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\begin{document}

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\section*{Grothendieck context}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - 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{QuasicoeherntSheavesOnSchemes}{Quasicoherent sheaves on schemes}\dotfill \pageref*{QuasicoeherntSheavesOnSchemes} \linebreak
\noindent\hyperlink{quasicoherent_sheaves_in_geometry}{Quasicoherent sheaves in $E_\infty$-geometry}\dotfill \pageref*{quasicoherent_sheaves_in_geometry} \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}

A \emph{Grothendieck context} is a pair of two [[symmetric monoidal categories]] $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an [[adjoint triple]] of [[functors]] such that the leftmost one is a [[closed monoidal functor]].

This is the variant/special case of the [[yoga of six operations]] with two [[adjoint pairs]] $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ for $f_! \simeq f_\ast$.

\begin{displaymath}
f^\ast \dashv (f_\ast = f_!) \dashv f^!
  \;\colon\;
  \mathcal{X}
   \;
    \itexarray{
     \overset{f^\ast}{\longleftarrow}
     \\
     \overset{f_\ast = f_! }{\longrightarrow}
     \\
     \overset{f^!}{\longleftarrow}
    }
    \;
  \mathcal{Y}
  \,.
\end{displaymath}
(The other specialization of the [[six operations]] where $f^\ast \simeq f^!$ is called the \emph{[[Wirthmüller context]]}).

The existence of the (derived) right adjoint $f^!$ to $f_\ast$ is what is called \emph{[[Grothendieck duality]]}.

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

\hypertarget{QuasicoeherntSheavesOnSchemes}{}\subsubsection*{{Quasicoherent sheaves on schemes}}\label{QuasicoeherntSheavesOnSchemes}

A [[homomorphism]] of [[schemes]] $f \;\colon\; X \longrightarrow Y$ induces an [[inverse image]] $\dashv$ [[direct image]] [[adjunction]] on the [[derived categories]] $QCoh(-)$ of [[quasicoherent sheaves]]

\begin{displaymath}
(f^\ast \dashv f_\ast) 
   \;\colon\;
   QCoh(X)
    \underoverset 
     \overset{f^\ast}{\longleftarrow}
     \overset{f_\ast}{\longrightarrow}
     {\bot}
   QCoh(Y)
  \,.
\end{displaymath}
(all [[derived functors]]) If $f$ is a [[proper morphism of schemes]] then under mild further conditions there is a further [[right adjoint]] $f^!$

\begin{displaymath}
(f^\ast \dashv f_\ast \dashv f^!) 
   \;\colon\;
   QCoh(X)
    \;
    \itexarray{
      \overset{f^\ast}{\longleftarrow}
      \\
      \overset{f_\ast}{\longrightarrow}
      \\
      \overset{f^!}{\longleftarrow}
     }
   \;
   QCoh(Y)
  \,.
\end{displaymath}
This is originally due to [[Grothendieck]], whence the name. Refined accounts are in (\hyperlink{Deligne66}{Deligne 66}, \href{Verdier68}{Verdier 68}, \hyperlink{Neeman96}{Neeman 96}).

\hypertarget{quasicoherent_sheaves_in_geometry}{}\subsubsection*{{Quasicoherent sheaves in $E_\infty$-geometry}}\label{quasicoherent_sheaves_in_geometry}

Generalization of the pull-push [[adjoint triple]] to [[E-∞ geometry]] is in (\hyperlink{LurieQC}{LurieQC, prop. 2.5.12}) and the [[projection formula]] for this is in (\hyperlink{LurieProper}{LurieProp, remark 1.3.14}).

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

\begin{itemize}%
\item [[Wirthmüller context]]


\item [[Verdier-Grothendieck context]]



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

The original construction for \hyperlink{QuasicoeherntSheavesOnSchemes}{quasicoherent sheaves on schemes} is due to [[Alexander Grothendieck]], whence the name ``Grothendieck context''.

Further stream-lined accounts then appeared in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Cohomology \`a{} support propre en construction du foncteur $f^!$}, Appendix to: \emph{Residues and Duality}, Lecture Notes in Math., vol. 20, Springer-Verlag, Heidelberg, 1966, pp. 404\{421. MR 36:5145

\end{itemize}
\begin{itemize}%
\item [[Jean-Louis Verdier]], \emph{Base change for twisted inverse images of coherent sheaves}, Collection: Algebraic Geometry (Internat. Colloq.), Tata Inst. Fund. Res., Bombay, 1968, pp. 393-408. MR 43:227

\end{itemize}
Further refinement and highlighting of the close relation to the \href{Brown+representability+theorem#CategoricalBrownRepresentability}{categorical Brown representability theorem} is in

\begin{itemize}%
\item [[Amnon Neeman]], \emph{The Grothendieck duality theorem via Bousfield's techniques and Brown representability}, J. Amer. Math. Soc. 9 (1996), 205-236 (\href{http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/}{web})

\end{itemize}
Discussion of [[integral transforms]] in Grothendieck contexts is in

\begin{itemize}%
\item [[Alexander Polishchuk]], \emph{Kernel algebras and generalized Fourier-Mukai transforms} (\href{http://arxiv.org/abs/0810.1542}{arXiv:0810.1542})

\end{itemize}
Generalization of the pull-push [[adjoint triple]] to [[E-∞ geometry]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 2.5 of \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}

\end{itemize}
and the [[projection formula]] for this triple appears as remark 1.3.14 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]}

\end{itemize}
A clear discussion of axioms of [[six operations]], their specialization to Grothendieck context and [[Wirthmüller context]] and their consequences is in

\begin{itemize}%
\item H. Fausk, P. Hu, [[Peter May]], \emph{Isomorphisms between left and right adjoints}, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (\href{http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html}{TAC}, \href{http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], \emph{Grothendieck-Neeman duality and the Wirthm\"u{}ller isomorphism}, \href{http://arxiv.org/abs/1501.01999}{arXiv:1501.01999}.

\end{itemize}
[[!redirects Grothendieck contexts]]



\end{document}