\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Grothendieck duality}

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

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

[[!include duality - contents]]

\hypertarget{grothendieck_duality}{}\section*{{Grothendieck duality}}\label{grothendieck_duality}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{dualizing_complexes}{Dualizing Complexes}\dotfill \pageref*{dualizing_complexes} \linebreak
\noindent\hyperlink{rigid_dualizing_complexes}{Rigid Dualizing Complexes}\dotfill \pageref*{rigid_dualizing_complexes} \linebreak
\noindent\hyperlink{noncommutative_grothendieck_duality}{Noncommutative Grothendieck Duality}\dotfill \pageref*{noncommutative_grothendieck_duality} \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}

Given a [[homomorphism]] $f$ of [[schemes]], one says that it satisfies \emph{Grothendieck duality} if the ([[derived functor|derived]]) [[direct image]] functor $f_\ast$ on [[quasicoherent sheaves]] has a (derived) [[right adjoint]] $f^!$. This is [[Verdier duality]] in a ``[[Grothendieck context]]'' of [[six operations]].

Grothendieck duality is intimately connected to \textbf{dualizing complexes}. This was the original approach of Grothendieck in the book \emph{Residues and Duality}.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

Suppose $f\colon X \to Y$ is a [[quasi-compact map|quasi-compact]] and [[quasi-separated map|quasi-separated]] [[morphism]] of [[schemes]]; then the [[triangulated functor]] $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below [[right adjoint]]. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a [[natural isomorphism]].

\hypertarget{dualizing_complexes}{}\subsection*{{Dualizing Complexes}}\label{dualizing_complexes}

Let $X$ be a noetherian scheme. A dualizing complex on $X$ is a complex $\mathcal{R} \in \mathsf{D}(\mathsf{Mod} X)$ that has these three properties:

\begin{itemize}%
\item $\mathcal{R} \in \mathsf{D}^{\mathrm{b}}_{\mathrm{c}}(\mathsf{Mod} X)$ (i.e. $\mathcal{R}$ has bounded coherent cohomology sheaves).


\item $\mathcal{R}$ has finite injective dimension.


\item The canonical morphism $\mathcal{O}_X \to \mathrm{R} \mathcal{Hom}_{X}(\mathcal{R}, \mathcal{R})$ in $\mathsf{D}(\mathsf{Mod} X)$ is an isomorphism.



\end{itemize}
The following two structures are basically equivalent to each other, for a given category of noetherian schemes $\mathsf{S}$:

\begin{itemize}%
\item A psudofunctor $f \mapsto f^!$, called the \textbf{twisted inverse image}, that assigns a functor

\begin{displaymath}
f^! : \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} Y)
\to \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X)
\end{displaymath}
to each map of schemes $f : X \to Y$ in $\mathsf{S}$, and has several known properties.


\item A dualizing complex $\mathcal{R}_X$ for every scheme $X$ in the category $\mathsf{S}$, with several known functorial properties.



\end{itemize}
The relation between these two structures is demonstrated in the following Example.

\textbf{Example}. Suppose $K$ is a regular finite dimensional noetherian ring, and let $\mathsf{S}$ be the category of finite type $K$-schemes. Given a twisted inverse image psudofunctor $f \mapsto f^!$, we define dualizing complexes as follows: on any $X \in \mathsf{S}$ with structural morphism $\pi_X : X \to \operatorname{Spec} K$, we let $\mathcal{R}_X := \pi_X^!(K)$.

Conversely, suppose we are given a dualizing complex $\mathcal{R}_X$ on each $X \in \mathsf{S}$. This gives rise to a duality (contrvariant equivalence) $D_X$ of $\mathsf{D}^{}_{\mathrm{c}}(\mathsf{Mod} X)$, exchanging $\mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X)$ with $\mathsf{D}^{-}_{\mathrm{c}}(\mathsf{Mod} X)$, with formula

\begin{displaymath}
D_X(\mathcal{M}) := \mathrm{R} \mathcal{Hom}_{X}(\mathcal{M}, \mathcal{R}_X) .
\end{displaymath}
We then define

\begin{displaymath}
f^!(\mathcal{M}) := D_Y( \mathrm{L} f^* ( D_X(\mathcal{M}) )) .
\end{displaymath}
\hypertarget{rigid_dualizing_complexes}{}\subsubsection*{{Rigid Dualizing Complexes}}\label{rigid_dualizing_complexes}

The notion of \emph{rigid dualizing complex} was introduced by Van den Bergh in 1997, for a \emph{noncommutative ring} $A$ over a base field $K$.

Yekutieli and Zhang have shown how to define a rigid dualizing complex $R_{A/K}$, when $K$ is a regular finite dimensional noetherian ring, and $A$ is an essentially finite type $K$-ring (both commutative). A refined variant of the rigid dualizing complex, namely the \emph{rigid residue complex} $\mathcal{K}_{A/K}$, was shown to exist, and to be unique (up to a unique isomorphism of complexes).

These rigid residue complexes have all the good functorial properties alluded to above, and even more. Specifically, they are covariant for essentially etale ring homomorphisms $A \to A'$ (via the rigid localization homomorphism), and contravariant (as graded modules) for all ring homomorphisms $A \to B$ (via the ind-rigid trace homomorphism).

The rigid localization homomorphism permits the gluing of the rigid residue complexes $\mathcal{K}_{A/K}$ on affine open sets $U = \operatorname{Spec} A$ of a scheme $X$ into a rigid residue complex $\mathcal{K}_{X/K}$ on $X$. In this way one obtains a collection of dualizing complexes $\mathcal{K}_{X/K}$ on all essentially finite type $K$-schemes $X$, consisting of quasi-coherent injective sheaves. For any map of scheme $f : X \to Y$ there is the ind-rigid trace homomoprhism

\begin{displaymath}
\mathrm{Tr}_f : f_*(\mathcal{K}_{Y/K}) \to \mathcal{K}_{X/K} ,
\end{displaymath}
which is a homomorphism of graded quasi-coherent sheaves. The \emph{Residue Theorem} says that when $f$ is proper, $\mathrm{Tr}_f$ is a homomorphism of complexes. The \emph{Duality Theorem} says that when $f$ is proper, $\mathrm{Tr}_f$ induces global duality. As explained above, there is a corresponding functor $f^!$; and the Duality Theorem says that $\mathrm{R} f_*$ and $f^!$ are adjoint functors.

Moreover, the rigidity method works also for finite type \textbf{Deligne-Mumford stacks} over $K$. The key observation is that the rigid residue complexes are complexes of quasi-coherent sheaves in the etale site over $K$. Details of this extension of the theory are still under preparation.

\hypertarget{noncommutative_grothendieck_duality}{}\subsection*{{Noncommutative Grothendieck Duality}}\label{noncommutative_grothendieck_duality}

(To be added later)

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

\begin{itemize}%
\item [[six operations]]


\item [[Poincare duality]], [[Verdier duality]]


\item [[Fourier-Mukai transform]]



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

\begin{itemize}%
\item [[Robin Hartshorne]], \emph{Residues and duality} (Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.) Springer LNM 20, 1966 \href{http://www.ams.org/mathscinet-getitem?mr=222093}{MR222093}


\item Domingo Toledo, Yue Lin L. Tong, \emph{Duality and intersection theory in complex manifolds. I.}, Math. Ann. \textbf{237} (1978), no. 1, 41--77, \href{http://www.ams.org/mathscinet-getitem?mr=506654}{MR80d:32008}, \href{http://dx.doi.org/10.1007/BF01351557}{doi}


\item Mitya Boyarchenko, [[Vladimir Drinfeld]], \emph{A duality formalism in the spirit of Grothendieck and Verdier}, \href{http://arxiv.org/abs/1108.6020}{arxiv/1108.6020}


\item Z. Mebkhout, \emph{Le formalisme des six op\'e{}rations de Grothendieck pour les $\mathcal{D}_X$-modules coh\'e{}rents}, Travaux en Cours \textbf{35}. Hermann, Paris, 1989. x+254 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1008245}{MR90m:32026}


\item [[Amnon Neeman]], \emph{Derived categories and Grothendieck duality}, in: Triangulated categories, 290--350, London Math. Soc. Lecture Note Ser. \textbf{375}, Cambridge Univ. Press 2010


\item Amnon Neeman, \emph{The Grothendieck duality theorem via Bousfield's techniques and Brown representability}, J. Amer. Math. Soc. \textbf{9} (1996), no. 1, 205--236, \href{http://www.ams.org/mathscinet-getitem?mr=1308405}{MR96c:18006}, \href{http://dx.doi.org/10.1090/S0894-0347-96-00174-9}{doi}


\item Brian Conrad, \emph{Grothendieck duality and base change}, Springer Lec. Notes Math. \textbf{1750} (2000) vi+296 pp.


\item [[Joseph Lipman]], \emph{Notes on derived functors and Grothendieck duality}, in: Foundations of Grothendieck duality for diagrams of schemes, 1--259, Lecture Notes in Math. \textbf{1960}, Springer 2009, \href{http://dx.doi.org/10.1007/978-3-540-85420-3}{doi}, \href{http://www.math.purdue.edu/~lipman/Duality.pdf}{draft pdf}


\item J. Lipman, \emph{Grothendieck operations and coherence in categories}, conference slides, 2009, \href{http://www.math.purdue.edu/~lipman/Madrid.pdf}{pdf}


\item Alonso Tarr\'i{}o, Leovigildo; Jerem\'i{}as L\'o{}pez, Ana; Joseph Lipman, \emph{Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes}, Contemporary Mathematics \textbf{244} Amer. Math. Soc. 1999. x+126L. \href{http://www.ams.org/mathscinet-getitem?mr=1716706}{MR2000h:14017}; \emph{Duality and flat base change on formal schemes}, Contemporary Math. \textbf{244} (1999), pp. 3--90.


\item J. Ayoub, \emph{Les six op\'e{}rations de Grothendieck et le formalisme des cycles \'e{}vanescents dans le monde motivique. I.}, Ast\'e{}risque No. 314 (2007), x+466 pp. (2008) \href{http://www.ams.org/mathscinet-getitem?mr=2423375}{MR2009h:14032}; \emph{II.} Ast\'e{}risque No. 315 (2007), vi+364 pp. (2008) \href{http://www.ams.org/mathscinet-getitem?mr=2438151}{MR2009m:14007}; also a file at K-theory archive \href{http://www.math.uiuc.edu/K-theory/0761/THESE.pdf}{THESE.pdf}


\item [[Amnon Yekutieli]], James Zhang, Rings with Auslander Dualizing Complexes, J. Algebra 213 (1999), 1-51; \emph{Rigid dualizing complexes over commutative rings}, Algebr. Represent. Theory \textbf{12} (2009), no. 1, 19--52, \href{http://dx.doi.org/10.1007/s10468-008-9102-9}{doi}; Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes, Selecta Math. 12 (2006), 137-177; Dualizing Complexes and Perverse Modules over Differential Algebras, Compositio Mathematica 141 (2005), 620-654.


\item [[Amnon Yekutieli]], An Explicit Construction of the Grothendieck Residue Complex, Ast\'e{}risque \textbf{208} (1992); The residue complex of a noncommutative graded algebra, J. Algebra \textbf{186} (1996), no. 2, 522--543; \emph{Smooth formal embeddings and the residue complex}, Canad. J. Math. \textbf{50} (1998), no. 4, 863--896, \href{http://www.ams.org/mathscinet-getitem?mr=1638635}{MR99i:14004}; \emph{Rigid dualizing complexes via differential graded algebras (survey)}, in: Triangulated categories, 452--463, London Math. Soc. Lecture Note Ser. \textbf{375}, Cambridge Univ. Press 2010, \href{http://www.ams.org/mathscinet-getitem?mr=2681716}{MR2011h:18015}; Residues and Differential Operators on Schemes, Duke Math. J. 95 (1998), 305-341; Duality and Tilting for Commutative DG Rings, \href{http://arxiv.org/abs/1312.6411}{arXiv:1312.6411}; Residues and Duality for Schemes and Stacks (\href{http://www.math.bgu.ac.il/~amyekut/lectures/resid-stacks/handout.pdf}{lecture notes}).


\item [[Roy Joshua]], \emph{Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories} ([[JoshuaDuality.pdf:file]])


\item [[Pieter Belmans]], section 2.2 of \emph{Grothendieck duality: lecture 3}, 2014 ([[BelmansDuality.pdf:file]])


\item [[Amnon Neeman]], \emph{An improvement on the base-change theorem and the functor $f^!$}, \href{http://arxiv.org/abs/1406.7599}{arXiv}.



\end{itemize}


\end{document}