almost-free

Let $R$ be a ring. Let $M^\bullet$ be a complex of modules over $R$ with $M^i = 0$ for $i > 0$ and $M^0$ a projective $R$-module. Let $K^\bullet$ be a second complex. \begin{enumerate} \item Assume $K^i = 0$ for $i \leq -2$. Then $\Hom_{D(R)}(M^\bullet, K^\bullet) = \Hom_{K(R)}(M^\bullet, K^\bullet)$. \item Assume $K^i = 0$ for $i \not \in [-1, 0]$ and $K^0$ a projective $R$-module. Then for a map of complexes $a^\bullet : M^\bullet \to K^\bullet$, the following are equivalent \begin{enumerate} \item $a^\bullet$ induces the zero map $\Ext^1_R(K^\bullet, N) \to \Ext^1_R(M^\bullet, N)$ for all $R$-modules $N$, and \item there is a map $h^0 : M^0 \to K^{-1}$ such that $a^{-1} + h^0 \circ d^{-1}_K = 0$. \end{enumerate} \item Assume $K^i = 0$ for $i \leq -3$. Let $\alpha \in \Hom_{D(R)}(M^\bullet, K^\bullet)$. If the composition of $\alpha$ with $K^\bullet \to K^{-2}[2]$ comes from an $R$-module map $a : M^{-2} \to K^{-2}$ with $a \circ d_M^{-3} = 0$, then $\alpha$ can be represented by a map of complexes $a^\bullet : M^\bullet \to K^\bullet$ with $a^{-2} = a$. \item In (2) for any second map of complexes $(a')^\bullet : M^\bullet \to K^\bullet$ representing $\alpha$ with $a = (a')^{-2}$ there exist $h^i : M^i \to K^{i - 1}$ for $i = 0, -1$ such that $$ h^{-1} \circ d_M^{-2} = 0, \quad (a')^{-1} = a^{-1} + d_K^{-2} \circ h^{-1} + h^0 \circ d_M^{-1},\quad (a')^0 = a^0 + d_K^{-1} \circ h^0 $$ \end{enumerate}


more-algebra-lemma-ext-1-annihilated-definite

Let $R$ be a ring and let $I \subset R$ be an ideal. Let $K \in D(R)$. Assume $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. The following are equivalent \begin{enumerate} \item $\Ext^1_R(K, N)$ is annihilated by $I$ for all $R$-modules $N$, \item $K$ can be represented by a complex $K^{-1} \to K^0$ with $K^0$ free such that for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_K^{-1} : K^{-1} \to K^0$, \item whenever $K$ is represented by a two term complex $K^{-1} \to K^0$ with $K^0$ projective, then for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_K^{-1} : K^{-1} \to K^0$. \end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \begin{enumerate} \item[(4)] $\Ext^1_R(K, N)$ is annihilated by $I$ for every finite $R$-module $N$, \item[(5)] $K$ can be represented by a complex $K^{-1} \to K^0$ with $K^0$ finite free and $K^{-1}$ finite such that for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_K^{-1} : K^{-1} \to K^0$. \end{enumerate}


more-algebra-lemma-two-term-base-change

Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. Let $K^{-1} \to K^0$ be a two term complex of $R$-modules representing $K$ such that $K^0$ is a flat $R$-module (for example projective or free). Let $R \to R'$ be a ring map. Then the complex $K^\bullet \otimes_R R'$ represents $\tau_{\geq -1}(K \otimes_R^\mathbf{L} R')$.


more-algebra-lemma-base-change-property-ext-1-annihilated

Let $I$ be an ideal of a ring $R$. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. Let $R \to R'$ be a ring map. If $K$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-ext-1-annihilated-definite} with respect to $(R, I)$, then $\tau_{\geq -1}(K \otimes_R^\mathbf{L} R')$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-ext-1-annihilated-definite} with respect to $(R', IR')$


more-algebra-lemma-two-term-surjection-map-zero

Let $R$ be a ring. Let $\alpha : K \to K'$ be a morphism of $D(R)$. Assume \begin{enumerate} \item $H^i(K) = H^i(K') = 0$ for $i \not \in \{-1, 0\}$ \item $H^0(\alpha)$ is an isomorphism and $H^{-1}(\alpha)$ is surjective. \end{enumerate} For any $f \in R$ if $f : K \to K$ is $0$, then $f : K' \to K'$ is $0$.


more-algebra-lemma-surjection-property-ext-1-annihilated

Let $I$ be an ideal of a ring $R$. Let $\alpha : K \to K'$ be a morphism of $D(R)$. Assume \begin{enumerate} \item $H^i(K) = H^i(K') = 0$ for $i \not \in \{-1, 0\}$ \item $H^0(\alpha)$ is an isomorphism and $H^{-1}(\alpha)$ is surjective. \end{enumerate} If $K$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-ext-1-annihilated-definite}, then $K'$ does too.


more-algebra-lemma-ext-1-annihilated

Let $R$ be ring and let $I \subset R$ be an ideal. Let $K \in D(R)$ with $H^i(K) = 0$ for $i \not \in \{-1, 0\}$. The following are equivalent \begin{enumerate} \item there exists a $c \geq 0$ such that the equivalent conditions (1), (2), (3) of Lemma \ref{lemma-ext-1-annihilated-definite} hold for $K$ and the ideal $I^c$, \item there exists a $c \geq 0$ such that (a) $I^c$ annihilates $H^{-1}(K)$ and (b) $H^0(K)$ is an $I^c$-projective module (see Section \ref{section-near-projective}). \end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \begin{enumerate} \item[(3)] there exists a $c \geq 0$ such that the equivalent conditions (4), (5) of Lemma \ref{lemma-ext-1-annihilated-definite} hold for $K$ and the ideal $I^c$, \item[(4)] $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \ldots, f_s \in R$ with $V(f_1, \ldots, f_s) \subset V(I)$ such that the localizations $H^0(K)_{f_i}$ are projective $R_{f_i}$-modules, \item[(5)] $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \ldots, f_s \in I$ with $V(f_1, \ldots, f_s) = V(I)$ such that the localizations $H^0(K)_{f_i}$ are projective $R_{f_i}$-modules. \end{enumerate}


more-algebra-lemma-zero-in-derived

Let $R$ be a ring. Let $K_j \in D(R)$, $j = 1, 2, 3$ with $H^i(K_j) = 0$ for $i \not \in \{-1, 0\}$. Let $\varphi : K_1 \to K_2$ and $\psi : K_2 \to K_3$ be maps in $D(R)$. If $H^0(\varphi) = 0$ and $H^{-1}(\psi) = 0$, then $\varphi \circ \psi = 0$.


more-algebra-lemma-silly

Let $R$ be a ring. Let $K \in D(R)$ be given by a two term complex of the form $R^{\oplus n} \to R^{\oplus n}$. Denote $A \in \text{Mat}(n \times n, R)$ the matrix of the differential. Then $\det(a) : K \to K$ is zero in $D(R)$.


more-algebra-lemma-tensor-NL

Let $R \to S$ and $S \to S'$ be ring maps. The canonical map $\NL_{S/R} \otimes_S^\mathbf{L} S' \to \NL_{S/R} \otimes_S S'$ induces an isomorphism $\tau_{\geq -1}(\NL_{S/R} \otimes_S^\mathbf{L} S') \to \NL_{S/R} \otimes_S S'$ in $D(S')$. Similarly, given a presentation $\alpha$ of $S$ over $R$ the canonical map $\NL(\alpha) \otimes_S^\mathbf{L} S' \to \NL(\alpha) \otimes_S S'$ induces an isomorphism $\tau_{\geq -1}(\NL(\alpha) \otimes_S^\mathbf{L} S') \to \NL(\alpha) \otimes_S S'$ in $D(S')$.


more-algebra-lemma-base-change-NL

Let $R \to S$ and $R \to R'$ be ring maps. Let $\alpha : P \to S$ be a presentation of $S$ over $R$. Then $\alpha' : P \otimes_R R' \to S \otimes_R R'$ is a presentation of $S' = S \otimes_R R'$ over $R'$. The canonical map $$ NL(\alpha) \otimes_S S' \to \NL(\alpha') $$ is an isomorphism on $H^0$ and surjective on $H^{-1}$. In particular, the canonical map $$ \NL_{S/R} \otimes_S S' \to \NL_{S'/R'} $$ is an isomorphism on $H^0$ and surjective on $H^{-1}$.


more-algebra-lemma-base-change-NL-flat

Consider a cocartesian diagram of rings $$ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u] & A' \ar[u] } $$ If $B$ is flat over $A$, then the canonical map $\NL_{B/A} \otimes_B B' \to \NL_{B'/A'}$ is a quasi-isomorphism. If in addition $\NL_{B/A}$ has tor-amplitude in $[-1, 0]$ then $\NL_{B/A} \otimes_B^\mathbf{L} B' \to \NL_{B'/A'}$ is a quasi-isomorphism too.


more-algebra-lemma-lci-NL

Let $A \to B$ be a local complete intersection as in Definition \ref{definition-local-complete-intersection}. Then $\NL_{B/A}$ is a perfect object of $D(B)$ with tor amplitude in $[-1, 0]$.


more-algebra-lemma-base-change-lci-bis

Consider a cocartesian diagram of rings $$ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u] & A' \ar[u] } $$ If $A \to B$ and $A' \to B'$ are local complete intersections as in Definition \ref{definition-local-complete-intersection}, then the kernel of $H^{-1}(\NL_{B/A} \otimes_B B') \to H^{-1}(\NL_{B'/A'})$ is a finite projective $B'$-module.


more-algebra-lemma-compute-Rlim

The functor $\lim : \textit{Ab}(\mathbf{N}) \to \textit{Ab}$ has a right derived functor \begin{equation} \label{equation-Rlim} R\lim : D(\textit{Ab}(\mathbf{N})) \longrightarrow D(\textit{Ab}) \end{equation} As usual we set $R^p\lim(K) = H^p(R\lim(K))$. Moreover, we have \begin{enumerate} \item for any $(A_n)$ in $\textit{Ab}(\mathbf{N})$ we have $R^p\lim A_n = 0$ for $p > 1$, \item the object $R\lim A_n$ of $D(\textit{Ab})$ is represented by the complex $$ \prod A_n \to \prod A_n,\quad (x_n) \mapsto (x_n - f_{n + 1}(x_{n + 1})) $$ sitting in degrees $0$ and $1$, \item if $(A_n)$ is ML, then $R^1\lim A_n = 0$, i.e., $(A_n)$ is right acyclic for $\lim$, \item every $K^\bullet \in D(\textit{Ab}(\mathbf{N}))$ is quasi-isomorphic to a complex whose terms are right acyclic for $\lim$, and \item if each $K^p = (K^p_n)$ is right acyclic for $\lim$, i.e., of $R^1\lim_n K^p_n = 0$, then $R\lim K$ is represented by the complex whose term in degree $p$ is $\lim_n K_n^p$. \end{enumerate}


more-algebra-lemma-apply-Mittag-Leffler-again

Let $$ (A^{-2}_n \to A^{-1}_n \to A^0_n \to A^1_n) $$ be an inverse system of complexes of abelian groups and denote $A^{-2} \to A^{-1} \to A^0 \to A^1$ its limit. Denote $(H_n^{-1})$, $(H_n^0)$ the inverse systems of cohomologies, and denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \to A^{-1} \to A^0 \to A^1$. If \begin{enumerate} \item $(A^{-2}_n)$ and $(A^{-1}_n)$ have vanishing $R^1\lim$, \item $(H^{-1}_n)$ has vanishing $R^1\lim$, \end{enumerate} then $H^0 = \lim H_n^0$.


more-algebra-lemma-distinguished-triangle-Rlim

Let $K = (K_n^\bullet)$ be an object of $D(\textit{Ab}(\mathbf{N}))$. There exists a canonical distinguished triangle $$ R\lim K \to \prod\nolimits_n K_n^\bullet \to \prod\nolimits_n K_n^\bullet \to R\lim K[1] $$ in $D(\textit{Ab})$. In other words, $R\lim K$ is a derived limit of the inverse system $(K_n^\bullet)$ of $D(\textit{Ab})$, see Derived Categories, Definition \ref{derived-definition-derived-limit}.


more-algebra-lemma-break-long-exact-sequence

With notation as in Lemma \ref{lemma-distinguished-triangle-Rlim} the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences $$ 0 \to R^1\lim_n H^{p - 1}(K_n^\bullet) \to H^p(R\lim K) \to \lim_n H^p(K_n^\bullet) \to 0 $$


more-algebra-lemma-lift-to-system-complexes-Ab

Let $(K_n)$ be an inverse system of objects of $D(\textit{Ab})$. Then there exists an object $M = (M_n^\bullet)$ of $D(\textit{Ab}(\mathbf{N}))$ and isomorphisms $M_n^\bullet \to K_n$ in $D(\textit{Ab})$ such that the diagrams $$ \xymatrix{ M_{n + 1}^\bullet \ar[d] \ar[r] & M_n^\bullet \ar[d] \\ K_{n + 1} \ar[r] & K_n } $$ commute in $D(\textit{Ab})$.


more-algebra-lemma-Rlim-pro-equal

Let $E \to D$ be a morphism of $D(\textit{Ab}(\mathbf{N}))$. Let $(E_n)$, resp.\ $(D_n)$ be the system of objects of $D(\textit{Ab})$ associated to $E$, resp.\ $D$. If $(E_n) \to (D_n)$ is an isomorphism of pro-objects, then $R\lim E \to R\lim D$ is an isomorphism in $D(\textit{Ab})$.


more-algebra-lemma-emmanouil

\begin{reference} Taken from \cite{Emmanouil}. \end{reference} Let $(A_n)$ be an inverse system of abelian groups. The following are equivalent \begin{enumerate} \item $(A_n)$ is Mittag-Leffler, \item $R^1\lim A_n = 0$ and the same holds for $\bigoplus_{i \in \mathbf{N}} (A_n)$. \end{enumerate}


more-algebra-lemma-Rlim-zero-of-direct-sums

Let $(A_n)$ be an inverse system of abelian groups. The following are equivalent \begin{enumerate} \item $(A_n)$ is zero as a pro-object, \item $\lim A_n = 0$ and $R^1\lim A_n = 0$ and the same holds for $\bigoplus_{i \in \mathbf{N}} (A_n)$. \end{enumerate}


more-algebra-lemma-compute-Rlim-modules

In the situation above. The functor $\lim : \textit{Mod}(\mathbf{N}, (A_n)) \to \text{Mod}_A$ has a right derived functor $$ R\lim : D(\textit{Mod}(\mathbf{N}, (A_n))) \longrightarrow D(A) $$ As usual we set $R^p\lim(K) = H^p(R\lim(K))$. Moreover, we have \begin{enumerate} \item for any $(M_n)$ in $\textit{Mod}(\mathbf{N}, (A_n))$ we have $R^p\lim M_n = 0$ for $p > 1$, \item the object $R\lim M_n$ of $D(\text{Mod}_A)$ is represented by the complex $$ \prod M_n \to \prod M_n,\quad (x_n) \mapsto (x_n - f_{n + 1}(x_{n + 1})) $$ sitting in degrees $0$ and $1$, \item if $(M_n)$ is ML, then $R^1\lim M_n = 0$, i.e., $(M_n)$ is right acyclic for $\lim$, \item every $K^\bullet \in D(\textit{Mod}(\mathbf{N}, (A_n)))$ is quasi-isomorphic to a complex whose terms are right acyclic for $\lim$, and \item if each $K^p = (K^p_n)$ is right acyclic for $\lim$, i.e., of $R^1\lim_n K^p_n = 0$, then $R\lim K$ is represented by the complex whose term in degree $p$ is $\lim_n K_n^p$. \end{enumerate}


more-algebra-lemma-distinguished-triangle-Rlim-modules

Let $K = (K_n^\bullet)$ be an object of $D(\textit{Mod}(\mathbf{N}, (A_n)))$. There exists a canonical distinguished triangle $$ R\lim K \to \prod\nolimits_n K_n^\bullet \to \prod\nolimits_n K_n^\bullet \to R\lim K[1] $$ in $D(A)$. In other words, $R\lim K$ is a derived limit of the inverse system $(K_n^\bullet)$ of $D(A)$, see Derived Categories, Definition \ref{derived-definition-derived-limit}.


more-algebra-lemma-break-long-exact-sequence-modules

With notation as in Lemma \ref{lemma-distinguished-triangle-Rlim-modules} the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences $$ 0 \to R^1\lim_n H^{p - 1}(K_n^\bullet) \to H^p(R\lim K) \to \lim_n H^p(K_n^\bullet) \to 0 $$ of $A$-modules.


more-algebra-lemma-lift-to-system-complexes

Let $(A_n)$ be an inverse system of rings. Suppose that we are given \begin{enumerate} \item for every $n$ an object $K_n$ of $D(A_n)$, and \item for every $n$ a map $\varphi_n : K_{n + 1} \to K_n$ of $D(A_{n + 1})$ where we think of $K_n$ as an object of $D(A_{n + 1})$ by restriction via $A_{n + 1} \to A_n$. \end{enumerate} There exists an object $M = (M_n^\bullet) \in D(\textit{Mod}(\mathbf{N}, (A_n)))$ and isomorphisms $\psi_n : M_n^\bullet \to K_n$ in $D(A_n)$ such that the diagrams $$ \xymatrix{ M_{n + 1}^\bullet \ar[d]_{\psi_{n + 1}} \ar[r] & M_n^\bullet \ar[d]^{\psi_n} \\ K_{n + 1} \ar[r]^{\varphi_n} & K_n } $$ commute in $D(A_{n + 1})$.


more-algebra-lemma-get-ML-system

Let $(A_n)$ be an inverse system of rings. Every $K \in D(\textit{Mod}(\mathbf{N}, (A_n)))$ can be represented by a system of complexes $(M_n^\bullet)$ such that all the transition maps $M_{n + 1}^\bullet \to M_n^\bullet$ are surjective.


more-algebra-lemma-get-K-flat-system

Let $(A_n)$ be an inverse system of rings. Every $K \in D(\textit{Mod}(\mathbf{N}, (A_n)))$ can be represented by a system of complexes $(K_n^\bullet)$ such that each $K_n^\bullet$ is K-flat.


more-algebra-lemma-derived-tensor-product-systems

Let $(A_n)$ be an inverse system of rings. Given $K, L \in D(\textit{Mod}(\mathbf{N}, (A_n)))$ there is a canonical derived tensor product $K \otimes^\mathbf{L} L$ in $D(\mathbf{N}, (A_n))$ compatible with the maps to $D(A_n)$. The construction is symmetric in $K$ and $L$ and an exact functor of triangulated categories in each variable.


more-algebra-lemma-tensor-Rlim-exact

Let $A$ be a ring. Let $E \to D \to F \to E[1]$ be a distinguished triangle of $D(\mathbf{N}, A)$. Let $(E_n)$, resp.\ $(D_n)$, resp.\ $(F_n)$ be the system of objects of $D(A)$ associated to $E$, resp.\ $D$, resp.\ $F$. Then for every $K \in D(A)$ there is a canonical distinguished triangle $$ R\lim (K \otimes^\mathbf{L}_A E_n) \to R\lim (K \otimes^\mathbf{L}_A D_n) \to R\lim (K \otimes^\mathbf{L}_A F_n) \to R\lim (K \otimes^\mathbf{L}_A E_n)[1] $$ in $D(A)$ with notation as in Remark \ref{remark-constructing-tensor-with-limits-functorially}.


more-algebra-lemma-tensor-Rlim-pro-equal

Let $A$ be a ring. Let $E \to D$ be a morphism of $D(\mathbf{N}, A)$. Let $(E_n)$, resp.\ $(D_n)$ be the system of objects of $D(A)$ associated to $E$, resp.\ $D$. If $(E_n) \to (D_n)$ is an isomorphism of pro-objects, then for every $K \in D(A)$ the corresponding map $$ R\lim (K \otimes^\mathbf{L}_A E_n) \longrightarrow R\lim (K \otimes^\mathbf{L}_A D_n) $$ in $D(A)$ is an isomorphism (notation as in Remark \ref{remark-constructing-tensor-with-limits-functorially}).


more-algebra-lemma-I-power-torsion-presentation

Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $I$-power torsion module. Then $M$ admits a resolution $$ \ldots \to K_2 \to K_1 \to K_0 \to M \to 0 $$ with each $K_i$ a direct sum of copies of $R/I^n$ for $n$ variable.


more-algebra-lemma-torsion-free

Let $R$ be a ring. Let $I$ be an ideal of $R$. For any $R$-module $M$ set $M[I^n] = \{m \in M \mid I^nm = 0\}$. If $I$ is finitely generated then the following are equivalent \begin{enumerate} \item $M[I] = 0$, \item $M[I^n] = 0$ for all $n \geq 1$, and \item if $I = (f_1, \ldots, f_t)$, then the map $M \to \bigoplus M_{f_i}$ is injective. \end{enumerate}


more-algebra-lemma-divide-by-torsion

Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$. \begin{enumerate} \item For any $R$-module $M$ we have $(M/M[I^\infty])[I] = 0$. \item An extension of $I$-power torsion modules is $I$-power torsion. \end{enumerate}


more-algebra-lemma-I-power-torsion

Let $I$ be a finitely generated ideal of a ring $R$. The $I$-power torsion modules form a Serre subcategory of the abelian category $\text{Mod}_R$, see Homology, Definition \ref{homology-definition-serre-subcategory}.


more-algebra-lemma-local-cohomology-closed

Let $R$ be a ring and let $I \subset R$ be a finitely generated ideal. The subcategory $I^\infty\text{-torsion} \subset \text{Mod}_R$ depends only on the closed subset $Z = V(I) \subset \Spec(R)$. In fact, an $R$-module $M$ is $I$-power torsion if and only if its support is contained in $Z$.


more-algebra-lemma-derived-vanishing-mod-I

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$ such hat $K \otimes_R^\mathbf{L} R/I = 0$ in $D(R)$. Then \begin{enumerate} \item $K \otimes_R^\mathbf{L} R/I^n = 0$ for all $n \geq 1$, \item $K \otimes_R^\mathbf{L} N = 0$ for any $I$-power torsion $R$-module $N$, \item $K \otimes_R^\mathbf{L} M = 0$ for any $M \in D^b(R)$ whose cohomology modules are $I$-power torsion. \end{enumerate}


more-algebra-lemma-characterize-flatness-on-torsion

Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ be an ideal. The following are equivalent \begin{enumerate} \item $\varphi$ is flat and $R/I \to S/IS$ is faithfully flat, \item $\varphi$ is flat, and the map $\Spec(S/IS) \to \Spec(R/I)$ is surjective. \item $\varphi$ is flat, and the base change functor $M \mapsto M \otimes_R S$ is faithful on modules annihilated by $I$, and \item $\varphi$ is flat, and the base change functor $M \mapsto M \otimes_R S$ is faithful on $I$-power torsion modules. \end{enumerate}


more-algebra-lemma-neighbourhood-isomorphism

Assume $(\varphi : R \to S, I)$ satisfies the equivalent conditions of Lemma \ref{lemma-characterize-flatness-on-torsion}. The following are equivalent \begin{enumerate} \item for any $I$-power torsion module $M$, the natural map $M \to M \otimes_R S$ is an isomorphism, and \item $R/I \to S/IS$ is an isomorphism. \end{enumerate}


more-algebra-lemma-neighbourhood-equivalence

Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. Then \begin{enumerate} \item for any $R$-module $M$ the map $M \to M \otimes_R S$ induces an isomorphism $M[I^\infty] \to (M \otimes_R S)[(IS)^\infty]$ of $I$-power torsion submodules, \item the natural map $$ \Hom_R(M, N) \longrightarrow \Hom_S(M \otimes_R S, N \otimes_R S) $$ is an isomorphism if either $M$ or $N$ is $I$-power torsion, and \item the base change functor $M \mapsto M \otimes_R S$ defines an equivalence of categories between $I$-power torsion modules and $IS$-power torsion modules. \end{enumerate}


more-algebra-lemma-map-identifies-koszul-and-cech-complexes

Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. For any $f_1, \ldots, f_r \in R$ such that $V(f_1, \ldots, f_r) = V(I)$ \begin{enumerate} \item the map of Koszul complexes $K(R, f_1, \ldots, f_r) \to K(S, f_1, \ldots, f_r)$ is a quasi-isomorphism, and \item The map of extended alternating {\v C}ech complexes $$ \xymatrix{ R \to \prod_{i_0} R_{f_{i_0}} \to \prod_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_r} \ar[d] \\ S \to \prod_{i_0} S_{f_{i_0}} \to \prod_{i_0 < i_1} S_{f_{i_0}f_{i_1}} \to \ldots \to S_{f_1\ldots f_r} } $$ is a quasi-isomorphism. \end{enumerate}


more-algebra-lemma-naive-Koszul-complex

Let $R$ be a ring. Let $I = (f_1, \ldots, f_n)$ be a finitely generated ideal of $R$. Let $M$ be the $R$-module generated by elements $e_1, \ldots, e_n$ subject to the relations $f_i e_j - f_j e_i = 0$. There exists a short exact sequence $$ 0 \to K \to M \to I \to 0 $$ such that $K$ is annihilated by $I$.


more-algebra-lemma-explicit-ext

Let $R$ be a ring. Let $I = (f_1, \ldots, f_n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ set $$ H_1(N, f_\bullet) = \frac{\{(x_1, \ldots, x_n) \in N^{\oplus n} \mid f_i x_j = f_j x_i \}} {\{f_1x, \ldots, f_nx) \mid x \in N\}} $$ For any $R$-module $N$ there exists a canonical short exact sequence $$ 0 \to \Ext_R(R/I, N) \to H_1(N, f_\bullet) \to \Hom_R(K, N) $$ where $K$ is as in Lemma \ref{lemma-naive-Koszul-complex}.


more-algebra-lemma-koszul-homology-annihilated

Let $R$ be a ring. Let $I = (f_1, \ldots, f_n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ the Koszul homology group $H_1(N, f_\bullet)$ defined in Lemma \ref{lemma-explicit-ext} is annihilated by $I$.


more-algebra-lemma-neighbourhood-extensions

Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. Let $M$, $N$ be $R$-modules. Assume $M$ is $I$-power torsion. Given an short exact sequence $$ 0 \to N \otimes_R S \to \tilde E \to M \otimes_R S \to 0 $$ there exists a commutative diagram $$ \xymatrix{ 0 \ar[r] & N \ar[r] \ar[d] & E \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\ 0 \ar[r] & N \otimes_R S \ar[r] & \tilde E \ar[r] & M \otimes_R S \ar[r] & 0 } $$ with exact rows.


more-algebra-lemma-recover-module-from-glueing-data

Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots, f_t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Let $M$ be an $R$-module. Then the complex (\ref{equation-glueing-complex}) is exact.


more-algebra-lemma-H0-inverse

Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots, f_t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Then the functor $H^0$ is a left quasi-inverse to the functor $\text{Can}$ of Remark \ref{remark-glueing-data}.


more-algebra-lemma-exact

Assume $\varphi : R \to S$ is a flat ring map and let $I = (f_1, \ldots, f_t) \subset R$ be an ideal. Then $\text{Glue}(R \to S, f_1, \ldots, f_t)$ is an abelian category, and the functor $\text{Can}$ is exact and commutes with arbitrary colimits.


more-algebra-lemma-equivalence-I-unit

Let $\varphi : R \to S$ be a flat ring map and $(f_1, \ldots, f_t) = R$. Then $\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories $$ \text{Mod}_R = \text{Glue}(R \to S, f_1, \ldots, f_t) $$


more-algebra-lemma-base-change-glue

Let $\varphi : R \to S$ be a flat ring map and $I = (f_1, \ldots, f_t)$ and ideal. Let $R \to R'$ be a flat ring map, and set $S' = S \otimes_R R'$. Then we obtain a commutative diagram of categories and functors $$ \xymatrix{ \text{Mod}_R \ar[r]_-{\text{Can}} \ar[d]_{-\otimes_R R'} & \text{Glue}(R \to S, f_1, \ldots, f_t) \ar[r]_-{H^0} \ar[d]^{-\otimes_R R'} & \text{Mod}_R \ar[d]^{-\otimes_R R'} \\ \text{Mod}_{R'} \ar[r]^-{\text{Can}} & \text{Glue}(R' \to S', f_1, \ldots, f_t) \ar[r]^-{H^0} & \text{Mod}_{R'} } $$


more-algebra-lemma-application-formal-glueing

Let $\varphi : R \to S$ be a flat ring map and let $I \subset R$ be a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism.  \begin{enumerate} \item Given an $R$-module $N$, an $S$-module $M'$ and an $S$-module map $\varphi : M' \to N \otimes_R S$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\psi : M \to N$ and an isomorphism $M \otimes_R S = M'$ compatible with $\varphi$ and $\psi$. \item Given an $R$-module $M$, an $S$-module $N'$ and an $S$-module map $\varphi : M \otimes_R S \to N'$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\psi : M \to N$ and an isomorphism $N \otimes_R S = N'$ compatible with $\varphi$ and $\psi$. \end{enumerate} In both cases we have $\Ker(\varphi) \cong \Ker(\psi)$ and $\Coker(\varphi) \cong \Coker(\psi)$.


more-algebra-lemma-same-quotients

Let $R$ be a ring and let $f \in R$. For every positive integer $n$ the map $R/f^nR \to R^\wedge/f^n R^\wedge$ is an isomorphism.


more-algebra-lemma-torsion-completion

\begin{reference} Slight generalization of \cite[Lemme~1]{Beauville-Laszlo}. \end{reference} Let $R$ be a ring, let $f \in R$ be an element, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^nR \to R'/f^nR'$ for $n > 0$. For any $f$-power torsion $R$-module $M$ the map $M \to M \otimes_R R'$ is an isomorphism. For example, we have $M \cong M \otimes_R R^\wedge$.


more-algebra-lemma-BL-faithful

Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^nR \to R'/f^nR'$ for $n > 0$. The $R$-module $R' \oplus R_f$ is faithful: for every nonzero $R$-module $M$, the module $M \otimes_R (R' \oplus R_f)$ is also nonzero. For example, if $M$ is nonzero, then $M \otimes_R (R^\wedge \oplus R_f)$ is nonzero.


more-algebra-lemma-faithful-descent

\begin{reference} Slight generalization of \cite[Lemme~2(a)]{Beauville-Laszlo}. \end{reference} Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^nR \to R'/f^nR'$ for $n > 0$. An $R$-module $M$ is finitely generated if and only if the ($R' \oplus R_f$)-module $M \otimes_R (R' \oplus R_f)$ is finitely generated. For example, if $M \otimes_R (R^\wedge \oplus R_f)$ is finitely generated as a module over $R^\wedge \oplus R_f$, then $M$ is a finitely generated $R$-module.


more-algebra-lemma-same-f-torsion

Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^nR \to R'/f^nR'$ for $n > 0$. The sequence (\ref{equation-BL-cech-re}) is \begin{enumerate} \item exact on the right, \item exact on the left if and only if $R[f^\infty] \to R'[f^\infty]$ is injective, and \item exact in the middle if and only if $R[f^\infty] \to R'[f^\infty]$ is surjective. \end{enumerate} In particular, $(R \to R', f)$ is a glueing pair if and only if $R[f^\infty] \to R'[f^\infty]$ is bijective. For example, $(R, f)$ is a glueing pair if and only if $R[f^\infty] \to R^\wedge[f^\infty]$ is bijective.


more-algebra-lemma-same-f-torsion-module

Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^nR \to R'/f^nR'$ for $n > 0$. The sequence (\ref{equation-BL-cech-mod-re}) is \begin{enumerate} \item exact on the right, \item exact on the left if and only if $M[f^\infty] \to (M \otimes_R R')[f^\infty]$ is injective, and \item exact in the middle if and only if $M[f^\infty] \to (M \otimes_R R')[f^\infty]$ is surjective. \end{enumerate} Thus $M$ is glueable for $(R \to R', f)$ if and only if $M[f^\infty] \to (M \otimes_R R')[f^\infty]$ is bijective. If $(R \to R', f)$ is a glueing pair, then $M$ is glueable for $(R \to R', f)$ if and only if $M[f^\infty] \to (M \otimes_R R')[f^\infty]$ is injective. For example, if $(R, f)$ is a glueing pair, then $M$ is glueable if and only if $M[f^\infty] \to (M \otimes_R R^\wedge)[f^\infty]$ is injective.


more-algebra-lemma-first-tor

Let $(R \to R', f)$ be a glueing pair. Then $\text{Tor}^R_1(R', f^n R) = 0$ for each $n > 0$.


more-algebra-lemma-first-tor-total

Let $(R \to R',f)$ be a glueing pair. Then $\text{Tor}^R_1(R', R/R[f^\infty]) = 0$.


more-algebra-lemma-BL3

\begin{reference} Slight generalization of \cite[Lemme 3(a)]{Beauville-Laszlo} \end{reference} Let $(R \to R', f)$ be a glueing pair. For every $R$-module $M$, we have $\text{Tor}^R_1(R', \Coker(M \to M_f)) = 0$.


more-algebra-lemma-BL-flat

Let $(R \to R', f)$ be a glueing pair. Let $M$ be an $R$-module which is not necessarily glueable for $(R \to R', f)$. Then $M$ is flat over $R$ if and only if $M \otimes_R R'$ is flat over $R'$ and $M_f$ is flat over $R_f$.


more-algebra-lemma-BL-properties

Let $(R \to R', f)$ be a glueing pair. Let $M$ be an $R$-module which is not necessarily glueable for $(R \to R', f)$. Then $M$ is a finite projective $R$-module if and only if $M \otimes_R R'$ is finite projective over $R'$ and $M_f$ is finite projective over $R_f$.


more-algebra-lemma-hom-from-Af

Let $A$ be a ring. Let $f \in A$. Let $K \in D(A)$. The following are equivalent \begin{enumerate} \item $\Ext^n_A(A_f, K) = 0$ for all $n$, \item $\Hom_{D(A)}(E, K) = 0$ for all $E$ in $D(A_f)$, \item $T(K, f) = 0$, \item for every $p \in \mathbf{Z}$ we have $T(H^p(K), f) = 0$, \item for every $p \in \mathbf{Z}$ we have $\Hom_A(A_f, H^p(K)) = 0$ and $\Ext^1_A(A_f, H^p(K)) = 0$, \item $R\Hom_A(A_f, K) = 0$, \item the map $\prod_{n \geq 0} K \to \prod_{n \geq 0} K$, $(x_0, x_1, \ldots) \mapsto (x_0 - fx_1, x_1 - fx_2, \ldots)$ is an isomorphism in $D(A)$, and \item add more here. \end{enumerate}


more-algebra-lemma-ideal-of-elements-complete-wrt

Let $A$ be a ring. Let $K \in D(A)$. The set $I$ of $f \in A$ such that $T(K, f) = 0$ is a radical ideal of $A$.


more-algebra-lemma-complete-derived-complete

Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $M$ be an $A$-module. \begin{enumerate} \item If $M$ is $I$-adically complete, then $T(M, f) = 0$ for all $f \in I$. \item Conversely, if $T(M, f) = 0$ for all $f \in I$ and $I$ is finitely generated, then $M \to \lim M/I^nM$ is surjective. \end{enumerate}


more-algebra-lemma-serre-subcategory

Let $I$ be an ideal of a ring $A$. \begin{enumerate} \item The derived complete $A$-modules form a weak Serre subcategory $\mathcal{C}$ of $\text{Mod}_A$. \item $D_\mathcal{C}(A) \subset D(A)$ is the full subcategory of derived complete objects. \end{enumerate}


more-algebra-lemma-derived-complete-zero

Let $I$ be a finitely generated ideal of a ring $A$. Let $M$ be a derived complete $A$-module. If $M/IM = 0$, then $M = 0$.


more-algebra-lemma-pseudo-coherent-is-derived-complete

Let $A$ be a ring and $I \subset A$ an ideal. If $A$ is $I$-adically complete then any pseudo-coherent object of $D(A)$ is derived complete.


more-algebra-lemma-double-localize

Let $A$ be a ring. Let $f, g \in A$. Then for $K \in D(A)$ we have $R\Hom_A(A_f, R\Hom_A(A_g, K)) = R\Hom_A(A_{fg}, K)$.


more-algebra-lemma-derived-completion

\begin{slogan} Derived completions along finitely generated ideals exist, and can be computed by a {\v C}ech procedure. \end{slogan} Let $I$ be a finitely generated ideal of a ring $A$. The inclusion functor $D_{comp}(A, I) \to D(A)$ has a left adjoint, i.e., given any object $K$ of $D(A)$ there exists a map $K \to K^\wedge$ of $K$ into a derived complete object of $D(A)$ such that the map $$ \Hom_{D(A)}(K^\wedge, E) \longrightarrow \Hom_{D(A)}(K, E) $$ is bijective whenever $E$ is a derived complete object of $D(A)$. In fact, if $I$ is generated by $f_1, \ldots, f_r \in A$, then we have $$ K^\wedge = R\Hom\left((A \to \prod\nolimits_{i_0} A_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_r}), K\right) $$ functorially in $K$.


more-algebra-lemma-derived-completion-vanishes

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet$ is an isomorphism for some $f \in I$, i.e., $K^\bullet$ is a complex of $A_f$-modules. Then the derived completion of $K^\bullet$ is zero.


more-algebra-lemma-completion-RHom

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K, L \in D(A)$. Then $$ R\Hom_A(K, L)^\wedge = R\Hom_A(K, L^\wedge) = R\Hom_A(K^\wedge, L^\wedge) $$


more-algebra-lemma-naive-derived-completion

Let $A$ be a ring and let $I \subset A$ be an ideal. Let $(K_n)$ be an inverse system of objects of $D(A)$ such that for all $f \in I$ and $n$ there exists an $e = e(n, f)$ such that $f^e$ is zero on $K_n$. Then for $K \in D(A)$ the object $K' = R\lim (K \otimes_A^\mathbf{L} K_n)$ is derived complete with respect to $I$.


more-algebra-lemma-koszul-derived-completion-complete

In Situation \ref{situation-koszul}. For $K \in D(A)$ the object $K' = R\lim (K \otimes_A^\mathbf{L} K_n^\bullet)$ is derived complete with respect to $I$.


more-algebra-lemma-characterize-derived-complete-Koszul

In Situation \ref{situation-koszul}. Let $K \in D(A)$. The following are equivalent \begin{enumerate} \item $K$ is derived complete with respect to $I$, and \item the canonical map $K \to R\lim (K \otimes_A^\mathbf{L} K_n^\bullet)$ is an isomorphism of $D(A)$. \end{enumerate}


more-algebra-lemma-derived-completion-koszul

In Situation \ref{situation-koszul}. The functor which sends $K \in D(A)$ to the derived limit $K' = R\lim( K \otimes_A^\mathbf{L} K_n^\bullet )$ is the left adjoint to the inclusion functor $D_{comp}(A) \to D(A)$ constructed in Lemma \ref{lemma-derived-completion}.


more-algebra-lemma-derived-completion-finite-cohomological-dimension

Let $A$ be a ring and let $I \subset A$ be an ideal which can be generated by $r$ elements. Then derived completion has finite cohomological dimension: \begin{enumerate} \item Let $K \to L$ be a morphism in $D(A)$ such that $H^i(K) \to H^i(L)$ is an isomorphism for $i \geq 1$ and surjective for $i = 0$. Then $H^i(K^\wedge) \to H^i(L^\wedge)$ is an isomorphism for $i \geq 1$ and surjective for $i = 0$. \item Let $K \to L$ be a morphism of $D(A)$ such that $H^i(K) \to H^i(L)$ is an isomorphism for $i \leq -1$ and injective for $i = 0$. Then $H^i(K^\wedge) \to H^i(L^\wedge)$ is an isomorphism for $i \leq -r - 1$ and injective for $i = -r$. \end{enumerate}


more-algebra-lemma-restriction-derived-complete

Let $A \to B$ be a ring map. Let $I \subset A$ be an ideal. The inverse image of $D_{comp}(A, I)$ under the restriction functor $D(B) \to D(A)$ is $D_{comp}(B, IB)$.


more-algebra-lemma-lift-universally

Let $A$ be a ring. Let $f \in A$. If there exists an integer $c \geq 1$ such that $A[f^c] = A[f^{c + 1}] = A[f^{c + 2}] = \ldots$ (for example if $A$ is Noetherian), then for all $n \geq 1$ there exist maps $$ (A \xrightarrow{f^n} A) \longrightarrow A/(f^n), \quad\text{and}\quad A/(f^{n + c}) \longrightarrow (A \xrightarrow{f^n} A) $$ in $D(A)$ inducing an isomorphism of the pro-objects $\{A/(f^n)\}$ and $\{(f^n : A \to A)\}$ in $D(A)$.


more-algebra-lemma-kernel-to-completion-square-zero

Let $f \in A$ be an element of a ring. Set $J = \bigcap f^nA$. Let $M$ be an $A$-module derived complete with respect to $f$. Then $JM' = 0$ where $M' = \Ker(M \to \lim M/f^nM)$. In particular, if $A$ is derived complete then $J$ is an ideal of square zero.


more-algebra-lemma-kernel-to-completion-nilpotent

Let $A$ be a ring derived complete with respect to an ideal $I$. Set $J = \bigcap I^n$. If $I$ can be generated by $r$ elements then $J^N = 0$ where $N = 2^r$.


more-algebra-lemma-sequence-Koszul-complexes

In Situation \ref{situation-koszul}. If $A$ is Noetherian, then the pro-objects $\{K_n^\bullet\}$ and $\{A/(f_1^n, \ldots, f_r^n)\}$ of $D(A)$ are isomorphic\footnote{In particular, for every $n$ there exists an $m \geq n$ such that $K_m^\bullet \to K_n^\bullet$ factors through the map $K_m^\bullet \to A/(f_1^m, \ldots, f_r^m)$.}.


more-algebra-lemma-noetherian-calculate

Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module with derived completion $M^\wedge$. Then there are short exact sequences $$ 0 \to R^1\lim \text{Tor}_{i + 1}^A(M, A/I^n) \to H^{-i}(M^\wedge) \to \lim \text{Tor}_i^A(M, A/I^n) \to 0 $$ A similar result holds for $M \in D^-(A)$.


more-algebra-lemma-derived-completion-pseudo-coherent

Let $A$ be a Noetherian ring and $I \subset A$ an ideal. Let $K$ be an object of $D(A)$ such that $H^n(K)$ a finite $A$-module for all $n \in \mathbf{Z}$. Then the cohomology modules $H^n(K^\wedge)$ of the derived completion are the $I$-adic completions of the cohomology modules $H^n(K)$.


more-algebra-lemma-when-derived-completion-is-completion

Let $I$ be an ideal in a Noetherian ring $A$. \begin{enumerate} \item If $M$ is a finite $A$-module and $N$ is a flat $A$-module, then the derived $I$-adic completion of $M \otimes_A N$ is the usual $I$-adic completion of $M \otimes_A N$. \item If $M$ is a finite $A$-module and $f \in A$, then the derived $I$-adic completion of $M_f$ is the usual $I$-adic completion of $M_f$. \end{enumerate}


more-algebra-lemma-eta-first-property

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. There is a canonical isomorphism $$ f^i : H^i(M^\bullet)/H^i(M^\bullet)[f] \longrightarrow H^i(\eta_fM^\bullet) $$ given by multiplication by $f^i$.


more-algebra-lemma-eta-qis

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. If $M^\bullet \to N^\bullet$ is a quasi-isomorphism of complexes of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$ and $N^i$, then the induced map $\eta_fM^\bullet \to \eta_fN^\bullet$ is a quasi-isomorphism too.


more-algebra-lemma-eta-third-property

Let $A$ be a ring and let $f, g \in A$ be nonzerodivisors. Let $M^\bullet$ be a complex of $A$-modules such that $fg$ is a nonzerodivisor on all $M^i$. Then $\eta_f\eta_gM^\bullet = \eta_{fg}M^\bullet$.


more-algebra-lemma-eta-locally-free

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. Assume \begin{enumerate} \item $M^i$, $M^{i + 1}$ free of ranks $r_i, r_{i + 1}$, \item the $r_i \times r_i$ minors of $(f, d^i) : M^i \to M^i \oplus M^{i + 1}$ generate a principal ideal. \end{enumerate} Then $(\eta_fM)^i$ is locally free of rank $r_i$ and the canonical map $(\eta_fM)^i \to f^iM^i \oplus f^{i + 1}M^{i + 1}$ is the inclusion of a direct summand.


more-algebra-lemma-eta-base-change

Let $A \to B$ be a ring map. Let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules. Assume \begin{enumerate} \item $f$ maps to a nonzerodivisor $g$ in $B$, \item $f$ is a nonzerodivisor on $M^i$, \item $M^i$ is finite free of rank $r_i$, \item the $r_i \times r_i$ minors of $(f, d^i) : M^i \to M^i \oplus M^{i + 1}$ generate a principal ideal. \end{enumerate} for all $i \in \mathbf{Z}$. Then there is a canonical isomorphism $\eta_fM^\bullet \otimes_A B = \eta_g(M^\bullet \otimes_A B)$.


more-algebra-lemma-vanishing-beta

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. For $i \in \mathbf{Z}$ the following are equivalent \begin{enumerate} \item $\Ker(d^i \bmod f^2)$ surjects onto $\Ker(d^i \bmod f)$, \item $\beta : H^i(M^\bullet \otimes_A f^iA/f^{i + 1}A) \to H^{i + 1}(M^\bullet \otimes_A f^{i + 1}A/f^{i + 2}A)$ is zero. \end{enumerate} These equivalent conditions are implied by the condition $H^{i + 1}(M^\bullet)[f] = 0$.


more-algebra-lemma-eta-vanishing-beta

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. If $\Ker(d^i \bmod f^2)$ surjects onto $\Ker(d^i \bmod f)$, then the canonical map $$ (\eta_fM)^i / f(\eta_fM)^i \longrightarrow f^iM^i/f^{i + 1}M^i \oplus f^{i + 1}M^{i + 1}/f^{i + 2}M^{i + 1}, x \longmapsto (x, d^i(x)) $$ identifies the left hand side with a direct sum of submodules of the right hand side.


more-algebra-lemma-ideal-direct-summand

Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules. Let $s : M \to N_1 \oplus N_2$ be a split injection. There exists a finitely generated ideal $I \subset A$ with the following property: a ring map $A \to B$ factors through $A/I$ if and only if $s \otimes \text{id}_B$ identifies $M \otimes_A B$ with a direct sum of submodules of $N_1 \otimes_A B \oplus N_2 \otimes_A B$.


more-algebra-lemma-eta-vanishing-beta-plus

Let $A$ be a ring and let $f \in \mathfrak m_A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. Assume \begin{enumerate} \item $M^i$ is finite free of rank $r_i$ and $M^i = 0$ for $|i| \gg 0$, \item the $r_i \times r_i$ minors of $(f, d^i) : M^i \to M^i \oplus M^{i + 1}$ generate a principal ideal \end{enumerate} for all $i \in \mathbf{Z}$. Consider the set of prime ideals $$ E = \{f \in \mathfrak p \subset A \mid \Ker(d^i \bmod f^2)_\mathfrak p \text{ surjects onto } \Ker(d^i \bmod f)_\mathfrak p \text{ for all }i \in \mathbf{Z}\} $$ There exists a finitely generated ideal $f \in I$ such that $I_\mathfrak p = fA_\mathfrak p$ for all $\mathfrak p \in E$ and such that the cohomology modules of $\eta_f M^\bullet \otimes_A A/I$ are finite free.


more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent

Let $A = \lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \in D(A_n)$ and maps $K_{n + 1} \to K_n$ in $D(A_{n + 1})$. Assume \begin{enumerate} \item the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels, \item $K_1$ is pseudo-coherent, and \item the maps induce isomorphisms $K_{n + 1} \otimes_{A_{n + 1}}^\mathbf{L} A_n \to K_n$. \end{enumerate} Then $K = R\lim K_n$ is a pseudo-coherent object of $D(A)$ and $K \otimes_A^\mathbf{L} A_n \to K_n$ is an isomorphism for all $n$.


more-algebra-lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent

Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_n \in D(A/I^n)$ and maps $K_{n + 1} \to K_n$ in $D(A/I^{n + 1})$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, \item $K_1$ is pseudo-coherent, and \item the maps induce isomorphisms $K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$. \end{enumerate} Then $K = R\lim K_n$ is a pseudo-coherent, derived complete object of $D(A)$ and $K \otimes_A^\mathbf{L} A/I^n \to K_n$ is an isomorphism for all $n$.


more-algebra-lemma-Rlim-perfect-gives-perfect

\begin{reference} \cite[Lemma 4.2]{Bhatt-Algebraize} \end{reference} Let $A = \lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \in D(A_n)$ and maps $K_{n + 1} \to K_n$ in $D(A_{n + 1})$. Assume \begin{enumerate} \item the transition maps $A_{n + 1} \to A_n$ are surjective with locally nilpotent kernels, \item $K_1$ is a perfect object, and \item the maps induce isomorphisms $K_{n + 1} \otimes_{A_{n + 1}}^\mathbf{L} A_n \to K_n$. \end{enumerate} Then $K = R\lim K_n$ is a perfect object of $D(A)$ and $K \otimes_A^\mathbf{L} A_n \to K_n$ is an isomorphism for all $n$.


more-algebra-lemma-Rlim-perfect-gives-complete

Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_n \in D(A/I^n)$ and maps $K_{n + 1} \to K_n$ in $D(A/I^{n + 1})$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, \item $K_1$ is a perfect object, and \item the maps induce isomorphisms $K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$. \end{enumerate} Then $K = R\lim K_n$ is a perfect, derived complete object of $D(A)$ and $K \otimes_A^\mathbf{L} A/I^n \to K_n$ is an isomorphism for all $n$.


more-algebra-lemma-kollar-kovacs

\begin{reference} Email from Kovacs of 23/02/2018. \end{reference} Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D(A)$. Set $K_n = K \otimes_A^\mathbf{L} A/I^n$. Assume for all $i \in \mathbf{Z}$ we have \begin{enumerate} \item $H^i(K)$ is a finite $A$-module, and \item the system $H^i(K_n)$ satisfies Mittag-Leffler. \end{enumerate} Then $\lim H^i(K)/I^nH^i(K)$ is equal to $\lim H^i(K_n)$ for all $i \in \mathbf{Z}$.


more-algebra-lemma-internal-hom-evaluate-isomorphism

Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. the map $$ R\Hom_R(L, M) \otimes_R^\mathbf{L} K \longrightarrow R\Hom_R(R\Hom_R(K, L), M) $$ of Lemma \ref{lemma-internal-hom-evaluate} is an isomorphism in the following two cases \begin{enumerate} \item $K$ perfect, or \item $K$ is pseudo-coherent, $L \in D^+(R)$, and $M$ finite injective dimension. \end{enumerate}


more-algebra-lemma-internal-hom-evaluate-isomorphism-technical

Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. the map $$ R\Hom_R(L, M) \otimes_R^\mathbf{L} K \longrightarrow R\Hom_R(R\Hom_R(K, L), M) $$ of Lemma \ref{lemma-internal-hom-evaluate} is an isomorphism if the following three conditions are satisfied \begin{enumerate} \item $L, M$ have finite injective dimension, \item $R\Hom_R(L, M)$ has finite tor dimension, \item for every $n \in \mathbf{Z}$ the truncation $\tau_{\leq n}K$ is pseudo-coherent \end{enumerate}


more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism

Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. The map $$ K \otimes_R^\mathbf{L} R\Hom_R(M, L) \longrightarrow R\Hom_R(M, K \otimes_R^\mathbf{L} L) $$ of Lemma \ref{lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \begin{enumerate} \item $M$ perfect, or \item $K$ is perfect, or \item $M$ is pseudo-coherent, $L \in D^+(R)$, and $K$ has tor amplitude in $[a, \infty]$. \end{enumerate}


more-algebra-lemma-upgrade-adjoint-tensor-RHom

Let $R \to R'$ be a ring map. For $K \in D(R)$ and $M \in D(R')$ there is a canonical isomorphism $$ R\Hom_R(K, M) = R\Hom_{R'}(K \otimes_R^\mathbf{L} R', M) $$


more-algebra-lemma-base-change-RHom

Let $R \to R'$ be a ring map. Let $K, M \in D(R)$. The map (\ref{equation-base-change-RHom}) $$ R\Hom_R(K, M) \otimes_R^\mathbf{L} R' \longrightarrow R\Hom_{R'}(K \otimes_R^\mathbf{L} R', M \otimes_R^\mathbf{L} R') $$ is an isomorphism in $D(R')$ in the following cases \begin{enumerate} \item $K$ is perfect, \item $R'$ is perfect as an $R$-module, \item $R \to R'$ is flat, $K$ is pseudo-coherent, and $M \in D^{+}(R)$, or \item $R'$ has finite tor dimension as an $R$-module, $K$ is pseudo-coherent, and $M \in D^{+}(R)$ \end{enumerate}


more-algebra-lemma-consequence-Artin-Rees

Let $I$ be an ideal of a Noetherian ring $A$. Let $ K \xrightarrow{\alpha} L \xrightarrow{\beta} M $ be a complex of finite $A$-modules. Set $H = \Ker(\beta)/\Im(\alpha)$. For $n \geq 0$ let $$ K/I^nK \xrightarrow{\alpha_n} L/I^nL \xrightarrow{\beta_n} M/I^nM $$ be the induced complex. Set $H_n = \Ker(\beta_n)/\Im(\alpha_n)$. Then there are canonical $A$-module maps giving a commutative diagram $$ \xymatrix{ & & & H \ar[lld] \ar[ld] \ar[d] \\ \ldots \ar[r] & H_3 \ar[r] & H_2 \ar[r] & H_1 } $$ Moreover, there exists a $c > 0$ and canonical $A$-module maps $H_n \to H/I^{n - c}H$ for $n \geq c$ such that the compositions $$ H/I^n H \to H_n \to  H/I^{n - c}H \quad\text{and}\quad H_n \to H/I^{n - c}H \to H_{n - c} $$ are the canonical ones. Moreover, we have \begin{enumerate} \item $(H_n)$ and $(H/I^nH)$ are isomorphic as pro-objects of $\text{Mod}_A$, \item $\lim H_n = \lim H/I^n H$, \item the inverse system $(H_n)$ is Mittag-Leffler, \item the image of $H_{n + c} \to H_n$ is equal to the image of $H \to H_n$, \item the composition $I^cH_n \to H_n \to H/I^{n - c}H \to H_n/I^{n - c}H_n$ is the inclusion $I^cH_n \to H_n$ followed by the quotient map $H_n \to H_n/I^{n - c}H_n$, and \item the kernel and cokernel of $H/I^nH \to H_n$ is annihilated by $I^c$. \end{enumerate}


more-algebra-lemma-kollar-kovacs-pseudo-coherent

\begin{reference} Email from Kovacs of 23/02/2018. \end{reference} Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D(A)$ be pseudo-coherent. Set $K_n = K \otimes_A^\mathbf{L} A/I^n$. Then for all $i \in \mathbf{Z}$ the system $H^i(K_n)$ satisfies Mittag-Leffler and $\lim H^i(K)/I^nH^i(K)$ is equal to $\lim H^i(K_n)$.


more-algebra-lemma-derived-completion-plain-completion

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M^\bullet$ be a bounded complex of finite $A$-modules. The inverse system of maps $$ M^\bullet \otimes_A^\mathbf{L} A/I^n \longrightarrow M^\bullet/I^nM^\bullet $$ defines an isomorphism of pro-objects of $D(A)$.


more-algebra-lemma-hom-systems-ML

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$, $N$ be finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. Then \begin{enumerate} \item the systems $(\Hom_A(M_n, N_n))$ and $(\text{Isom}_A(M_n, N_n))$ are Mittag-Leffler, \item there exists a $c \geq 0$ such that the kernels and cokernels of $$ \Hom_A(M, N)/I^n\Hom_A(M, N) \to \Hom_A(M_n, N_n) $$ are killed by $I^c$ for all $n$, \item we have $\lim \Hom_A(M_n, N_n) =\Hom_A(M, N)^\wedge = \Hom_{A^\wedge}(M^\wedge, N^\wedge)$ \item $\lim \text{Isom}_A(M_n, N_n) = \text{Isom}_{A^\wedge}(M^\wedge, N^\wedge)$. \end{enumerate} Here ${}^\wedge$ denotes usual $I$-adic completion.


more-algebra-lemma-isomorphic-completions

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$, $N$ be finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. If $M_n \cong N_n$ for all $n$, then $M^\wedge \cong N^\wedge$ as $A^\wedge$-modules.


more-algebra-lemma-iso

A morphism $(c, \varphi_n)$ of the category of Remark \ref{remark-weird-systems} is an isomorphism if and only if there exists a $c' \geq 0$ such that $\Ker(\varphi_n)$ and $\Coker(\varphi_n)$ are $I^{c'}$-torsion for all $n \gg 0$.


more-algebra-lemma-dejong-kollar-kovacs

\begin{reference} Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018. \end{reference} Let $I$ be an ideal of the Noetherian ring $A$. Let $M$ and $N$ be finite $A$-modules. Write $A_n = A/I^n$, $M_n = M/I^nM$, and $N_n = N/I^nN$. For every $i \geq 0$ the objects $$ \{\Ext^i_A(M, N)/I^n\Ext^i_A(M, N)\}_{n \geq 1} \quad\text{and}\quad \{\Ext^i_{A_n}(M_n, N_n)\}_{n \geq 1} $$ are isomorphic in the category $\mathcal{C}$ of Remark \ref{remark-weird-systems}.


more-algebra-lemma-consequence-Artin-Rees-bis

Let $I$ be an ideal of a Noetherian ring $A$. Let $ K \xrightarrow{\alpha} L \xrightarrow{\beta} M $ be a complex of finite $A$-modules. Set $H = \Ker(\beta)/\Im(\alpha)$. For $n \geq 0$ let $$ I^nK \xrightarrow{\alpha_n} I^nL \xrightarrow{\beta_n} I^nM $$ be the induced complex. Set $H_n = \Ker(\beta_n)/\Im(\alpha_n)$. Then there are canonical $A$-module maps $$ \ldots \to H_3 \to H_2 \to H_1 \to H $$ There exists a $c > 0$ such that for $n \geq c$ the image of $H_n \to H$ is contained in $I^{n - c}H$ and there is a canonical $A$-module map $I^nH \to H_{n - c}$ such that the compositions $$ I^n H \to H_{n - c} \to  I^{n - 2c}H \quad\text{and}\quad H_n \to I^{n - c}H \to H_{n - 2c} $$ are the canonical ones. In particular, the inverse systems $(H_n)$ and $(I^nH)$ are isomorphic as pro-objects of $\text{Mod}_A$.


more-algebra-lemma-ext-factors

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$, $N$ be $A$-modules with $M$ finite. For each $p > 0$ there exists a $c \geq 0$ such that for $n \geq c$ the map $\Ext_A^p(M, N) \to \Ext_A^p(I^nM, N)$ factors through $\Ext^p_A(I^nM, I^{n - c}N) \to \Ext_A^p(I^nM, N)$.


more-algebra-lemma-ext-annihilated

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$, $N$ be $A$-modules with $M$ finite and $N$ annihilated by a power of $I$. For each $p > 0$ there exists an $n$ such that the map $\Ext_A^p(M, N) \to \Ext_A^p(I^nM, N)$ is zero.


more-algebra-lemma-ext-induced-toplogy

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $K \in D(A)$ be pseudo-coherent and let $M$ be a finite $A$-module. For each $p \in \mathbf{Z}$ there exists an $c$ such that the image of $\Ext_A^p(K, I^nM) \to \Ext_A^p(K, M)$ is contained in $I^{n - c}\Ext_A^p(K, M)$ for $n \geq c$.


more-algebra-lemma-sequence-powers-pro-bounded

In Situation \ref{situation-koszul} assume $A$ is Noetherian. With notation as above, the inverse system $(I^n)$ is pro-isomorphic in $D(A)$ to the inverse system $(I_n^\bullet)$.


more-algebra-lemma-tensoring-Deligne-system

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M^\bullet$ be a bounded complex of finite $A$-modules. The inverse system of maps $$ I^n \otimes_A^\mathbf{L} M^\bullet \longrightarrow I^nM^\bullet $$ defines an isomorphism of pro-objects of $D(A)$.