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\begin{document}

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\section*{perfect infinity-stack}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{equivalent_reformulations}{Equivalent reformulations}\dotfill \pageref*{equivalent_reformulations} \linebreak
\noindent\hyperlink{geometric_function_theory}{Geometric $\infty$-function theory}\dotfill \pageref*{geometric_function_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the context of [[dg-geometry]] an [[∞-stack]] $X$ is called \emph{perfect} if its [[(∞,1)-category]] $QC(X)$ of [[quasicoherent ∞-stack]]s (of [[module]]s over the [[structure sheaf]] $\mathcal{O}(X)$) is generated from compact objects/dualizable objects: modules that are locally [[perfect chain complex]]es.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $k$ be a [[field]] of [[characteristic]] 0. Let $T$ be the [[Lawvere theory]] of commutative [[associative algebra]]s over $k$. When this is regarded as an [[(∞,1)-algebraic theory]], the $T$-$\infty$-algebras are modeled (by the [[monoidal Dold-Kan correspondence]] equivalently) by the

\begin{itemize}%
\item [[model structure on simplicial T-algebras]];


\item [[model structure on dg-algebras]] (over $k$, in non-positive degree, with positively graded differential).



\end{itemize}
The [[higher geometry]]/[[derived geometry]] over formal duals of these algebras is sometimes called [[dg-geometry]]: a general space in this context is given by an [[∞-stack]] over a full sub-[[(∞,1)-site]]

\begin{displaymath}
C \subset T Alg_\infty^{op}
\end{displaymath}
of the [[opposite (∞,1)-category]] of these $\infty$-algebras.

\begin{udefn}
The [[(∞,2)-presheaf]] of [[quasicoherent ∞-stack]]s is

\begin{displaymath}
Mod : C^{op} \to (\infty,1)Cat
\end{displaymath}
given by

\begin{displaymath}
Spec A \mapsto A Mod
  \,,
\end{displaymath}
where on the right we take the $(\infty,1)$-category of $\infty$-modules over the $\infty$-algebra $A$, regarded as an unbounded dg-algebra.

\end{udefn}
\begin{udefn}
For $X \in Sh_{(\infty,1)}(C)$ an [[∞-stack]] in [[dg-geometry]], write

\begin{displaymath}
QC(X) := PSh_{(\infty,2)}(C)\left( X, Mod \right)
\end{displaymath}
for the $(\infty,1)$-category of \textbf{quasicoherent $\infty$-stacks} on $X$.

\end{udefn}
\begin{uremark}
By the [[co-Yoneda lemma]] we may express every $X \in Sh_{(\infty,1)}(C)$ as an [[(∞,1)-colimit]] of [[representable functor|representables]]

\begin{displaymath}
X \simeq {\lim_\to}_i U_i = {\lim_\to}_i Spec A_i
  \,.
\end{displaymath}
We have then

\begin{displaymath}
QC(X) \simeq {\lim_\leftarrow}_i QC(U_i) \simeq {\lim_\leftarrow}_i A_i Mod
  \,.
\end{displaymath}
\end{uremark}
This appears as (\hyperlink{Ben-ZviFrancisNadler}{Ben-ZviFrancisNadler, section 3.1}).

\begin{uprop}
For all $X \in \mathbf{H}$, we have that $QC(X)$

\begin{itemize}%
\item is a [[stable (∞,1)-category]]


\item that has all [[(∞,1)-colimit]]s.



\end{itemize}
\end{uprop}
(\hyperlink{Ben-ZviFrancisNadler}{Ben-ZviFrancisNadler, section 3.1}).

\begin{udefn}
Let $A \in T Alg_\infty$ . An $A$-module is a \textbf{perfect module} if it lies in the smallest [[sub-(∞,1)-category]] of $A Mod$ containing $A$ and closed under finite [[(∞,1)-colimit]]s and [[retract]]s.

For a [[∞-stack]] $X \in Sh_{(\infty,1)}(C)$, the $\infty$-category $Perf(X)$ is the full sub-$(\infty,1)$-category of $QC(X)$ consisting of those modules that are prefect over every affine $U\to X$.

\end{udefn}
This appears as (\hyperlink{Ben-ZviFrancisNadler}{Ben-ZviFrancisNadler, definition 3.1}).

\begin{udefn}
A [[∞-stack]] $X \in Sh_{(\infty,1)}(C)$ is called a \textbf{perfect stack} if

\begin{itemize}%
\item it has affine diagonal $X \to X \times X$;


\item and $QC(X)$ is the [[ind-object in an (infinity,1)-category|(∞,1)-category of ind-objects]]

\begin{displaymath}
QC(X) \simeq \Ind \Perf(X)
\end{displaymath}
of the full [[sub-(∞,1)-category]] $Perf(X) \subset QC(X)$ of perfect complexes of modules on $X$.



\end{itemize}
A morphism $X \rightarrow Y$ is said to be \textbf{perfect morphism} if its fibers $X \times_Y U$ over affines $U \rightarrow Y$ are perfect.

\end{udefn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{equivalent_reformulations}{}\subsubsection*{{Equivalent reformulations}}\label{equivalent_reformulations}

\begin{udef}
A [[stable (∞,1)-category]] $C$ is \textbf{compactly generated} if it has a [[small set]] $\{c_i\}_{i \in I}$ of [[compact object]] that are [[generator]]s in the sense that if for $N \in C$ we have that $C(c_i, N)$ is equivalent to the [[zero morphism]], then $N$ is the [[zero object]].

\end{udef}
\begin{utheorem}
For a [[∞-stack]] $X \in Sh_{(\infty,1)}(C)$ with affine diagonal, the following are equivalent:

\begin{itemize}%
\item $X$ is perfect


\item $QC(X)$ is

\begin{itemize}%
\item compactly generated,


\item and its [[compact object|compact]] and [[dualizable object|dualizable]] objects coincide.



\end{itemize}


\end{itemize}
\end{utheorem}
\hypertarget{geometric_function_theory}{}\subsubsection*{{Geometric $\infty$-function theory}}\label{geometric_function_theory}

The assigmnent

\begin{displaymath}
QC : X \mapsto QC(X)
\end{displaymath}
of the $(\infty,2)$-algebras $QC(X)$ of [[quasicoherent ∞-stack]]s to perfect \$$\infty$-stacks $X$ constitutes a [[geometric ∞-function theory]]: this assignment commutes with [[(∞,1)-pullback]]s and admits a ggood pull-push theory of [[integral transforms on sheaves]].

(\hyperlink{Ben-ZviFrancisNadler}{Ben-ZviFrancisNadler}

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

Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the ([[constant ∞-stack]] on a) [[classifying space]] $\mathcal{B}G$ of a [[topological group]] $G$ such as the [[circle]] $S^1 \simeq \mathcal{B} \mathbb{Z}$ or the classifying spaces of most [[algebraic group]]s in non-zero [[characteristic]]. This is because if $X$ is perfect, then the [[global section]]s functor $\Gamma$ must preserve colimits, which fails when the global sections $\Gamma(X, \mathcal{O}_X)$ of the structure sheaf is `too large', as in the previous cases.

But the following are examples of perfect $\infty$-stacks

\begin{itemize}%
\item quasi-compact [[derived scheme]]s with affine diagonal;


\item the total space of a quasi-projective morphism over a perfect base;


\item a quasi-projective [[derived scheme]];


\item the quotient $X/G$ of a quasi-projective [[derived scheme]] $X$ by a linear action of an affine group (for $k$ of [[characteristic]] 0).



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

The concept of a \emph{perfect stack} in the context of [[dg-geometry]] is considered in

\begin{itemize}%
\item [[David Ben-Zvi|Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral transforms and Drinfeld centers in derived algebraic geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv})

\end{itemize}
[[!redirects perfect infinity-stacks]] [[!redirects perfect ∞-stack]] [[!redirects perfect ∞-stacks]]



\end{document}