Homological Algebra

en · official · sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6

\begin{definition} \label{definition-preadditive} A category $\mathcal{A}$ is called {\it preadditive} if each morphism set $\Mor_\mathcal{A}(x, y)$ is endowed with the structure of an abelian group such that the compositions $$ \Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z) $$ are bilinear. A functor $F : \mathcal{A} \to \mathcal{B}$ of preadditive categories is called {\it additive} if and only if $F : \Mor(x, y) \to \Mor(F(x), F(y))$ is a homomorphism of abelian groups for all $x, y \in \Ob(\mathcal{A})$. \end{definition}

Referencing norms: AdditiveByFiniteProducts, AdditiveByHomGroups, homomorphism_on_hom_groups, preadditive_category

en · official · sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30

\begin{definition} \label{definition-additive-category} A category $\mathcal{A}$ is called {\it additive} if it is preadditive and finite products exist, in other words it has a zero object and direct sums. \end{definition}

Referencing norms: AdditiveByFiniteProducts

en · official · sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22

\begin{definition} \label{definition-abelian-category} A category $\mathcal{A}$ is {\it abelian} if it is additive, if all kernels and cokernels exist, and if the natural map $\Coim(f) \to \Im(f)$ is an isomorphism for all morphisms $f$ of $\mathcal{A}$. \end{definition}

Referencing norms: abelian_category, abelian_category/necessary, abelian_category/sufficient, coimage_to_image_isomorphism, has_all_cokernels, has_all_kernels

en · official · sha256:8cfae125b32f40b6c59de9809280a2477df867a4fb7dfdc10432b00228fe6b98

\begin{definition} \label{definition-exact} Let $\mathcal{A}$ be an additive category. Consider a sequence of morphisms $$ \ldots \to x \to y \to z \to \ldots \quad\text{or}\quad x_1 \to x_2 \to \ldots \to x_n $$ in $\mathcal{A}$. We say such a sequence is a {\it complex} if the composition of any two consecutive (drawn) arrows is zero. If $\mathcal{A}$ is abelian then we say a complex of the first type above is {\it exact at $y$} if $\Im(x \to y) = \Ker(y \to z)$ and we say a complex of the second kind is {\it exact at $x_i$} where $1 < i < n$ if $\Im(x_{i - 1} \to x_i) = \Ker(x_i \to x_{i + 1})$. We a sequence as above is {\it exact} or is an {\it exact sequence} or is an {\it exact complex} if it is a complex and exact at every object (in the first case) or exact at $x_i$ for all $1 < i < n$ (in the second case). There are variants of these notions for sequences of the form $$ \ldots \to x_{-3} \to x_{-2} \to x_{-1} \quad\text{and}\quad x_1 \to x_2 \to x_3 \to \ldots $$ A {\it short exact sequence} is an exact complex of the form $$ 0 \to A \to B \to C \to 0. $$ \end{definition}

Referencing norms: short_exact_sequence

en · official · sha256:e66f034eeeacfcb7b11738065188279ef03410097d4c35d3837c698b10ba0cab

\begin{lemma} \label{lemma-exact-functor} Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item If $F$ is either left or right exact, then it is additive. \item $F$ is left exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $0 \to F(A) \to F(B) \to F(C)$ is exact. \item $F$ is right exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $F(A) \to F(B) \to F(C) \to 0$ is exact. \item $F$ is exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $0 \to F(A) \to F(B) \to F(C) \to 0$ is exact. \end{enumerate} \end{lemma}

Referencing norms: LeftExactBySES, LeftExactIsAdditive, preserves_left_exactness

en · official · sha256:4d94a8c285e62eb525882ae9eadf2584c813bde419530fe169fa37a99ba07572

\begin{definition} \label{definition-cohomological-delta-functor} Let $\mathcal{A}, \mathcal{B}$ be abelian categories. A {\it cohomological $\delta$-functor} or simply a {\it $\delta$-functor} from $\mathcal{A}$ to $\mathcal{B}$ is given by the following data: \begin{enumerate} \item a collection $F^n : \mathcal{A} \to \mathcal{B}$, $n \geq 0$ of additive functors, and \item for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ a collection $\delta_{A \to B \to C} : F^n(C) \to F^{n + 1}(A)$, $n \geq 0$ of morphisms of $\mathcal{B}$. \end{enumerate} These data are assumed to satisfy the following axioms \begin{enumerate} \item for every short exact sequence as above the sequence $$ \xymatrix{ 0 \ar[r] & F^0(A) \ar[r] & F^0(B) \ar[r] & F^0(C) \ar[lld]^{\delta_{A \to B \to C}} \\ & F^1(A) \ar[r] & F^1(B) \ar[r] & F^1(C) \ar[lld]^{\delta_{A \to B \to C}} \\ & F^2(A) \ar[r] & F^2(B) \ar[r] & \ldots } $$ is exact, and \item for every morphism $(A \to B \to C) \to (A' \to B' \to C')$ of short exact sequences of $\mathcal{A}$ the diagrams $$ \xymatrix{ F^n(C) \ar[d] \ar[rr]_{\delta_{A \to B \to C}} & & F^{n + 1}(A) \ar[d] \\ F^n(C') \ar[rr]^{\delta_{A' \to B' \to C'}} & & F^{n + 1}(A') } $$ are commutative. \end{enumerate} \end{definition}

Referencing norms: DerivedFormDeltaFunctor, LongExactSequence

en · official · sha256:2623fbb984bcd0a9c4431c334a750882551e9e7ecb2317676261159e79b5b902

\begin{definition} \label{definition-morphism-delta-functors} Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $(F^n, \delta_F)$ and $(G^n, \delta_G)$ be $\delta$-functors from $\mathcal{A}$ to $\mathcal{B}$. A {\it morphism of $\delta$-functors from $F$ to $G$} is a collection of transformation of functors $t^n : F^n \to G^n$, $n \geq 0$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ the diagrams $$ \xymatrix{ F^n(C) \ar[d]_{t^n} \ar[rr]_{\delta_{F, A \to B \to C}} & & F^{n + 1}(A) \ar[d]^{t^{n + 1}} \\ G^n(C) \ar[rr]^{\delta_{G, A \to B \to C}} & & G^{n + 1}(A) } $$ are commutative. \end{definition}

Referencing norms: DerivedUniversalDeltaFunctor

en · official · sha256:91ce552a862af1f6011c0531ff67bbb2072d48573bb4672f60019ce6905bf7ab

\begin{definition} \label{definition-universal-delta-functor} Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^n, \delta_F)$ be a $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$. We say $F$ is a {\it universal $\delta$-functor} if and only if for every $\delta$-functor $G = (G^n, \delta_G)$ and any morphism of functors $t : F^0 \to G^0$ there exists a unique morphism of $\delta$-functors $\{t^n\}_{n \geq 0} : F \to G$ such that $t = t^0$. \end{definition}

Referencing norms: DerivedUniversalDeltaFunctor

en · official · sha256:3b91442b6a83fe54c0144799446ea4e30d2c23f4f254154a4942fbe3fda8bb46

\begin{definition} \label{definition-injective} Let $\mathcal{A}$ be an abelian category. An object $J \in \Ob(\mathcal{A})$ is called {\it injective} if for every injection $A \hookrightarrow B$ and every morphism $A \to J$ there exists a morphism $B \to J$ making the following diagram commute $$ \xymatrix{ A \ar[r] \ar[d] & B \ar@{-->}[ld] \\ J & } $$ \end{definition}

Referencing norms: InjectiveByLifting, lifting_property

en · official · sha256:7ab9cf5a23a4b66b3ff6bfdb8a5b4147c2310114dd153f177517d28ebf878baa

\begin{lemma} \label{lemma-characterize-injectives} Let $\mathcal{A}$ be an abelian category. Let $I$ be an object of $\mathcal{A}$. The following are equivalent: \begin{enumerate} \item The object $I$ is injective. \item The functor $B \mapsto \Hom_\mathcal{A}(B, I)$ is exact. \item Any short exact sequence $$ 0 \to I \to A \to B \to 0 $$ in $\mathcal{A}$ is split. \item We have $\Ext_\mathcal{A}(B, I) = 0$ for all $B \in \Ob(\mathcal{A})$. \end{enumerate} \end{lemma}

Referencing norms: InjectiveByExactHom, InjectiveBySplitExtensions, every_extension_of_splits, hom_into_is_exact

en · official · sha256:db418977b164a669fc7d1d7db62d4c9d2ec9568b87dd6176b73e93ee0de025ee

\begin{definition} \label{definition-enough-injectives} Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has {\it enough injectives} if every object $A$ has an injective morphism $A \to J$ into an injective object $J$. \end{definition}

Referencing norms: enough_injectives, InjectiveResolutionsExist