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\begin{document}

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\section*{homological resolution}

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

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

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[homological algebra]] a \emph{homological resolution} of a [[vector space]] or more generally of a [[module]] $N$ is a [[chain complex]] $V_\bullet$ whose [[chain homology]] $H_\bullet(V)$ reproduces $N$, in that

\begin{displaymath}
H_n(V)
  \;\simeq\;
  \left\{
    \itexarray{
       N &\vert& n = 0
       \\
       0 &\vert& \text{otherwise}
    }
  \right.
\end{displaymath}
Of course if one regards $N$ itself as a chain complex $N_\bullet$ which is concentrated in degree 0, then $N_\bullet$ has this property, trivially. Therefore, typically when asking for a homological resolution $V_\bullet$ of $N$, it is understood that the entries of $V_\bullet$ have certain nice properties that $N$ is lacking.

For instance if $V_\bullet$ consists of [[projective modules]], then it is called a \emph{[[projective resolution]]} of $N$, or if it consists of [[injective modules]] then it is called an \emph{[[injective resolution]]}, or if it consists of [[free modules]], then it is called a \emph{[[free resolution]]}, etc. The module $N$ itself may be far from being projective or injective or free, etc. and so the corresponding resolution, if it exists, allows to nevertheless regard $N$ with tools applicable to these particularly nice classes of modules, up to chain homology

Typically in these constructions one demands not just that the [[chain homology]] of the resolution reproduces $N$ via \emph{any} isomorphism $H_0(V) \simeq N$, but that there exists a [[chain map]] $f_\bullet$ from $V$ to $N$ or from $N$ to $V$, such that it is this chain map which under passage to [[chain homology]] induces this isomorphism. A chain map which induces isomorphisms on chain homology groups is called a \emph{[[quasi-isomorphism]]}, and so typically a \emph{homological resolution} means a choice of quasi-isomorphism from a module to a (particularly nice) chain complex.

Famous classes of examples of such resolutions are the injective and projective resolutions that are used to construct \emph{[[derived functors in homological algebra]]}, see there for more.

Phrased this way, there is nothing special about starting with a single module, and more generally one may speak of [[resolutions]] of one chain complex by another chain complex with some better properties. The theory of [[homotopical categories]] such as [[model categories]] or [[fibration categories]]/[[cofibration categories]] is used to handle homological resolutions in this generality, see at \emph{[[model structure on chain complexes]]} for more on this.

Viewed from this general perspective of [[homotopical categories]], homological resolutions are a special case of the general concept of [[resolutions]] in [[homotopy theory]]. (In fact [[homological algebra]] may be understood as a fragment of [[stable homotopy theory]], see at \emph{[[Dold-Kan correspondence]]} and \emph{[[stable Dold-Kan correspondence]]} for more on this.)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The ur examples of homological resolutions are the [[Koszul complexes]] or more generally [[Koszul-Tate resolutions]] of an $A$-[[module]] by [[free modules]] ove or of an $A$-[[associative algebra|algebra]] by free $A$-algebras.

For example, consider $A$ is a commutative [[augmented algebra]] over a [[field]] $k$ and $I$ an ideal in $A$. Resolve the [[quotient]] $A/I$ by a free $A$-algebra $R$ with a [[derivation]] [[differential]] so that $H(R) = A/I$ is concentrated in degree $0$ and all other homology vanishes. Iff $I$ is a [[regular ideal]], the [[Koszul complex]] will do; if $I$ is not regular, continue the process forming the [[Koszul-Tate resolution]], the algebraic analog of a [[Moore-Postnikov system]], which was indeed Tate's inspiration.

If the original object is itself [[graded object|graded]] or differential graded, the resolution will be bigraded by resolution degree and \emph{internal degree}.

By \textbf{homological resolution of a quotient}, one means a \emph{weak quotient} or \emph{homotopy quotient} in some $\infty$-categorical [[homotopy theory]] context which is equivalent to a category of (co)chain complexes. This means: a homological resolution of a quotient is a (co)chain complex $C^\bullet$ of abelian groups whose \emph{(co)homology} $H^\bullet(C^\bullet)$ in some degree, usually in degree 0, is the desired quotient, $H^0(C) = desired quotient$. Depending on the situation one may want to demand that the (co)homology in all other degrees vanish, in which case $C^\bullet$ would be [[homotopy theory|weakly equivalent]] to the desired quotient.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

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

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion for an audience of physicists in the context of [[BV-BRST formalism]] is in

\begin{itemize}%
\item [[Marc Henneaux]], [[Claudio Teitelboim]], section 8.3 of \emph{[[Quantization of Gauge Systems]]}, Princeton University Press, 1992

\end{itemize}
[[!redirects homological resolutions]]

[[!redirects Homological resolution]] [[!redirects Homological resolutions]]



\end{document}