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\begin{document}

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\section*{pure morphism}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{pure_morphisms}{}\section*{{Pure morphisms}}\label{pure_morphisms}

\noindent\hyperlink{in_general_categorical_setup}{In general categorical setup}\dotfill \pageref*{in_general_categorical_setup} \linebreak
\noindent\hyperlink{in_ring_theory_and_for_schemes}{In ring theory and for schemes}\dotfill \pageref*{in_ring_theory_and_for_schemes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{in_general_categorical_setup}{}\subsection*{{In general categorical setup}}\label{in_general_categorical_setup}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[regular cardinal]] $\kappa$, a [[morphism]] $f: A\to B$ in a [[category]] $C$ is \textbf{$\kappa$-pure} (or \textbf{$\kappa$-universally injective}) if for every [[commutative square]]

\begin{displaymath}
\itexarray{
P & \stackrel{g}\to & Q\\
u\downarrow && \downarrow v \\
A &\stackrel{f}\to & B
}
\end{displaymath}
in which $P$ and $Q$ are $\kappa$-[[presentable objects]], the morphism $u:P\to A$ factors through $g$, i.e. there is some $h: Q\to A$ with $u = h\circ g$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Notice that the above definition does not require that also the morphism $v$ is factored, hence it does \emph{not} express a [[lifting property]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
In a $\kappa$-[[accessible category]] $C$ every $\kappa$-pure morphism is [[monic]], hence exhibits a [[pure subobject]]. In a locally $\kappa$-[[locally presentable category|presentable category]] $\kappa$-pure morphisms are, moreover, [[regular monomorphisms]], and in fact coincide with the $\kappa$-[[directed colimits]] of [[split monomorphism]]s in the category of arrows $C^2 = Arr(C)$; more generally this characterization holds in all $\kappa$-accessible categories admiting [[pushouts]].

\end{prop}
(\hyperlink{AHT96}{Adámek-Hub-Tholen 1996}).

\hypertarget{in_ring_theory_and_for_schemes}{}\subsection*{{In ring theory and for schemes}}\label{in_ring_theory_and_for_schemes}

We work with unital, possibly commutative, [[rings]] and [[modules]]. Given a ring $R$, a morphism $f: M\to M'$ of left $R$-modules is pure if the tensoring the [[exact sequence]] of left $R$-modules

\begin{displaymath}
0\to Ker f \to M\stackrel{f}\to M'\to Coker f\to 0
\end{displaymath}
with any right $R$-module $N$ (from the left) yields an exact sequence of abelian groups.

Grothendieck has proved that faithfully flat morphisms of commutative schemes are of effective descent for the categories of quasicoherent $\mathcal{O}$-modules. But this was not entirely optimal, as there is in fact a more general class than faithfully flat morphisms which satisfy the effective descent. For a local case of commutative rings, [[Joyal]] and [[Tierney]] have then proved (unpublished) that the [[effective descent morphisms]] for modules are precisely the pure morphisms of rings (or dually of affine schemes). The result can be extracted also from their Memoirs volume on Galois theory. [[Janelidze]] and [[Tholen]] have reproved the theorem as a corollary of a result for noncommutative rings obtained using the Beck's [[comonadicity theorem]].

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

\begin{itemize}%
\item [[pure subobject]]

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

\begin{itemize}%
\item The \href{https://stacks.math.columbia.edu}{Stacks Project}, 34.4. Descent for universally injective morphisms, \href{https://stacks.math.columbia.edu/tag/08WE}{tag/08WE}, 28.10. Radicial and universally injective morphisms (of schemes) \href{https://stacks.math.columbia.edu/tag/01S2}{tag/01S2}


\item Bachuki Mesablishvili, \emph{Pure morphisms of commutative rings are effective descent morphisms for modules -- a new proof}, Theory and Appl. of Categories \textbf{7}, 2000, No. 3, 38-42, \href{http://www.tac.mta.ca/tac/volumes/7/n3/7-03abs.html}{tac}


\item [[T. Brzeziński]], R. Wisbauer, \emph{Corings and comodules}, London Math. Soc. Lec. Note Series \textbf{309}, Cambridge Univ. Press 2003; \href{http://maths.swan.ac.uk/staff/tb/corinerr.pdf}{errata pdf}


\item [[George Janelidze]], [[Walter Tholen]], \emph{Facets of descent III: monadic descent for rings and algebras}, Appl. Categ. Structures \textbf{12} (2004), no. 5-6, 461--477, \href{http://www.ams.org/mathscinet-getitem?mr=2107397}{MR2005i:13019}, \href{http://dx.doi.org/10.1023/B:APCS.0000049312.36783.0a}{doi}


\item [[Jiří Adámek]], H. Hub, [[Walter Tholen]], \emph{On pure morphisms in accessible categories}, J. Pure Appl. Alg. \textbf{107}, 1 (1996), pp 1-8, 



\end{itemize}
\begin{itemize}%
\item Michel H\'e{}bert, \emph{Purity and injectivity in accessible categories}, 


\item W.W. Crawley-Boevey, \emph{Locally finitely presented additive categories}, Communications in Algebra \textbf{22}(5)(1994), 1641-1674.


\item [[Mike Prest]], \emph{Purity, spectra and localisation}, Enc. Math. Appl. \textbf{121}, Camb. Univ. Press 2011, 798 pages; publishers book \href{http://www.cambridge.org/gb/knowledge/isbn/item2327409/?site_locale=en_GB}{page}


\item Christian U. Jensen, Helmut Lenzing, \emph{Model theoretic algebra: with particular emphasis on fields, rings, modules}, Algebra, Logic and Applications \textbf{2}, Gordon and Breach 1989.


\item Ivo Herzog, \emph{Pure-injective envelopes}, Journal of Algebra and Its Applications 2(4) (2003), 397-402 \href{http://lima.osu.edu/people/iherzog/env.pdf}{pdf}


\item [[André Joyal]], [[Myles Tierney]], \emph{An extension of the Galois theory of Grothendieck}, Mem. Amer. Math. Soc. \textbf{309} (1984) volume 51, pages vii+71



\end{itemize}
The following paper was the first with the result on that pure morhisms are of the effective descent but the proof has been omitted:

\begin{itemize}%
\item Jean-Pierre Olivier, \emph{Descente par morphismes purs}, C. R. Acad. Sci. Paris Sér. A-B \textbf{271} (1970) A821–A823

\end{itemize}
[[!redirects pure morphisms]] [[!redirects purity]] [[!redirects universally injective morphism]]



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