Derived Categories

en · official · sha256:7330949afd1629e09a53e8833569efe91f5d8ca6b7368e0a68a87d0caa530934

\begin{definition} \label{definition-injective-resolution} Let $\mathcal{A}$ be an abelian category. Let $A \in \Ob(\mathcal{A})$. An {\it injective resolution of $A$} is a complex $I^\bullet$ together with a map $A \to I^0$ such that: \begin{enumerate} \item We have $I^n = 0$ for $n < 0$. \item Each $I^n$ is an injective object of $\mathcal{A}$. \item The map $A \to I^0$ is an isomorphism onto $\Ker(d^0)$. \item We have $H^i(I^\bullet) = 0$ for $i > 0$. \end{enumerate} Hence $A[0] \to I^\bullet$ is a quasi-isomorphism. In other words the complex $$ \ldots \to 0 \to A \to I^0 \to I^1 \to \ldots $$ is acyclic. Let $K^\bullet$ be a complex in $\mathcal{A}$. An {\it injective resolution of $K^\bullet$} is a complex $I^\bullet$ together with a map $\alpha : K^\bullet \to I^\bullet$ of complexes such that \begin{enumerate} \item We have $I^n = 0$ for $n \ll 0$, i.e., $I^\bullet$ is bounded below. \item Each $I^n$ is an injective object of $\mathcal{A}$. \item The map $\alpha : K^\bullet \to I^\bullet$ is a quasi-isomorphism. \end{enumerate} \end{definition}

Referencing norms: InjectiveResolutionComputesRF, InjectiveResolutionResolves, ResolutionByQuasiIsomorphism, RightDerivedByResolution, acyclic_in_positive_degrees, augmentation_onto_kernel, bounded_below, injective_resolution, injective_resolution/necessary, injective_resolution/sufficient, quasi_isomorphism_from_shift, terms_injective, vanishes_in_negative_degrees

en · official · sha256:869cffaf0235a6f9d2d38c26d042d9ff1c0bbfae733cd1cdcc72af28b5bc8fa4

\begin{lemma} \label{lemma-injective-resolutions-exist} Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. \begin{enumerate} \item Any object of $\mathcal{A}$ has an injective resolution. \item If $H^n(K^\bullet) = 0$ for all $n \ll 0$ then $K^\bullet$ has an injective resolution. \item If $K^\bullet$ is a complex with $K^n = 0$ for $n < a$, then there exists an injective resolution $\alpha : K^\bullet \to I^\bullet$ with $I^n = 0$ for $n < a$ such that each $\alpha^n : K^n \to I^n$ is injective. \end{enumerate} \end{lemma}

Referencing norms: InjectiveResolutionsExist

en · official · sha256:4f7053d7253056bd73774874a7759dc3c2be34cd7e4d6223cfeddd34df6fb0bc

\begin{definition} \label{definition-derived-functor} In Situation \ref{situation-classical}. \begin{enumerate} \item The {\it right derived functors of $F$} are the partial functors $RF$ associated to cases (1) and (2) of Situation \ref{situation-classical}. \item The {\it left derived functors of $F$} are the partial functors $LF$ associated to cases (3) and (4) of Situation \ref{situation-classical}. \item An object $A$ of $\mathcal{A}$ is said to be {\it right acyclic for $F$}, or {\it acyclic for $RF$} if $A[0]$ computes $RF$. \item An object $A$ of $\mathcal{A}$ is said to be {\it left acyclic for $F$}, or {\it acyclic for $LF$} if $A[0]$ computes $LF$. \end{enumerate} \end{definition}

Referencing norms: RightAcyclicByComputing, shift_zero

en · official · sha256:7ecd153a29139bad7c85e08acd4818c3aedb736903c0a95c5400227dea7d508d

\begin{definition} \label{definition-higher-derived-functors} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. Assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $i \in \mathbf{Z}$. The {\it $i$th right derived functor $R^iF$ of $F$} is the functor $$ R^iF = H^i \circ RF : \mathcal{A} \longrightarrow \mathcal{B} $$ \end{definition}

Referencing norms: RightDerivedByResolution, cohomology_of

en · official · sha256:dfebd11201136d7893be34773ff7fda188a8b1936ac0ddeff5a25c3f02c5f5a0

\begin{lemma} \label{lemma-higher-derived-functors} Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor. \begin{enumerate} \item For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of complexes in $\text{Comp}^{+}(\mathcal{A})$ there is an associated long exact sequence $$ \ldots \to H^i(RF(A^\bullet)) \to H^i(RF(B^\bullet)) \to H^i(RF(C^\bullet)) \to H^{i + 1}(RF(A^\bullet)) \to \ldots $$ \item The functors $R^iF : \mathcal{A} \to \mathcal{B}$ are zero for $i < 0$. Also $R^0F = F : \mathcal{A} \to \mathcal{B}$. \item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective. \item The sequence $(R^iF, \delta)$ forms a universal $\delta$-functor (see Homology, Definition \ref{homology-definition-universal-delta-functor}) from $\mathcal{A}$ to $\mathcal{B}$. \end{enumerate} \end{lemma}

Referencing norms: DerivedUniversalDeltaFunctor, LongExactSequence

en · official · sha256:d0947fabef80aaa804f6f4635edca13dba8e17d1cdf28aacd25709217ffafa46

\begin{lemma} \label{lemma-F-acyclic} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $A$ be an object of $\mathcal{A}$. \begin{enumerate} \item $A$ is right acyclic for $F$ if and only if $F(A) \to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$, \item if $F$ is left exact, then $A$ is right acyclic for $F$ if and only if $R^iF(A) = 0$ for all $i > 0$. \end{enumerate} \end{lemma}

Referencing norms: HigherDerivedVanishForAcyclic

en · official · sha256:d50febff637ad15f523e1d151117f8b2d94f768d9f3d3ab188837d89f7ddcabf

\begin{lemma}[Leray's acyclicity lemma] \label{lemma-leray-acyclicity} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. Let $A^\bullet$ be a bounded below complex of right $F$-acyclic objects such that $RF$ is defined at $A^\bullet$\footnote{For example this holds if $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined.}. The canonical map $$ F(A^\bullet) \longrightarrow RF(A^\bullet) $$ is an isomorphism in $D^{+}(\mathcal{B})$, i.e., $A^\bullet$ computes $RF$. \end{lemma}

Referencing norms: LerayAcyclicity, ResolutionByQuasiIsomorphism, terms_right_acyclic_for

en · official · sha256:d107facc91c19d9cb0f08172be60381e0686a02742a3a98e6272ac70647e3f71

\begin{lemma} \label{lemma-compose-derived-functors} Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be left exact functors. Assume $\mathcal{A}$, $\mathcal{B}$ have enough injectives. The following are equivalent \begin{enumerate} \item $F(I)$ is right acyclic for $G$ for each injective object $I$ of $\mathcal{A}$, and \item the canonical map $$ t : R(G \circ F) \longrightarrow RG \circ RF. $$ is isomorphism of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$. \end{enumerate} \end{lemma}

Referencing norms: ComposeRightDerived, maps_injectives_to_acyclics

en · official · sha256:044a2efee16ab10a34e3bb64e608c8cf8292260bff8d6e4acf41d4d41c2dca12

\begin{definition} \label{definition-right-derived-functor-defined} Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X \in \Ob(\mathcal{D})$. \begin{enumerate} \item we say the {\it right derived functor $RF$ is defined at} $X$ if the ind-object $$ (X/S) \longrightarrow \mathcal{D}', \quad (s : X \to X') \longmapsto F(X') $$ is essentially constant\footnote{For a discussion of when an ind-object or pro-object of a category is essentially constant we refer to Categories, Section \ref{categories-section-essentially-constant}.}; in this case the value $Y$ in $\mathcal{D}'$ is called the {\it value of $RF$ at $X$}. \item we say the {\it left derived functor $LF$ is defined at} $X$ if the pro-object $$ (S/X) \longrightarrow \mathcal{D}', \quad (s: X' \to X) \longmapsto F(X') $$ is essentially constant; in this case the value $Y$ in $\mathcal{D}'$ is called the {\it value of $LF$ at $X$}. \end{enumerate} By abuse of notation we often denote the values simply $RF(X)$ or $LF(X)$. \end{definition}

en · official · sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45

\begin{definition} \label{definition-everywhere-defined} In Situation \ref{situation-derived-functor}. We say $F$ is {\it right derivable}, or that $RF$ {\it everywhere defined} if $RF$ is defined at every object of $\mathcal{D}$. We say $F$ is {\it left derivable}, or that $LF$ {\it everywhere defined} if $LF$ is defined at every object of $\mathcal{D}$. \end{definition}

Referencing norms: RFEverywhereDefined

en · official · sha256:a12a542d90fce40869d878a5e1f1be4adaed0407a78b0a5d0febeb73e30ca0b5

\begin{definition} \label{definition-computes} In Situation \ref{situation-derived-functor}. \begin{enumerate} \item An object $X$ of $\mathcal{D}$ {\it computes} $RF$ if $RF$ is defined at $X$ and the canonical map $F(X) \to RF(X)$ is an isomorphism. \item An object $X$ of $\mathcal{D}$ {\it computes} $LF$ if $LF$ is defined at $X$ and the canonical map $LF(X) \to F(X)$ is an isomorphism. \end{enumerate} \end{definition}

Referencing norms: InjectiveComplexComputesRF, RightAcyclicByComputing, RightDerivedByResolution

en · official · sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6

\begin{situation} \label{situation-classical} Here $F : \mathcal{A} \to \mathcal{B}$ is an additive functor between abelian categories. This induces exact functors $$ F : K(\mathcal{A}) \to K(\mathcal{B}), \quad K^{+}(\mathcal{A}) \to K^{+}(\mathcal{B}), \quad K^{-}(\mathcal{A}) \to K^{-}(\mathcal{B}). $$ See Lemma \ref{lemma-additive-exact-homotopy-category}. We also denote $F$ the composition $K(\mathcal{A}) \to D(\mathcal{B})$, $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$, and $K^{-}(\mathcal{A}) \to D^-(\mathcal{B})$ of $F$ with the localization functor $K(\mathcal{B}) \to D(\mathcal{B})$, etc. This situation leads to four derived functors we will consider in the following. \begin{enumerate} \item The right derived functor of $F : K(\mathcal{A}) \to D(\mathcal{B})$ relative to the multiplicative system $\text{Qis}(\mathcal{A})$. \item The right derived functor of $F : K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ relative to the multiplicative system $\text{Qis}^{+}(\mathcal{A})$. \item The left derived functor of $F : K(\mathcal{A}) \to D(\mathcal{B})$ relative to the multiplicative system $\text{Qis}(\mathcal{A})$. \item The left derived functor of $F : K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B})$ relative to the multiplicative system $\text{Qis}^-(\mathcal{A})$. \end{enumerate} Each of these cases is an example of Situation \ref{situation-derived-functor}. \end{situation}

Referencing norms: InjectiveIsRightAcyclic, RFEverywhereDefined, applied_complex

en · official · sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7

\begin{lemma} \label{lemma-left-exact-higher-derived} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. \begin{enumerate} \item We have $R^iF = 0$ for $i < 0$, \item $R^0F$ is left exact, \item the map $F \to R^0F$ is an isomorphism if and only if $F$ is left exact. \end{enumerate} \end{lemma}

Referencing norms: NegativeDerivedVanish, R0AgreesIfLeftExact

en · official · sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4

\begin{lemma} \label{lemma-right-derived-delta-functor} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. \begin{enumerate} \item The functors $R^iF$, $i \geq 0$ come equipped with a canonical structure of a $\delta$-functor from $\mathcal{A} \to \mathcal{B}$, see Homology, Definition \ref{homology-definition-cohomological-delta-functor}. \item If every object of $\mathcal{A}$ is a subobject of a right acyclic object for $F$, then $\{R^iF, \delta\}_{i \geq 0}$ is a universal $\delta$-functor, see Homology, Definition \ref{homology-definition-universal-delta-functor}. \end{enumerate} \end{lemma}

Referencing norms: DerivedFormDeltaFunctor

en · official · sha256:e032a20e4fb0e18232eca1300982b8c867068569810bf8e4474e48073cab1465

\begin{lemma} \label{lemma-injective-acyclic} Let $\mathcal{A}$ be an abelian category. Let $I \in \Ob(\mathcal{A})$ be an injective object. Let $I^\bullet$ be a bounded below complex of injectives in $\mathcal{A}$. \begin{enumerate} \item $I^\bullet$ computes $RF$ relative to $\text{Qis}^{+}(\mathcal{A})$ for any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into any triangulated category $\mathcal{D}$. \item $I$ is right acyclic for any additive functor $F : \mathcal{A} \to \mathcal{B}$ into any abelian category $\mathcal{B}$. \end{enumerate} \end{lemma}

Referencing norms: InjectiveComplexComputesRF, InjectiveIsRightAcyclic, InjectiveResolutionComputesRF

en · official · sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651

\begin{lemma} \label{lemma-enough-injectives-right-derived} Let $\mathcal{A}$ be an abelian category with enough injectives. \begin{enumerate} \item For any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into a triangulated category $\mathcal{D}$ the right derived functor $$ RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D} $$ is everywhere defined. \item For any additive functor $F : \mathcal{A} \to \mathcal{B}$ into an abelian category $\mathcal{B}$ the right derived functor $$ RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B}) $$ is everywhere defined. \end{enumerate} \end{lemma}

Referencing norms: RFEverywhereDefined