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\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*{quasicoherent sheaf} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos theory - 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{LocPres}{As locally presentable modules}\dotfill \pageref*{LocPres} \linebreak \noindent\hyperlink{AsSheaves}{As sheaves on $Aff/X$}\dotfill \pageref*{AsSheaves} \linebreak \noindent\hyperlink{AsSheavesII}{Direct definition for presheaves of sets on Aff}\dotfill \pageref*{AsSheavesII} \linebreak \noindent\hyperlink{AsCocycles}{As homs into the stack of modules}\dotfill \pageref*{AsCocycles} \linebreak \noindent\hyperlink{AsFibHoms}{As cartesian morphisms of fibrations}\dotfill \pageref*{AsFibHoms} \linebreak \noindent\hyperlink{Higher}{Quasicoherent modules in higher/derived geometry}\dotfill \pageref*{Higher} \linebreak \noindent\hyperlink{by_maps_into_the_stack_}{By maps into the stack $QCoh$}\dotfill \pageref*{by_maps_into_the_stack_} \linebreak \noindent\hyperlink{HigherGeometryAsExtensionsOfStructureSheaf}{As extensions of the structure sheaf}\dotfill \pageref*{HigherGeometryAsExtensionsOfStructureSheaf} \linebreak \noindent\hyperlink{SyntheticDescription}{Synthetic definition using the internal language}\dotfill \pageref*{SyntheticDescription} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{quasicoherent_sheaves_over_affine_schemes}{Quasicoherent sheaves over affine schemes}\dotfill \pageref*{quasicoherent_sheaves_over_affine_schemes} \linebreak \noindent\hyperlink{the_category_of_quasicoherent_sheaves}{The category of quasicoherent sheaves}\dotfill \pageref*{the_category_of_quasicoherent_sheaves} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{dmodules}{D-Modules}\dotfill \pageref*{dmodules} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} A \emph{quasicoherent sheaf of modules} (often just ``quasicoherent sheaf'', for short) is a [[sheaf of modules]] over the [[structure sheaf]] of a [[ringed space]] that is \emph{locally [[presentable module|presentable]]} in that it is locally the [[cokernel]] of a morphism of [[free modules]]. For comparison, by the [[Serre-Swan theorem]] a [[vector bundle]] on a suitable [[ringed space]] is equivalently encoded in its [[sheaf]] of [[sections]] which is even locally [[free module|free]] and [[projective module|projective]]. In this sense quasicoherent sheaves of modules are a generalization of [[vector bundles]]. The [[category]] of vector bundles is too small to be closed under various natural operations like [[kernels]], [[direct images]] and alike. In particular, it is not an [[abelian category]]. The category of all $\mathcal{O}$-modules and especially its [[full subcategory]] of quasicoherent sheaves of $\mathcal{O}$-modules are better behaved in that respect. There are several different but equivalent ways to define and think of quasicoherent sheaves. A very concrete definition characterizes quasicoherent sheaves as those that are, while not locally free, locally the [[cokernel]] of a morphism of free module sheaves. This is the definition given in the section \begin{itemize}% \item \hyperlink{LocPres}{As locally presentable modules} \end{itemize} below. It makes very manifest how passing from vector bundles to quasicoherent sheaves adds in the [[cokernel]]s that are missing in the category of [[vector bundle]]s. But it turns out that there is a more abstract, more [[sheaf and topos theory|sheaf theoretic]] reformulation of this definition: if we think of the underlying [[space]] as a ([[presheaf|pre]])[[sheaf]] (as motivated at [[motivation for sheaves, cohomology and higher stacks]]) we find that a quasicoherent sheaf on a space is given by an assignment of a module to each plot, such that the pullback of these modules is given, up to coherent isomorphism, by tensoring over the corresponding rings. This is described in the section \begin{itemize}% \item \hyperlink{AsSheaves}{As sheaves over Aff/X}. \end{itemize} and in more details in the section \begin{itemize}% \item \hyperlink{AsSheavesII}{Definition as sheaves over Aff}. \end{itemize} The tensoring operation appearing here is that defining the pullback operations in the [[stack]] that classifies the canonical [[bifibration]] $Mod \to CRing$ of [[module]]s over rings. In view of this, one finds that this definition, in turn, is equivalent to a very fundamental definition: with $QC := (-)Mod : Ring \to Cat$ the functor that sends a ring to its category of modules, one finds that the category of quasicoherent sheaves on a [[space]] $X$ is simply the [[hom-object]] \begin{displaymath} QC(X) := Hom(X,QC) \end{displaymath} in the corresponding 2-category of category-valued (pre)sheaves, i.e (pre)[[stack]]s. This is the perspective described in \begin{itemize}% \item \hyperlink{AsCocycles}{As hom objects} \end{itemize} below. By the equivalence between [[Grothendieck fibration]]s and [[pseudofunctor]]s this in turn is directly equivalent to the identification of $QC(X)$ with the category of cartesian functors between the [[category of elements]] of $X$ and $Mod$. This is described in \begin{itemize}% \item \hyperlink{AsFibHoms}{As cartesian morphisms of fibered categories} \end{itemize} This definition, finally, provides a powerful [[nPOV]] on quasicoherent sheaves: all notions involved, sheaf, stack, morphism of stacks, have natural, immediate and well understood generalizations to [[higher category theory]]. Therefore this last definition immediately generalizes to a definition of quasicoherent $\infty$-sheaves or ``derived'' quasicoherent sheaves, such as they appear for instance in [[geometric ∞-function theory]]. This is discussed in the section \begin{itemize}% \item \hyperlink{Higher}{Higher quasicoherent sheaves}. \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{LocPres}{}\subsubsection*{{As locally presentable modules}}\label{LocPres} Let $(X,O_X)$ be a [[ringed space]] or, more generally, a [[ringed site]]. A \textbf{quasicoherent sheaf of $O_X$-modules} on $X$ is a sheaf $\mathcal{E}$ of $O_X$-modules that is locally a [[cokernel]] of a [[morphism]] of [[free module]]s. This means: there is a [[cover]] $\{U_\alpha\}_{\alpha\in A}$ of $X$ by open sets such that for every $\alpha$ there exist $I_\alpha$ and $J_\alpha$ (not necessarily finite) and an [[exact sequence]] of sheaves of $O_X$-modules of the form \begin{displaymath} O_X^{I_\alpha}|_{U_\alpha} \to O_X^{J_\alpha}|_{U_\alpha} \to \mathcal{E}|_{U_\alpha}\to 0, \end{displaymath} This should be viewed as a \emph{local presentation} of $\mathcal{E}$. If $I_\alpha, J_\alpha$ can be chosen finite and $\mathcal{E}$ is of finite type then the quasicoherent sheaf is a \textbf{[[coherent sheaf]]}. (See there for details.) However , coherent sheaves are ill-behaved for a general ringed space, and even general [[scheme]]s; they behave well on [[Noetherian scheme]]s. Replacing covers by open sets, by covers of a terminal object in a site, the definition extends to [[ringed site]]s with a terminal object; the restrictions of $O_X$-modules should be replaced by pullbacks. There are generalizations for [[algebraic stack]]s, ind-schemes, diagrams of schemes (for example \href{http://arxiv.org/abs/math/0012061}{configuration schemes} of V. Lunts, obtained by gluing along closed embeddings of schemes; simplicial schemes) and so on. \hypertarget{AsSheaves}{}\subsubsection*{{As sheaves on $Aff/X$}}\label{AsSheaves} There is an equivalent reformulation of the above in terms of [[sheaf|sheaves]] of $\mathcal{O}$-modules on the [[site]] $Aff/X$ of [[affine scheme]]s over $X$. This is the [[over category]] whose objects are morphism (of [[scheme]]s) of the form $Spec A \to X$ and whose morphisms are commuting triangles \begin{displaymath} \itexarray{ Spec A &&\stackrel{f}{\to}&& Spec B \\ & {}_a\searrow && \swarrow_b \\ && X } \,. \end{displaymath} Then: a quasicoherent sheaf on $(X, \mathcal{O}_X)$ is a [[sheaf]] $N$ of $\mathcal{O}_X$-[[module]]s on $Aff/X$ such that for each morphism $f$ as above the restriction morphism \begin{displaymath} N(f) : N(b) \to N(a) \end{displaymath} extends to an [[isomorphism]] \begin{displaymath} N(b) \otimes_{f^*} A \stackrel{\simeq}{\to} N(a) \end{displaymath} of $A$-[[module]]s. For a very explicit statement of this see for instance \href{http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf#page=13}{page 13} of \begin{itemize}% \item [[Paul Goerss]], \emph{Topological modular forms (aftern Hopkins, Miller, and Lurie)} (\href{http://arxiv.org/abs/0910.5130}{arXiv}) \end{itemize} See also a very precise and detailed treatment in \begin{itemize}% \item [[Dmitri Orlov]], \emph{Quasi-coherent sheaves in commutative and non-commutative geometry}, Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 3, Pages 119--138 (see also preprint version \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=57}{dvi}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=56}{ps}) \end{itemize} \hypertarget{AsSheavesII}{}\subsubsection*{{Direct definition for presheaves of sets on Aff}}\label{AsSheavesII} Here is a more detailed way to say again what the above paragraph said. Let $Aff = CRing^{op}$; recall the fibered category $Mod\to CRing^{op}$ where for each $f:A\to B$ in $CRing$ the inverse image functor is $f^*=B\otimes_A - :{}_A Mod\to {}_B Mod$. Then the identity functor $CRing\to CRing$ can be interpreted as the presheaf of rings and is denoted by $O$ (the ``structure sheaf''). An $O$-module is a presheaf of $O$-modules. Usually some Grothendieck topology is given and one asks for sheaves in fact. We can Yoneda extend $O$-modules to presheaves. We now define quasicoherent sheaves of $O$-modules on an arbitrary presheaf $X$ on $Aff$, viewed as a covariant functor on $CRing$. A \textbf{quasicoherent sheaf of $O$-modules} on $X$ is a rule assigning to any $\phi\in X(A)$ an $A$-module $M_\phi = M_{A,\phi}$ and to any morphism $f:A\to B$ in $CRing$ an isomorphism $\theta_{f,\phi}:f^*(M_\phi)\to M_{X(f)(\phi)}$ such that for any [[composable pair]] $A\stackrel{f}\to B\stackrel{g}\to C$ and any $\phi\in X(A)$ the cocycle condition \begin{displaymath} \theta_{g\circ f,\phi}\circ \alpha_{fg} = \theta_{g,X(f)(\phi)}\circ g^*(\theta_{f,\phi})\colon g^* f^*(M_\phi)\to M_{X(g\circ f)(\phi)} \end{displaymath} holds, where $\alpha_{fg}:g^* f^*(M_\phi)\to (g\circ f)^*(M_\phi)$ is the canonical isomorphism which is part of the data of the (covariant) pseudofunctor $A\to {}_A Mod$, $f\mapsto f^*$. Notice that if $X = h^C = h_{Spec C}$ is (co)representable presheaf, then $\phi\in [A,X]_{Pshv(Aff)}=[C,A]_{CRing}$ is the same as a morphism $\phi^{op}:C\to A$ of rings; restricting the quasicoherent sheaf to $Spec A$ along $\phi:Spec A\to X$ and taking the global sections over $A$, would give the $A$-module $M_\phi$. Clearly, $Aff$ and $O$ can be much generalized. For example, rings may be noncommutative or one can take category opposite to the category of monads in $Set$ and an arbitrary (not identity) presheaf $D$ of monads in $Set$; the extension of scalars for monads gives an inverse image functor for Eilenberg-Moore categories. Durov's construction of quasicoherent sheaves for monads in $Set$ is an example where commutative algebraic monads are used; the theory of quasicoherent sheaves of $D$-modules (``$O$-modules with integrable connection'') is another. Instead setups involving operads, higher operads and alike can be used as well; commutativity condition is useful if one wants a monoidal category of quasicoherent sheaves. \hypertarget{AsCocycles}{}\subsubsection*{{As homs into the stack of modules}}\label{AsCocycles} The above definition may be further re-interpreted as follows. \begin{uprop} On the [[site]] $Aff = CRing^{op}$, let \begin{displaymath} QC : CRing \to Cat \end{displaymath} \begin{displaymath} (Spec S \stackrel{f^{op}}{\to} Spec R) \mapsto (R Mod \stackrel{S \otimes_{f} -}{\to} S Mod) \end{displaymath} be the (pseudo)functor ([[stack]]) corresponding to the canonical [[Grothendieck fibration]] of [[module]]s $Mod \to CRing$. Its right [[Kan extension]] through the 2-[[Yoneda embedding]] $Y : CRing^{op} \hookrightarrow [CRing,Cat]$ is given on a presheaf $X : CRing \to Set$ by the [[hom-object]] \begin{displaymath} QC(X) := (Ran_Y QC)(X) := [CRing,Cat](X,QC) \,. \end{displaymath} When $X$ is the functor [[representable functor|represented]] by a scheme, then $QC(X)$ is the category of quasicoherent sheaves on $X$, as defined above. \end{uprop} We now explain the above statement in detail and thereby prove it. Let $C =$[[Ring]]${}^{op}$ be the category of (commutative, unital) [[ring]]s. For $R$ a [[ring]] write $Spec R$ for it regarded as an object of $C$. Write $Spec f = f^{op} : Spec(S) \to Spec(R)$ for the morphism in $Ring^{op}$ corresponding to the map $f : R \to S$ of commutative rings. Consider the [[2-category]] of (pre)[[stack]]s on $C$. The canonical [[module]] [[bifibration]] $p : Mod \to Ring$ of the category of modules over all rings is the bifibration whose fibered part corresponds to the (pre)stack $QC \in [C^{op},Cat]$ given on objects by \begin{displaymath} QC : Spec R \mapsto R Mod \end{displaymath} and on morphisms by \begin{displaymath} QC : (Spec S \stackrel{f^{op}}{\to} Spec R) \mapsto (R Mod \stackrel{S\otimes_{f} }{\to} S Mod) \,, \end{displaymath} where on the right we have the functor that sends any $R$-module $N$ to the [[tensor product]] over $S$ with the $R$-$S$-[[bimodule]] $S = {}_S S_R$ with its canonical left $S$-action and with the right $R$-action induced by the ring homomorphism $f$. One may think of this as the stack of generalized algebraic vector bundles: the operation $S \otimes_{f} - : R Mod \to S Mod$ corresponds to the [[pullback]] of [[bundle]]s along a morphism of the underlying spaces. (See for instance the discussion of [[monadic descent]] at [[Sweedler coring]] for more on this.) We may [[Kan extension|right Kan extend]] the 2-functor $QC : CRing^{op} \to Cat$ through the [[Yoneda embedding]] $CRing^{op} \hookrightarrow [CRing,Cat]$ to get a definition of $QC$ on arbitrary [[presheaf|presheaves]]. \begin{displaymath} \itexarray{ CRing^{op} &\stackrel{QC}{\to}& Cat \\ {}^{Y}\downarrow & \nearrow_{\mathrlap{Ran_Y QC}} \\ [CRing,Cat] } \,. \end{displaymath} Consider $X \in [C^{op},Set]$ any ([[presheaf|pre]])[[sheaf]] on $C$. This may be the presheaf [[representable functor|represented]] by a [[scheme]], but for the purposes of the definition of $QC$ it may be much more generally any presheaf. By the general formula for [[Kan extension]] in terms of a [[weighted limit]] given by an [[end]] we have \begin{displaymath} Ran_Y QC : X \mapsto \int_{R \in Ring} ([Ring,Cat]^{op}(X,Y(R)))^{QC(R)} \end{displaymath} which using the [[Yoneda lemma]] is \begin{displaymath} \cdots = \int_{R \in CRing} [CRing,Cat](X(R), QC(R)) \,. \end{displaymath} This is the [[end]]-formula for the [[hom-object]] in an [[enriched functor category]] $[C^{op},Cat]$, hence this is nothing but the category of (pseudo)[[natural transformation]]s between the 2-functor $X$ and the 2-functor $QC$. We write for short \begin{displaymath} QC(X) := (Ran_Y QC)(X) := [C^{op},Cat](X,QC) \,. \end{displaymath} This definition of ``generalized vector bundles'' on arbitrary presheaves is entirely analogous to the definition of differential forms on arbitrary presheaves, that is discussed in some detail for instance in the entry on [[rational homotopy theory]]. We claim that the category $QC(X)$ is the category of quasicoherent sheaves on $X$ as defined by other means above, whenever that other definition applies to $X$. To see this, straightforwardly unwrap the definition: an object $N$ in $QC(X) = [C^{op},Cat](X,QC)$ is a [[transfor|pseudonatural transformation]] of 2-functors $N : X \to QC$, where $X$ is regarded as a 2-functor by the canonical embedding $disc : Set \hookrightarrow Cat$ that regards a [[set]] as a [[discrete category]]. The components of $N$ are \begin{itemize}% \item for each $Spec R \in Ring^{op}$ a [[functor]] $N|_{Spec R} : X(R) \to QC(R) = R Mod$: this functor picks one $R$-module $N(r) \in R Mod$ for each plot $(r : Spec R \to X) \in X(Spec R)$; \item for each morphism $f : Spec A \to Spec B$ a pseudonaturality square \begin{displaymath} \itexarray{ X(Spec B) &\stackrel{X(f)}{\to}& X(Spec A) \\ {}^{\mathllap{N|_{Spec B}}}\downarrow &{}^{\Gamma(f)}\swArrow_{\simeq}& \downarrow^{\mathrlap{N|_{Spec A}}} \\ B Mod &\underset{- \otimes_{f^*} A}{\to}& A Mod } \end{displaymath} in [[Cat]] (these are subject to coherence conditions). This unwraps to the following data: \begin{itemize}% \item the component functors $N|_{Spec A}$ provide an assignment $a \mapsto N(a)$ of modules $N(a)$ to each plot $(a : Spec A \to X) \in X(Spec A)$; \item these assignments form a presheaf on the [[overcategory]] $Aff/X$ by taking the restriction morphism \begin{displaymath} N(f) : N(b) \to N(a) \end{displaymath} to be that underlying the components of the natural isomorphism in the above diagram \begin{displaymath} N(b)\otimes_{f^*} A \stackrel{\simeq}{\to} N(a) \,, \end{displaymath} i.e. the restriction of this morphism to $(n,1)$. \end{itemize} \item for each tuple of composable morphisms \begin{displaymath} \itexarray{ && Spec B \\ & {}^f\nearrow && \searrow^g \\ Spec A &&\to&& Spec C } \end{displaymath} a pseudo-naturality prism equation relating, $N(f)$, $N(g)$ and $N(g\circ g)$. The present author is too lazy to write out the diagram in detail, but it is of precisely the kind described in great detail for instance in the entry on [[group cohomology]]. Under the above identification, this yields the cocycle condition mentioned in the above definitions. \end{itemize} This way, the transformation $N : X \to QC$ defines manifestly a quasicoherent sheaf on $Aff/X$ in the sense of the definition in the above section \hyperlink{AsSheaves}{As sheaves on Aff/X}. Conversely, every quasicoherent sheaf according to that definition gives rise to a transformation $N : X \to QC$ under this prescription. \hypertarget{AsFibHoms}{}\subsubsection*{{As cartesian morphisms of fibrations}}\label{AsFibHoms} By the equivalence between [[pseudofunctor]]s $Ring \to Cat$ and [[Grothendieck fibration]]s $F \to Ring^{op}$ induced by the [[Grothendieck construction]], the above may equivalently be reformulated as follows. Recall from the discussion at [[Grothendieck fibration]] that the equivalence in question is between the following two [[bicategory|bicategories]]: \begin{itemize}% \item on the one hand the bicategory whose objects are [[pseudofunctors]] $Ring \to Cat$, whose morphisms are pseudonatural transformations, and whose 2-morphisms are modifications of these \item on the other hand the bicategory whose objects are [[Grothendieck fibration]]s $F \to Ring^{op}$, whose morphism are \textbf{cartesian functors} \begin{displaymath} \itexarray{ F_1 &&\to&& F_2 \\ & \searrow && \swarrow \\ && Ring^{op} } \end{displaymath} and whose 2-morphisms are [[natural transformation]]s between these. \end{itemize} Recall furthermore that for $X : Ring \to Cat$ an ordinary [[presheaf]], i.e. a pseudofunctor that factors through an ordinary functor $Ring \to Set$ via the inclusion $Set \to Cat$, the [[Grothendieck fibration]] associated with $X$ is the [[category of elements]] $Ring^{op}/X$ of $X$. Recall furthermore that by definition, the pseudofunctor $QC : Ring Cat$ is the one corresponding to the [[Grothendieck fibration]] $Mod^{op} \to Ring^{op}$. Therefore, by the above equivalence of 2-categories, we find that the category of functors $[Ring,Cat](X,QC)$ is equivalent to the category of cartesian functors over $Ring^{Op}$, $CartFunc_E(Ring^{op}/X,Mod^{op})$ \begin{displaymath} QC(X) \simeq [Ring,Cat](X,QX) \simeq CartFunc(Ring^{op}/X, Mod^{op}) \,. \end{displaymath} In this form quasicoherent sheaves on $X$ are conceived for instance in paragraph 1.1.5 of \begin{itemize}% \item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative stacks} (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2333}{ps}) \end{itemize} Here, as in the above discussion, the [[fibered category]] of modules can be replaced by a more general fibered category $\pi: \mathcal{F}\to\mathcal{B}$. Then the \textbf{category of quasicoherent modules in} this \textbf{fibered category} is the category opposite to the category of cartesian sections of $\pi$. This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2305}{dvi}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2333}{ps}). Given a category $\mathrm{Aff}$ of affine schemes (opposite to the category of rings) equipped with some [[subcanonical coverage|subcanonical pretopology]] one considers the [[stack]] of $O$-modules over $\mathrm{Aff}$: the fiber over a ring $R$, it assigns the category $Qcoh(\mathrm{Spec}\,R)$. Now given any stack on a subcanonical site, one defines the fiber over a sheaf on it so that the fiber over a representable sheaf is equivalent to the fiber over its representing object. There is a canonical way to do this (will write later about it -- Zoran); this is in particular a source of a definition $Qcoh$ on an ind-scheme. On ind-schemes Beilinson and Drinfel'd in \begin{itemize}% \item A. Beilinson, V. Drinfel'd, \emph{Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system}, preliminary version (\href{http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf}{pdf}) \end{itemize} consider two variants: a less important variant of quasicoherent $O_X^p$-modules (existing in bigger generality) and more delicate variant of quasicoherent $O^!_X$-modules defined for ``reasonable ind-schemes''; one of the differences is which functors play the role of pullbacks. In particular, these notions apply for a rather general variant of the category of formal schemes. \hypertarget{Higher}{}\subsubsection*{{Quasicoherent modules in higher/derived geometry}}\label{Higher} The last definition has a straightforward generalization to various [[higher geometry]] setups, such as [[derived schemes]] and other [[generalized schemes]]. \hypertarget{by_maps_into_the_stack_}{}\paragraph*{{By maps into the stack $QCoh$}}\label{by_maps_into_the_stack_} For instance the notion of quasicoherent sheaves generalized to [[derived stack]]s on the site of [[simplicial ring]]s as described at [[geometric ∞-function theory]] is obtained, we claim, simply by taking $QC : SRing \to (\infty,1)Cat$ to be the functor that assigns the [[(∞,1)-category]] for modules over a simplicial ring to any simplicial ring, and then setting for any derived stack $X$ \begin{displaymath} QC(X) := Hom(X,QC) \,. \end{displaymath} Moreover, using the theorem described at [[tangent (∞,1)-category]], that the [[bifibration]] of modules over simplicial rings is nothing but the [[tangent (∞,1)-category]] of $SRing$, one sees that all this is a special case of an even much more general abstract nonsense: for any presentable [[(∞,1)-category]] [[site]] $C$ whatsoever, we have the [[tangent (∞,1)-category]] fibration $T_C \to C$. With the [[(∞,1)-functor]] classifying it denoted $QC : C^{op} \to (\infty,1)Cat$ we may adopt for any [[∞-stack]] $X : C^{op} \to \infty Grpd$ the definition \begin{displaymath} QX(X) := [C^{op},\infty Grpd](X,QC) \end{displaymath} as a definition of generalized $\infty$-vector bundles on $X$. This general nonsense is considered further at [[schreiber:∞-vector bundle]]. Concrete realizations are discussed at [[quasicoherent ∞-stack]]. \hypertarget{HigherGeometryAsExtensionsOfStructureSheaf}{}\paragraph*{{As extensions of the structure sheaf}}\label{HigherGeometryAsExtensionsOfStructureSheaf} In (\hyperlink{LurieQC}{LurieQC, section 2.2, section 2.3}) the following definition is given. Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]]. Let \begin{displaymath} \mathcal{G}^{mod} \coloneqq (T (\mathcal{G}^{op}))_{cpt}^{op} \end{displaymath} be the opposite of the [[full sub-(∞,1)-category]] on the [[compact objects]] of the [[tangent (∞,1)-category]] of its [[opposite (∞,1)-category]]. For instance for [[E-∞ geometry]] we have $\mathcal{G} = CRing_\infty$ is the [[(∞,1)-category]] of [[E-∞ rings]] with [[etale morphisms]] as admissible maps. (\hyperlink{LurieQC}{LurieQC, above Notation 2.2.4}) Then the canonical [[(∞,1)-functor]] \begin{displaymath} \mathcal{G} \longrightarrow \mathcal{G}^{mod} \end{displaymath} is a morphism of discrete [[geometry (for structured (∞,1)-toposes)|geometries]]. For $\mathcal{X}$ an [[(∞,1)-topos]], a [[left exact (∞,1)-functor]] \begin{displaymath} \mathcal{O} \colon \mathcal{G} \longrightarrow \mathcal{X} \end{displaymath} constitutes a $\mathcal{G}$-[[structure sheaf]] and makes $(\mathcal{X}, \mathcal{O})$ be a $\mathcal{G}$-[[structured (∞,1)-topos]]. A left exact extension of this \begin{displaymath} \itexarray{ \mathcal{G} &\stackrel{\mathcal{O}}{\longrightarrow}& \mathcal{X} \\ \downarrow & \nearrow_{\mathrlap{(\mathcal{O}, \mathcal{F})}} \\ \mathcal{G}^{mod} } \end{displaymath} exhibits a sheaf $\mathcal{F}$ of $\mathcal{O}$-[[modules]] on $\mathcal{X}$. (\hyperlink{LurieQC}{LurieQC, notation 2.2.4}) Now if $(\mathcal{X},\mathcal{O})$ is [[locally representable structured (infinity,1)-topos]] then such an $\mathcal{O}$-module $\mathcal{F}$ is \emph{quasi-coherent} if also $(\mathcal{X}, (\mathcal{O}, \mathcal{F}))$ is locally representable. (\hyperlink{LurieQC}{LurieQC, def. 2.3.6}) \hypertarget{SyntheticDescription}{}\subsubsection*{{Synthetic definition using the internal language}}\label{SyntheticDescription} Let $X$ be a [[scheme]]. Recall that the [[big Zariski topos]] of $X$ is the topos of sheaves over $Sch/X$ (or, more precisely, the affine schemes over $X$ which are locally of finite presentation). In this topos, there is a [[local ring]] $\mathbb{A}^1$, the sheaf mapping an $X$-scheme $T$ to $\mathcal{O}_T(T)$. A sheaf $N$ of modules over $\mathbb{A}^1$ is quasicoherent if and only if, from the [[internal language|internal point of view]] of the big Zariski topos, the canonical map \begin{displaymath} N \otimes_{\mathbb{A}^1} A \longrightarrow Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N) \end{displaymath} is bijective for all finitely presented $\mathbb{A}^1$-algebras $A$. The outer Hom set is the set of all maps from the set $Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1)$ to the (underlying set of) $N$. This characterization has a geometric interpretation. The set $Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1)$ deserves the name ``spectrum of $A$'', since it consists of what classically is known as the ($\mathbb{A}^1$-)rational points of $A$. Furthermore, if $A$ is induced from a sheaf of $\mathcal{O}_X$-modules, then the object of the Zariski topos which is described by this set-theoretical expression coincides with the [[functor of points]] of the [[relative spectrum]] of that sheaf. The set $Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N)$ is therefore the set of all $N$-valued functions on the spectrum of $A$. An element of $N \otimes_{\mathbb{A}^1} A$ gives rise to such a function: associate to a pure tensor $x \otimes f$ the function $\varphi \mapsto f(\varphi) x$. In a [[synthetic differential geometry|synthetic]]/algebraic context, there should be no more functions than those which result from this construction. This is what the characterization expresses. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{quasicoherent_sheaves_over_affine_schemes}{}\subsubsection*{{Quasicoherent sheaves over affine schemes}}\label{quasicoherent_sheaves_over_affine_schemes} Given an affine scheme $X=\mathrm{Spec}\,R$ (where $R$ is a commutative unital ring), the affine Serre theorem establishes the equivalence of the category $Qcoh(\mathrm{Spec}\,R)$ of quasicoherent sheaves (in Zariski topology) and the category of $R$-modules. Similarly on a projective scheme of the type $Proj(A)$ where $A$ is a nonnegatively graded ring, the (projective) Serre theorem establishes the equivalence of $Qcoh(\mathrm{Proj}\,(A))$ and the localization of the category of graded $A$-modules by the subcategory of modules of finite length (and similarly, of coherent sheaves and graded $A$-modules of finite type modulo finite-length). These theorems are among basic motivating theorems for [[noncommutative algebraic geometry]]. An interesting in-depth comparison of the notions of quasi-coherent sheaves in commutative and noncommutative context are also in Orlov's article quoted above. \hypertarget{the_category_of_quasicoherent_sheaves}{}\subsubsection*{{The category of quasicoherent sheaves}}\label{the_category_of_quasicoherent_sheaves} In the case of general (commutative) schemes, every presheaf of $O_X$-modules which is quasicoherent in the sense of having local presentation as above, is in fact a sheaf. It is known that the category of quasicoherent sheaves of $O_X$-modules over any [[quasicompact]] quasiseparated scheme is a [[Grothendieck category]] and in particular has enough [[injective object]]s. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[coherent sheaf]] \item [[Bondal-Orlov reconstruction theorem]] \item [[module over a derived stack]] \end{itemize} \hypertarget{dmodules}{}\subsubsection*{{D-Modules}}\label{dmodules} The category of [[D-module]]s on a [[space]] $X$ is equivalently that of quasicoherent sheaves on the corresponding [[deRham space]]. \hypertarget{references}{}\subsection*{{References}}\label{references} Quasicoherent sheaves in [[E-∞ geometry]] (on ``[[Spectral Schemes]]'' over [[E-∞ rings]]) are discussed in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]} \end{itemize} Their [[descent]] properties are discussed in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Descent Theorems]]} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf}{pdf}) \end{itemize} and a [[Grothendieck existence theorem]] for [[coherent sheaves]] in this higher context is discussed in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf}{pdf}) \end{itemize} category: algebraic geometry [[!redirects quasicoherent sheaves]] [[!redirects quasicoherent module]] [[!redirects quasicoherent modules]] [[!redirects quasi-coherent sheaves]] [[!redirects quasi-coherent module]] [[!redirects quasi-coherent modules]] [[!redirects quasi-coherent sheaf]] [[!redirects quasicoherent sheaf of modules]] [[!redirects quasicoherent sheaves of modules]] \end{document}