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\section*{An Introduction to Homological Algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

This entry provides a hyperlinked index for the textbook

\begin{itemize}%
\item [[Charles Weibel]],

\emph{An Introduction to Homological Algebra}

Cambridge University Press (1994)



\end{itemize}
which gives a first exposition to central concepts in \emph{[[homological algebra]]}.

For a more comprehensive account of the theory see also chapters 8 and 12-18 of

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]}, Grundlehren der Mathematischen Wissenschaften \textbf{332}, Springer (2006)

\end{itemize}
and see the $n$Lab lecture notes

\begin{itemize}%
\item \emph{[[schreiber:Introduction to Homological Algebra]]} ([[IntroductionToHomologicalAlgebra-170509.pdf:file]])

\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{1_chain_complexes}{1 Chain complexes}\dotfill \pageref*{1_chain_complexes} \linebreak
\noindent\hyperlink{11_complexes_of_modules}{1.1 Complexes of $R$-modules}\dotfill \pageref*{11_complexes_of_modules} \linebreak
\noindent\hyperlink{12_operations_on_chain_complexes}{1.2 Operations on chain complexes}\dotfill \pageref*{12_operations_on_chain_complexes} \linebreak
\noindent\hyperlink{13_long_exact_sequences}{1.3 Long exact sequences}\dotfill \pageref*{13_long_exact_sequences} \linebreak
\noindent\hyperlink{14_chain_homotopies}{1.4 Chain homotopies}\dotfill \pageref*{14_chain_homotopies} \linebreak
\noindent\hyperlink{15_mapping_cones_and_cyclinders}{1.5 Mapping cones and cyclinders}\dotfill \pageref*{15_mapping_cones_and_cyclinders} \linebreak
\noindent\hyperlink{16_more_on_abelian_categories}{1.6 More on abelian categories}\dotfill \pageref*{16_more_on_abelian_categories} \linebreak
\noindent\hyperlink{2_derived_functors}{2 Derived functors}\dotfill \pageref*{2_derived_functors} \linebreak
\noindent\hyperlink{21_functor}{2.1 $\delta$-Functor}\dotfill \pageref*{21_functor} \linebreak
\noindent\hyperlink{22_projective_resolutions}{2.2 Projective resolutions}\dotfill \pageref*{22_projective_resolutions} \linebreak
\noindent\hyperlink{23_injective_resolutions}{2.3 Injective resolutions}\dotfill \pageref*{23_injective_resolutions} \linebreak
\noindent\hyperlink{24_left_derived_functors}{2.4 Left derived functors}\dotfill \pageref*{24_left_derived_functors} \linebreak
\noindent\hyperlink{25_right_derived_functors}{2.5 Right derived functors}\dotfill \pageref*{25_right_derived_functors} \linebreak
\noindent\hyperlink{26_adjoint_functors_and_leftright_exactness}{2.6 Adjoint functors and left/right exactness}\dotfill \pageref*{26_adjoint_functors_and_leftright_exactness} \linebreak
\noindent\hyperlink{27_balancing__and_}{2.7 Balancing $Tor$ and $Ext$}\dotfill \pageref*{27_balancing__and_} \linebreak
\noindent\hyperlink{3_tor_and_ext}{3 Tor and Ext}\dotfill \pageref*{3_tor_and_ext} \linebreak
\noindent\hyperlink{31__for_abelian_groups}{3.1 $Tor$ for abelian groups}\dotfill \pageref*{31__for_abelian_groups} \linebreak
\noindent\hyperlink{32__and_flatness}{3.2 $Tor$ and flatness}\dotfill \pageref*{32__and_flatness} \linebreak
\noindent\hyperlink{33__for_nice_rings}{3.3 $Ext$ for nice rings}\dotfill \pageref*{33__for_nice_rings} \linebreak
\noindent\hyperlink{34__and_extensions}{3.4 $Ext$ and extensions}\dotfill \pageref*{34__and_extensions} \linebreak
\noindent\hyperlink{35_derived_functors_of_the_inverse_limit}{3.5 Derived functors of the inverse limit}\dotfill \pageref*{35_derived_functors_of_the_inverse_limit} \linebreak
\noindent\hyperlink{36_universal_coefficient_theorem}{3.6 Universal coefficient theorem}\dotfill \pageref*{36_universal_coefficient_theorem} \linebreak
\noindent\hyperlink{4_homological_dimension}{4 Homological dimension}\dotfill \pageref*{4_homological_dimension} \linebreak
\noindent\hyperlink{41_dimensions}{4.1 Dimensions}\dotfill \pageref*{41_dimensions} \linebreak
\noindent\hyperlink{42_rings_of_small_dimension}{4.2 Rings of Small Dimension}\dotfill \pageref*{42_rings_of_small_dimension} \linebreak
\noindent\hyperlink{43_change_of_rings_theorem}{4.3 Change of Rings Theorem}\dotfill \pageref*{43_change_of_rings_theorem} \linebreak
\noindent\hyperlink{44_local_rings}{4.4 Local rings}\dotfill \pageref*{44_local_rings} \linebreak
\noindent\hyperlink{45_koszul_complexes}{4.5 Koszul Complexes}\dotfill \pageref*{45_koszul_complexes} \linebreak
\noindent\hyperlink{46_local_cohomology}{4.6 Local Cohomology}\dotfill \pageref*{46_local_cohomology} \linebreak
\noindent\hyperlink{5_spectral_sequences}{5 Spectral sequences}\dotfill \pageref*{5_spectral_sequences} \linebreak
\noindent\hyperlink{51_introduction}{5.1 Introduction}\dotfill \pageref*{51_introduction} \linebreak
\noindent\hyperlink{52_terminology}{5.2 Terminology}\dotfill \pageref*{52_terminology} \linebreak
\noindent\hyperlink{53_lerayserre_spectral_sequence}{5.3 Leray-Serre Spectral Sequence}\dotfill \pageref*{53_lerayserre_spectral_sequence} \linebreak
\noindent\hyperlink{54_spectral_sequence_of_a_filtration}{5.4 Spectral sequence of a filtration}\dotfill \pageref*{54_spectral_sequence_of_a_filtration} \linebreak
\noindent\hyperlink{55_convergence}{5.5 Convergence}\dotfill \pageref*{55_convergence} \linebreak
\noindent\hyperlink{56_spectral_sequence_of_a_double_complex}{5.6 Spectral sequence of a double complex}\dotfill \pageref*{56_spectral_sequence_of_a_double_complex} \linebreak
\noindent\hyperlink{57_hypercohomology}{5.7 Hypercohomology}\dotfill \pageref*{57_hypercohomology} \linebreak
\noindent\hyperlink{58_grothendieck_spectral_sequence}{5.8 Grothendieck spectral sequence}\dotfill \pageref*{58_grothendieck_spectral_sequence} \linebreak
\noindent\hyperlink{59_exact_couples}{5.9 Exact couples}\dotfill \pageref*{59_exact_couples} \linebreak
\noindent\hyperlink{6_group_homology_and_cohomology}{6 Group homology and cohomology}\dotfill \pageref*{6_group_homology_and_cohomology} \linebreak
\noindent\hyperlink{7_lie_algebra_homology_and_cohomology}{7 Lie algebra homology and cohomology}\dotfill \pageref*{7_lie_algebra_homology_and_cohomology} \linebreak
\noindent\hyperlink{8_simplicial_methods_in_homological_algebra}{8 Simplicial methods in homological algebra}\dotfill \pageref*{8_simplicial_methods_in_homological_algebra} \linebreak
\noindent\hyperlink{81_simplicial_object}{8.1 Simplicial object}\dotfill \pageref*{81_simplicial_object} \linebreak
\noindent\hyperlink{82_operations_on_simplicial_objects}{8.2 Operations on simplicial objects}\dotfill \pageref*{82_operations_on_simplicial_objects} \linebreak
\noindent\hyperlink{83_simplicial_homotopy_groups}{8.3 Simplicial homotopy groups}\dotfill \pageref*{83_simplicial_homotopy_groups} \linebreak
\noindent\hyperlink{84_the_doldkan_correspondence}{8.4 The Dold-Kan correspondence}\dotfill \pageref*{84_the_doldkan_correspondence} \linebreak
\noindent\hyperlink{85_the_eilenbergzilber_theorem}{8.5 The Eilenberg-Zilber theorem}\dotfill \pageref*{85_the_eilenbergzilber_theorem} \linebreak
\noindent\hyperlink{86_canonical_resolutions}{8.6 Canonical resolutions}\dotfill \pageref*{86_canonical_resolutions} \linebreak
\noindent\hyperlink{87_cotriple_homology}{8.7 Cotriple homology}\dotfill \pageref*{87_cotriple_homology} \linebreak
\noindent\hyperlink{88_andrequillen_homology_and_cohomology}{8.8 Andre-Quillen Homology and Cohomology}\dotfill \pageref*{88_andrequillen_homology_and_cohomology} \linebreak
\noindent\hyperlink{9_hochschild_and_cyclic_homology}{9 Hochschild and cyclic homology}\dotfill \pageref*{9_hochschild_and_cyclic_homology} \linebreak
\noindent\hyperlink{91_hochschild_homology_and_cohomology_of_algebras}{9.1 Hochschild Homology and Cohomology of Algebras}\dotfill \pageref*{91_hochschild_homology_and_cohomology_of_algebras} \linebreak
\noindent\hyperlink{92_derivations_differentials_and_separable_algebra}{9.2 Derivations, Differentials and Separable Algebra}\dotfill \pageref*{92_derivations_differentials_and_separable_algebra} \linebreak
\noindent\hyperlink{93__extensions_smooth_algebras}{9.3 $H^2$, Extensions, Smooth Algebras}\dotfill \pageref*{93__extensions_smooth_algebras} \linebreak
\noindent\hyperlink{94_hochschild_products}{9.4 Hochschild products}\dotfill \pageref*{94_hochschild_products} \linebreak
\noindent\hyperlink{95_morita_invariance}{9.5 Morita Invariance}\dotfill \pageref*{95_morita_invariance} \linebreak
\noindent\hyperlink{96_cyclic_homology}{9.6 Cyclic Homology}\dotfill \pageref*{96_cyclic_homology} \linebreak
\noindent\hyperlink{97_group_rings}{9.7 Group Rings}\dotfill \pageref*{97_group_rings} \linebreak
\noindent\hyperlink{98_mixed_complexes}{9.8 Mixed Complexes}\dotfill \pageref*{98_mixed_complexes} \linebreak
\noindent\hyperlink{99_graded_algebras}{9.9 Graded Algebras}\dotfill \pageref*{99_graded_algebras} \linebreak
\noindent\hyperlink{910_lie_algebras_of_matrices}{9.10 Lie Algebras of Matrices}\dotfill \pageref*{910_lie_algebras_of_matrices} \linebreak
\noindent\hyperlink{10_the_derived_category}{10 The derived category}\dotfill \pageref*{10_the_derived_category} \linebreak
\noindent\hyperlink{a_category_theory_language}{A Category Theory Language}\dotfill \pageref*{a_category_theory_language} \linebreak
\noindent\hyperlink{a1_categories}{A.1 Categories}\dotfill \pageref*{a1_categories} \linebreak
\noindent\hyperlink{a2_functors}{A.2 Functors}\dotfill \pageref*{a2_functors} \linebreak
\noindent\hyperlink{a3_natural_transformations}{A.3 Natural transformations}\dotfill \pageref*{a3_natural_transformations} \linebreak
\noindent\hyperlink{a4_abelian_categories}{A.4 Abelian categories}\dotfill \pageref*{a4_abelian_categories} \linebreak
\noindent\hyperlink{a5_limits_and_colimits}{A.5 Limits and colimits}\dotfill \pageref*{a5_limits_and_colimits} \linebreak
\noindent\hyperlink{a6_adjoint_functors}{A.6 Adjoint functors}\dotfill \pageref*{a6_adjoint_functors} \linebreak


\hypertarget{1_chain_complexes}{}\subsection*{{1 Chain complexes}}\label{1_chain_complexes}

\hypertarget{11_complexes_of_modules}{}\subsubsection*{{1.1 Complexes of $R$-modules}}\label{11_complexes_of_modules}

\begin{itemize}%
\item [[abelian group]], [[commutative ring]], [[module]]


\item [[exact sequence]]



\end{itemize}
\textbf{Definition 1.1.1} [[chain complex]]

\begin{itemize}%
\item [[chain map]], [[chain homotopy]]


\item [[category of chain complexes]]



\end{itemize}
\textbf{Exercise 1.1.2} \href{chain+map#OnHomology}{homology is functorial}

\textbf{Exercise 1.1.3} \href{split+exact+sequence#OfVectorSpaces}{exact sequences of chain complexes are split}

\textbf{Exercise 1.1.4} [[internal hom of chain complexes]]

\textbf{Definition 1.1.2} [[quasi-isomorphism]]

[[cochain complex]], [[bounded chain complex]]

\textbf{Exercise 1.1.5} \href{exact+sequence#ExactnessAndQuasiIsomorphisms}{exactness and weak nullity}

\textbf{Application 1.1.3} [[chain on a simplicial set]], [[simplicial homology]]

\textbf{Exercise 1.17} \href{simplicial+homology#OfTetrahedon}{simplicial homology of the tetrahedron}

\textbf{Application 1.1.4} [[singular homology]]

\hypertarget{12_operations_on_chain_complexes}{}\subsubsection*{{1.2 Operations on chain complexes}}\label{12_operations_on_chain_complexes}

\begin{itemize}%
\item [[additive and abelian categories]]


\item [[Ab-enriched category]]

\begin{itemize}%
\item [[pre-additive category]]


\item [[additive category]],

\begin{itemize}%
\item [[additive functor]]

\end{itemize}


\end{itemize}


\end{itemize}
\textbf{Exercise 1.2.1} \href{chain+homology+and+cohomology#ChainHomologyRespectsDirectProduct}{homology respects direct product}

\textbf{Definition 1.2.1} [[kernel]], [[cokernel]]

\begin{itemize}%
\item [[quotient]]

\end{itemize}
\textbf{Exercise 1.2.2} \href{Mod#RModIsAbelian}{in an abelian category kernels/cokernels are the monos/epis}

\textbf{Exercise 1.2.3} \href{category+of+chain+complexes#KernelsOfChainComplexes}{(co)kernels of chain maps are degreewise (co)kernels}

\textbf{Definition 1.2.2} [[abelian category]], [[abelian subcategory]]

\textbf{Theorem 1.2.3} \href{category+of+chain+complexes#IsAbelian}{a category of chain complexes is itself abelian}

\textbf{Exercise 1.2.4} \href{category+of+chain+complexes#ShortExactSequencesDegreewise}{exact sequence of chain complexes is degreewise exact}

$R$[[Mod]]

\textbf{Example 1.2.4} [[double complex]]

\textbf{Sing trick 1.2.5} \href{double+complex#EquivalenceOfTheTwoDefinitions}{double complex with commuting/anti-commuting differentials}

\textbf{Total complex 1.2.6} [[total complex]]

\textbf{Exercise 1.2.5} \href{total+complex#TotOfBoundedDegreewiseExactIsExact}{total complex of a bounded degreewise exact double complex is itself exact}

\textbf{Example 1.2.4} [[double complex]]

\textbf{Truncations 1.2.7} [[truncation of a chain complex]]

\textbf{Translation 1.2.8} [[suspension of a chain complex]]

\textbf{Exercise 1.2.8} [[mapping cone]]

\hypertarget{13_long_exact_sequences}{}\subsubsection*{{1.3 Long exact sequences}}\label{13_long_exact_sequences}

\textbf{Theorem 1.3.1} [[connecting homomorphism]], [[long exact sequences in homology]]

\textbf{Exercise 1.3.1} [[3x3 lemma]],

\textbf{Snake lemma 1.3.2} [[snake lemma]]

\textbf{Exercise 1.3.3} [[5 lemma]]

\textbf{Remark 1.3.5} [[exact triangle]]

\hypertarget{14_chain_homotopies}{}\subsubsection*{{1.4 Chain homotopies}}\label{14_chain_homotopies}

\begin{itemize}%
\item [[homotopy theory]]

\end{itemize}
\textbf{Definition 1.4.1} [[split exact sequence]]

\textbf{Exercise 1.4.1} \href{split+exact+sequence#OfVectorSpaces}{splitness of exact sequences of free modules}

\textbf{Definition 1.4.3} [[null homotopy]]

\textbf{Exercise 1.4.3} \href{split+exact+sequence#RelationToChainHomotopy}{split exact means identity is null homotopic}

\textbf{Definition 1.4.4} [[chain homotopy]]

\textbf{Lemma 1.4.5} \href{chain%20map#OnHomology}{chain homotopy respects homology}

\textbf{Exercise 1.4.5} [[homotopy category of chain complexes]]

\hypertarget{15_mapping_cones_and_cyclinders}{}\subsubsection*{{1.5 Mapping cones and cyclinders}}\label{15_mapping_cones_and_cyclinders}

\textbf{1.5.1} [[mapping cone]]

\textbf{1.5.5} [[mapping cylinder]]

\textbf{1.5.8} [[fiber sequence]]

\hypertarget{16_more_on_abelian_categories}{}\subsubsection*{{1.6 More on abelian categories}}\label{16_more_on_abelian_categories}

\textbf{Theorem 1.6.1} [[Freyd-Mitchell embedding theorem]]

\textbf{Functor categories 1.6.4} [[functor category]]

[[presheaf]]

\textbf{Definition 1.6.5} [[abelian sheaf]]

\textbf{Definition 1.6.6} left/right [[exact functor]]

\textbf{Yoneda embedding 1.6.10} [[Yoneda embedding]]

\textbf{Yoneda lemma 1.6.11} [[Yoneda lemma]]

\href{http://ncatlab.org/nlab/show/Freyd-Mitchell+embedding+theorem#Proof}{proof of the Freyd-Mitchell embedding theorem}

\hypertarget{2_derived_functors}{}\subsection*{{2 Derived functors}}\label{2_derived_functors}

[[derived functor in homological algebra]]

\hypertarget{21_functor}{}\subsubsection*{{2.1 $\delta$-Functor}}\label{21_functor}

\textbf{Definition 2.1.1} [[delta-functor]]

\hypertarget{22_projective_resolutions}{}\subsubsection*{{2.2 Projective resolutions}}\label{22_projective_resolutions}

[[projective module]] ([[cofibrant object]] in the [[model structure on chain complexes]])

\textbf{Definition 2.2.4} [[projective resolution]] ([[cofibrant replacement]])

\textbf{Horseshoe lemma 2.2.8} [[horseshoe lemma]]

\hypertarget{23_injective_resolutions}{}\subsubsection*{{2.3 Injective resolutions}}\label{23_injective_resolutions}

[[injective module]] ([[fibrant object]] in the other [[model structure on chain complexes]])

\textbf{Baer's criterion 2.3.1} [[Baer's criterion]]

\textbf{Definition 2.3.5} [[injective resolution]] ([[fibrant replacement]])

\textbf{Definition 2.3.9} [[adjoint functor]]

\hypertarget{24_left_derived_functors}{}\subsubsection*{{2.4 Left derived functors}}\label{24_left_derived_functors}

[[left derived functor]]

\hypertarget{25_right_derived_functors}{}\subsubsection*{{2.5 Right derived functors}}\label{25_right_derived_functors}

[[right derived functor]]

\textbf{Application 2.5.4} [[global section functor]], [[abelian sheaf cohomology]]

\hypertarget{26_adjoint_functors_and_leftright_exactness}{}\subsubsection*{{2.6 Adjoint functors and left/right exactness}}\label{26_adjoint_functors_and_leftright_exactness}

[[adjoint functor]]

\textbf{Definition 2.6.4} [[Tor]]

\textbf{Application 2.6.5} [[sheafification]]

\textbf{Application 2.6.6} [[direct image]], [[inverse image]]

\textbf{Application 2.6.7} [[colimit]]

\textbf{Variation 2.6.9} [[limit]]

\textbf{Definition 2.6.13} [[filtered category]], [[filtered colimit]]

\hypertarget{27_balancing__and_}{}\subsubsection*{{2.7 Balancing $Tor$ and $Ext$}}\label{27_balancing__and_}

\textbf{Tensor product of complexes 2.7.1} [[tensor product of chain complexes]]

\textbf{Lemma 2.7.3} [[acyclic assembly lemma]]

\hypertarget{3_tor_and_ext}{}\subsection*{{3 Tor and Ext}}\label{3_tor_and_ext}

[[Tor]] and [[Ext]]

\hypertarget{31__for_abelian_groups}{}\subsubsection*{{3.1 $Tor$ for abelian groups}}\label{31__for_abelian_groups}

\textbf{Proposition 3.1.2-3.1.3} \href{Tor#RelationToTorsionGroups}{relation to torsion subgroups}

\hypertarget{32__and_flatness}{}\subsubsection*{{3.2 $Tor$ and flatness}}\label{32__and_flatness}

\textbf{Definition 3.2.1} [[flat module]]

\textbf{Definition 3.2.3} [[Pontrjagin duality]]

\textbf{Flat resolution lemma 3.2.8} [[flat resolution lemma]]

\textbf{Corollary 3.2.13} \href{Tor#Localization}{Localization for Tor}

\hypertarget{33__for_nice_rings}{}\subsubsection*{{3.3 $Ext$ for nice rings}}\label{33__for_nice_rings}

\textbf{Corollary 3.3.11} \href{Ext#Localization}{Localization for Ext}

\hypertarget{34__and_extensions}{}\subsubsection*{{3.4 $Ext$ and extensions}}\label{34__and_extensions}

[[extension]]

[[group extension]]

\textbf{Vista 3.4.6} [[Yoneda extension group]]

\hypertarget{35_derived_functors_of_the_inverse_limit}{}\subsubsection*{{3.5 Derived functors of the inverse limit}}\label{35_derived_functors_of_the_inverse_limit}

[[tower]]

[[additive and abelian categories|(AB4)-category]]

[[directed limit]]

\textbf{Definition 3.5.1} [[lim{\tt \symbol{94}}1]]

\textbf{Definition 3.5.6} [[Mittag-Leffler condition]]

\textbf{Exercise 3.5.5} [[pullback]]

\hypertarget{36_universal_coefficient_theorem}{}\subsubsection*{{3.6 Universal coefficient theorem}}\label{36_universal_coefficient_theorem}

\textbf{Theorem 2.6.1} [[Künneth formula]]

\textbf{Universal cofficient theorem for homology 3.6.2} \href{universal+coefficient+theorem#InHomology}{universal coefficient theorem in homology}

\textbf{Theorem 3.6.3} [[Künneth formula for complexes]]

\textbf{Application 3.6.4} \href{universal+coefficient+theorem#InTopology}{universal coefficient theorem in topology}

\textbf{Universal coefficient theorem in cohomology 3.6.5} \href{universal+coefficient+theorem#InCohomology}{universal coefficient theorem in cohomology}

[[Eilenberg-Zilber theorem]]

\textbf{Exercise 3.6.2} [[hereditary ring]]

\hypertarget{4_homological_dimension}{}\subsection*{{4 Homological dimension}}\label{4_homological_dimension}

\hypertarget{41_dimensions}{}\subsubsection*{{4.1 Dimensions}}\label{41_dimensions}

\begin{itemize}%
\item [[dimension]]


\item [[global dimension theorem]]


\item [[homological dimension]]



\end{itemize}
\hypertarget{42_rings_of_small_dimension}{}\subsubsection*{{4.2 Rings of Small Dimension}}\label{42_rings_of_small_dimension}

\hypertarget{43_change_of_rings_theorem}{}\subsubsection*{{4.3 Change of Rings Theorem}}\label{43_change_of_rings_theorem}

\hypertarget{44_local_rings}{}\subsubsection*{{4.4 Local rings}}\label{44_local_rings}

\begin{itemize}%
\item [[local ring]]

\end{itemize}
\hypertarget{45_koszul_complexes}{}\subsubsection*{{4.5 Koszul Complexes}}\label{45_koszul_complexes}

\begin{itemize}%
\item [[Koszul complex]]

\end{itemize}
\hypertarget{46_local_cohomology}{}\subsubsection*{{4.6 Local Cohomology}}\label{46_local_cohomology}

\hypertarget{5_spectral_sequences}{}\subsection*{{5 Spectral sequences}}\label{5_spectral_sequences}

\hypertarget{51_introduction}{}\subsubsection*{{5.1 Introduction}}\label{51_introduction}

\hypertarget{52_terminology}{}\subsubsection*{{5.2 Terminology}}\label{52_terminology}

\begin{itemize}%
\item [[spectral sequence]]

\end{itemize}
\hypertarget{53_lerayserre_spectral_sequence}{}\subsubsection*{{5.3 Leray-Serre Spectral Sequence}}\label{53_lerayserre_spectral_sequence}

\begin{itemize}%
\item [[Leray-Serre spectral sequence]]

\end{itemize}
\hypertarget{54_spectral_sequence_of_a_filtration}{}\subsubsection*{{5.4 Spectral sequence of a filtration}}\label{54_spectral_sequence_of_a_filtration}

\begin{itemize}%
\item [[spectral sequence of a filtration]]

\end{itemize}
\hypertarget{55_convergence}{}\subsubsection*{{5.5 Convergence}}\label{55_convergence}

\hypertarget{56_spectral_sequence_of_a_double_complex}{}\subsubsection*{{5.6 Spectral sequence of a double complex}}\label{56_spectral_sequence_of_a_double_complex}

\begin{itemize}%
\item [[spectral sequence of a double complex]]

\end{itemize}
\hypertarget{57_hypercohomology}{}\subsubsection*{{5.7 Hypercohomology}}\label{57_hypercohomology}

\begin{itemize}%
\item [[hypercohomology]]

\end{itemize}
\hypertarget{58_grothendieck_spectral_sequence}{}\subsubsection*{{5.8 Grothendieck spectral sequence}}\label{58_grothendieck_spectral_sequence}

\begin{itemize}%
\item [[Grothendieck spectral sequence]]

\end{itemize}
\hypertarget{59_exact_couples}{}\subsubsection*{{5.9 Exact couples}}\label{59_exact_couples}

\begin{itemize}%
\item [[exact couple]]


\item [[Bockstein spectral sequence]]



\end{itemize}
\hypertarget{6_group_homology_and_cohomology}{}\subsection*{{6 Group homology and cohomology}}\label{6_group_homology_and_cohomology}

\begin{itemize}%
\item [[group cohomology]]

\end{itemize}
\hypertarget{7_lie_algebra_homology_and_cohomology}{}\subsection*{{7 Lie algebra homology and cohomology}}\label{7_lie_algebra_homology_and_cohomology}

\begin{itemize}%
\item [[Lie algebra homology]], [[Lie algebra cohomology]]

\end{itemize}
\hypertarget{8_simplicial_methods_in_homological_algebra}{}\subsection*{{8 Simplicial methods in homological algebra}}\label{8_simplicial_methods_in_homological_algebra}

\hypertarget{81_simplicial_object}{}\subsubsection*{{8.1 Simplicial object}}\label{81_simplicial_object}

\begin{itemize}%
\item [[simplicial object]]

\end{itemize}
\hypertarget{82_operations_on_simplicial_objects}{}\subsubsection*{{8.2 Operations on simplicial objects}}\label{82_operations_on_simplicial_objects}

\hypertarget{83_simplicial_homotopy_groups}{}\subsubsection*{{8.3 Simplicial homotopy groups}}\label{83_simplicial_homotopy_groups}

\begin{itemize}%
\item [[simplicial homotopy group]]

\end{itemize}
\hypertarget{84_the_doldkan_correspondence}{}\subsubsection*{{8.4 The Dold-Kan correspondence}}\label{84_the_doldkan_correspondence}

\begin{itemize}%
\item [[Dold-Kan correspondence]]

\end{itemize}
\hypertarget{85_the_eilenbergzilber_theorem}{}\subsubsection*{{8.5 The Eilenberg-Zilber theorem}}\label{85_the_eilenbergzilber_theorem}

\begin{itemize}%
\item [[Eilenberg-Zilber theorem]]

\end{itemize}
\hypertarget{86_canonical_resolutions}{}\subsubsection*{{8.6 Canonical resolutions}}\label{86_canonical_resolutions}

\begin{itemize}%
\item [[monad]](=triple) [[comonad]]

[[bar construction]],

[[classifying space]]



\end{itemize}
\hypertarget{87_cotriple_homology}{}\subsubsection*{{8.7 Cotriple homology}}\label{87_cotriple_homology}

\begin{itemize}%
\item [[bar construction]]


\item [[monadic descent]]



\end{itemize}
\hypertarget{88_andrequillen_homology_and_cohomology}{}\subsubsection*{{8.8 Andre-Quillen Homology and Cohomology}}\label{88_andrequillen_homology_and_cohomology}

\begin{itemize}%
\item [[Kähler differential]]


\item [[cotangent complex]]



\end{itemize}
\hypertarget{9_hochschild_and_cyclic_homology}{}\subsection*{{9 Hochschild and cyclic homology}}\label{9_hochschild_and_cyclic_homology}

\hypertarget{91_hochschild_homology_and_cohomology_of_algebras}{}\subsubsection*{{9.1 Hochschild Homology and Cohomology of Algebras}}\label{91_hochschild_homology_and_cohomology_of_algebras}

\begin{itemize}%
\item [[Hochschild cohomology]]

\end{itemize}
\hypertarget{92_derivations_differentials_and_separable_algebra}{}\subsubsection*{{9.2 Derivations, Differentials and Separable Algebra}}\label{92_derivations_differentials_and_separable_algebra}

\begin{itemize}%
\item [[derivation]]


\item [[differential]]


\item [[separable algebra]]



\end{itemize}
\hypertarget{93__extensions_smooth_algebras}{}\subsubsection*{{9.3 $H^2$, Extensions, Smooth Algebras}}\label{93__extensions_smooth_algebras}

\begin{itemize}%
\item [[smooth algebra]]

\end{itemize}
\hypertarget{94_hochschild_products}{}\subsubsection*{{9.4 Hochschild products}}\label{94_hochschild_products}

\begin{itemize}%
\item [[Hochschild-Kostant-Rosenberg theorem]]

\end{itemize}
\hypertarget{95_morita_invariance}{}\subsubsection*{{9.5 Morita Invariance}}\label{95_morita_invariance}

\hypertarget{96_cyclic_homology}{}\subsubsection*{{9.6 Cyclic Homology}}\label{96_cyclic_homology}

\begin{itemize}%
\item [[cyclic cohomology]]

\end{itemize}
\hypertarget{97_group_rings}{}\subsubsection*{{9.7 Group Rings}}\label{97_group_rings}

\hypertarget{98_mixed_complexes}{}\subsubsection*{{9.8 Mixed Complexes}}\label{98_mixed_complexes}

\hypertarget{99_graded_algebras}{}\subsubsection*{{9.9 Graded Algebras}}\label{99_graded_algebras}

\hypertarget{910_lie_algebras_of_matrices}{}\subsubsection*{{9.10 Lie Algebras of Matrices}}\label{910_lie_algebras_of_matrices}

\hypertarget{10_the_derived_category}{}\subsection*{{10 The derived category}}\label{10_the_derived_category}

\begin{itemize}%
\item [[derived category]]

\end{itemize}
\hypertarget{a_category_theory_language}{}\subsection*{{A Category Theory Language}}\label{a_category_theory_language}

\begin{itemize}%
\item [[category theory]]

\end{itemize}
\hypertarget{a1_categories}{}\subsubsection*{{A.1 Categories}}\label{a1_categories}

\begin{itemize}%
\item [[category]]

\end{itemize}
\hypertarget{a2_functors}{}\subsubsection*{{A.2 Functors}}\label{a2_functors}

\begin{itemize}%
\item [[functor]]

\end{itemize}
\hypertarget{a3_natural_transformations}{}\subsubsection*{{A.3 Natural transformations}}\label{a3_natural_transformations}

\begin{itemize}%
\item [[natural transformation]]

\end{itemize}
\hypertarget{a4_abelian_categories}{}\subsubsection*{{A.4 Abelian categories}}\label{a4_abelian_categories}

\begin{itemize}%
\item [[abelian category]]

\end{itemize}
\hypertarget{a5_limits_and_colimits}{}\subsubsection*{{A.5 Limits and colimits}}\label{a5_limits_and_colimits}

\begin{itemize}%
\item [[limit]], [[colimit]]

\end{itemize}
\hypertarget{a6_adjoint_functors}{}\subsubsection*{{A.6 Adjoint functors}}\label{a6_adjoint_functors}

\begin{itemize}%
\item [[adjoint functor]]

\end{itemize}
category: reference

[[!redirects An introduction to homological algebra]]



\end{document}