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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*{A Survey of Elliptic Cohomology - E-infinity rings and derived schemes} This is a sub-entry of \begin{itemize}% \item [[A Survey of Elliptic Cohomology]] \end{itemize} see there for background and context. This entry discusses the algebraic/homotopy theoretic prerequisites for [[derived algebraic geometry]]. Previous: \begin{itemize}% \item [[A Survey of Elliptic Cohomology - cohomology theories]] \item [[A Survey of Elliptic Cohomology - formal groups and cohomology]] \end{itemize} Next: \begin{itemize}% \item [[A Survey of Elliptic Cohomology - elliptic curves]] \end{itemize} \begin{quote}% the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: \textbf{somebody should go through this and polish} \end{quote} \hypertarget{contents}{}\section*{{contents}}\label{contents} \noindent\hyperlink{contents}{contents}\dotfill \pageref*{contents} \linebreak \noindent\hyperlink{part_1__the_sheaf_of_elliptic_cohomology_ring_spectra}{part 1 -- the sheaf of elliptic cohomology ring spectra}\dotfill \pageref*{part_1__the_sheaf_of_elliptic_cohomology_ring_spectra} \linebreak \noindent\hyperlink{part_2__the_stable_symmetric_monoidal_category_of_spectra}{part 2 - the stable symmetric monoidal $(\infty,1)$-category of spectra}\dotfill \pageref*{part_2__the_stable_symmetric_monoidal_category_of_spectra} \linebreak \noindent\hyperlink{part_3__brave_new_schemes}{part 3 - brave new schemes}\dotfill \pageref*{part_3__brave_new_schemes} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{part_1__the_sheaf_of_elliptic_cohomology_ring_spectra}{}\section*{{part 1 -- the sheaf of elliptic cohomology ring spectra}}\label{part_1__the_sheaf_of_elliptic_cohomology_ring_spectra} We will talk about a lifting problem that will lead to the formulation of [[tmf]]. This requires [[E-infinity ring]]s and [[derived algebraic geometry]]. \textbf{Definition} An $\Omega$-[[spectrum]] is a sequence of pointed [[topological space]]s $\{E_n\}$ and base-point preserving maps $\{\sigma_n : E_n \to \Omega E_{n+1}\}$ that are [[weak homotopy equivalence]]s. ($\Omega E_n$ is the [[loop space]] of $E_n$). if $\{E_n\}$ is an $\Omega$-spectrum, define $h^{-n}(X) := [X, E_n]$ ([[homotopy]] classes of continuous maps). Then this $h$ is a [[generalized (Eilenberg-Steenrod) cohomology theory]]. It should be noted that all our spaces are based and $h$ is a reduced cohomology theory. Define $\pi_n(E) := [S^0, E_n]$. $\pi_*(E)$ are the coefficients (i.e. the cohomology over the point of the corresponding unreduced theory) of $E$. \textbf{Brown's representability theory}: Any [[reduced cohomology theory]] on [[CW-complex]]es is [[representable functor|represented]] by an $\Omega$-[[spectrum]]. \textbf{examples} \begin{enumerate}% \item [[singular cohomology]] with coefficients in $A$: the [[Eilenberg-MacLane spectrum]] $H A$. \item complex [[K-theory]]: $K_n = \mathbb{Z} \times BU$ for $n$ even and $\cdots = U$ otherwise \end{enumerate} Le $M_{1,1}$ be the [[moduli stack]] of all [[elliptic curve]]s, then $Hom(Spec R, M_{1,1}) = \{elliptic curves over Spec R\}$. (we will construct this more rigorously later) If $\phi : Spec R \to M_{1,1}$ is a map that is a [[flat morphism]], then we obtain an [[elliptic cohomology]] theory called $A_{\phi}$. This assignment is a [[presheaf]] of [[cohomology theory|cohomology theories]]. To get a single [[cohomology theory]] from that we want to take global sections, but there is no good way to say what a global section of a cohomology-theoy valued functor would be. One reason is that there is not a good notion to say what a \emph{sheaf} of [[cohomology theory]]s is. But if we had an [[(infinity,1)-category]] valued functor, then [[Higher Topos Theory]] would provide all that technology. So that's what we try to get now. \textbf{goal} find lift \begin{displaymath} \itexarray{ && Spectra \\ & {}^{?}\nearrow & \;\;\;\downarrow^{represent} \\ \{\phi : Spec R \to M_{1,1}\} &\to& CohomologyTheories } \,. \end{displaymath} \textbf{Hopkins-Miller}: use the multiplicative nature of cohomology theories to solve this, i.e. instead look for a more refined lift \begin{displaymath} \itexarray{ && CommRingSpectra \\ & {}^{O_{M^{der}}}\nearrow & \downarrow \\ \{\phi : Spec R \to M_{1,1}\} &\to& MultiplicativeCohomologyTheories } \,. \end{displaymath} \textbf{theorem} There exists a [[symmetric monoidal category|symmetric monoidal]] [[model category]] $StTop$ of [[spectrum|spectra]] such that the [[homotopy category]] is the [[stable homotopy category]] as a symmetric monoidal category. This and the following is described in more detail at [[symmetric monoidal smash product of spectra]]. \textbf{Definition} An [[A-infinity ring]] is an ordinary [[monoid]] in $StTop$ and an [[E-infinity ring]] is an ordinray commutative monoid there. So an $E_\infty$-ring is an honest [[monoid]] with respect to the funny smash product that makes spectra a symmetric monoidal category, but it is just a monoid up to homotopy with respect to the ordinary product of spaces. For more on this see (for the time being) the literature referenced at [[stable homotopy theory]]. \textbf{proposition} Let $A$ be an [[A-infinity ring]] [[spectrum]]. \begin{enumerate}% \item the $\infty$-monoidal structure on the spectrum induces a [[multiplicative cohomology theory]]. \item $\pi_0(A)$ is a commutative ring \item $\pi_n(A)$ is a [[module]] over $\pi_0(A)$. \end{enumerate} \textbf{Definition} For $A$ an [[E-infinity ring]], $M$ with a map $A \wedge M \to M$ such that the obvious diagrams commute is a [[module]] for that [[E-infinity ring]]. \textbf{Proposition} $\pi_*(M)$ is a graded [[module]] over $\pi_*(A)$. \textbf{Definition} for $A$ an [[E-infinity ring]] and $M$ an $A$-module, we have that $M$ is \textbf{flat module} if \begin{enumerate}% \item $\pi_0(M)$ is flat over $\pi_0(A)$ in the ordinary sense \item $\forall n : \pi_n(A) \otimes_{\pi_0(A)} \pi_0(M) \to \pi_n(M)$ is an isomorphism of $\pi_0(M)$-modules \end{enumerate} \textbf{definition} a morphism $f : A \to B$ of [[E-infinity ring]]s is flat if $B$ regarded as an $A$-module using this morphism is flat. \textbf{Theorem (Goerss-Hopkins-Miller)}: A lift $O_{M_{1,1}, der}$ as indicated in the GOAL above (multiplicative version) does exists and is unique up to homotopy equivalence. The [[tmf]]-[[spectrum]] is the global sections of this: \begin{displaymath} tmf[\Delta^{-1}] = \Gamma(O_{M_{1,1}, der}) \end{displaymath} this is not elliptic (its not even nor has period 2), but is a multiplicative spectrum and hence defines a cohomology theory. The [[spectrum]] [[tmf]] is obtained in the same manner by replacing $M_{1,1}$ by its [[Deligne-Mumford compactification]]. \hypertarget{part_2__the_stable_symmetric_monoidal_category_of_spectra}{}\section*{{part 2 - the stable symmetric monoidal $(\infty,1)$-category of spectra}}\label{part_2__the_stable_symmetric_monoidal_category_of_spectra} recall that we want global sections of the [[presheaf]] \begin{displaymath} \{Spec R \to M_{1,1}\} \to CohomologyTheories \end{displaymath} (on the left we have something like the etale [[site]] of the moduli stack $M_{1,1}$ ) but there is no good notion of gluing in CohomologyTheories (lack of colimits) hence no good notion of sheaves with values in cohomology theories. $CohomologyTheories$ is the [[homotopy category]] of some other category, to be identified, and passage to homotopy categories may destroy existence of useful colimits. The category of CohomologyTheories ``is'' the stable homotopy category. A simple example: in the [[(infinity,1)-category]] [[Top]] we have the homotopy pushout \begin{displaymath} \itexarray{ S^1 &\to& D^2 \\ \downarrow && \downarrow \\ D^2 &\to& S^2 } \end{displaymath} but in the [[homotopy category]] the pushout is instead \begin{displaymath} \itexarray{ S^1 &\to& D^2 \\ \downarrow && \downarrow \\ D^2 &\to& * } \end{displaymath} The result is not even homotopy equivalent. In the homotopy category the pushout does not exist. So we want to refine $CohomologyTheories$ to the cateory of [[spectrum|spectra]] that they come from by the [[Brown representability theorem]]. In fact, we want to lift $MultiplicativeCohomologyTheories$ to that of [[E-infinity ring]]-spectra. The map \begin{displaymath} E_\infty Rings \to MultiplicativeCohomologyTheories \end{displaymath} should be that of taking the [[homotopy category of an (infinity,1)-category]]. \textbf{Approach A} (modern but traditional [[stable homotopy theory]]) choose a [[symmetric monoidal category|symmetric monoidal]] [[simplicial model category]] whose [[homotopy category]] is the [[stable homotopy category]] and whose [[tensor product]] is the [[smash product of spectra]]. For instance use the [[symmetric monoidal smash product of spectra]]. Then define [[E-infinity ring]] spectra to be ordinary [[monoid]] objects in this symmetric monoidal model category of spectra. \textbf{Approach B} ([[Jacob Lurie]]: be serious about working with [[(infinity,1)-category]] instead of just [[model category]] theory) . \begin{enumerate}% \item define [[(infinity,1)-category]] ([[Higher Topos Theory|chapter 1 of HTT]]) in this framework we'll have a [[stable (infinity,1)-category of spectra]], let's call that $Sp$ \item show that $Sp$ is a [[symmetric monoidal (infinity,1)-category]] \item show that the [[homotopy category of an (infinity,1)-category]] of $Sp$ is the [[stable homotopy category]], where the tensor product goes to the [[smash product of spectra]] \item define an [[E-infinity ring]] to be a [[commutative monoid in an (infinity,1)-category]] in $Sp$. \end{enumerate} These two approaches are equivalent is a suitable sense. See [[Noncommutative Algebra]], page 129 and [[Commutative Algebra]], Remark 0.0.2 and paragraph 4.3. [[derived algebraic geometry]] categorifies [[algebraic geometry]] [[E-infinity ring]] categoriefies commutative [[ring]] [[(infinity,1)-category]] catgeorifies [[category]] \textbf{Definition} An [[(infinity,1)-category]] is (for instance modeled by) \begin{itemize}% \item a [[Top]]-[[enriched category]] (essentially [[simplicially enriched category]]) \item a [[quasi-category]] (see there for details). In fact, see chapter 1 of [[Higher Topos Theory]] for lots of details. \end{itemize} use [[homotopy coherent nerve]] to go from a [[simplicially enriched category]] to its corresponding [[quasi-category]] \textbf{definition} [[homotopy category of an (infinity,1)-category]] (see there) \textbf{definition} morphism of [[(infinity,1)-category|(infinity,1)-categories]] is, when regarded as a [[quasi-category]], just a morphism of [[simplicial set]]s.: this is an [[(infinity,1)-functor]]. There is an [[(infinity,1)-category of (infinity,1)-functors]] between two [[(infinity,1)-categories]] \textbf{why simplicial sets?} because they provide a convenient calculus for doing [[homotopy coherent category theory]]. suppose some [[(infinity,1)-category]] $C$ and its homotopy category $C \to h C$. A commutative-up-to-homotopy diagram in $C$ is a functor $I \to h C$ \begin{displaymath} \itexarray{ && C \\ && \downarrow \\ I &\to& h C } \end{displaymath} for $I$ some diagram category. to get a \textbf{homotopy coherent} diagram instead take the [[nerve]] $N(I)$ of $I$ and then map $N(I) \to C$. The nerve automatically encodes the homotopy coherence. See [[Higher Topos Theory]] pages 37, 38 (but the general idea is well known from [[simplicial model category]] theory). Now let $C$ be an [[(infinity,1)-category]]. Suppose that it has a [[zero object]] $0 \in C$, i.e. an object that is both an [[initial object]] and a [[terminal object]]. Assume that $C$ admits [[kernel]]s and [[cokernel]]s, i.e. all [[homotopy pullback]]s and pushouts with $0$ in one corner. Then from this we get [[loop space object]]s $\Omega X$ and [[delooping]] objects $B X$ in $C$ (called suspension objects $\Sigma X$ in this context). \begin{displaymath} \itexarray{ X &\stackrel{f}{\to}& Y \\ \downarrow &\Downarrow& \downarrow \\ 0 &\to& coker f } \;\;\;\; \itexarray{ ker(g) &\stackrel{}{\to}& X \\ \downarrow &\Downarrow& \downarrow^g \\ 0 &\to& Y } \end{displaymath} in particular a [[loop space object]] $\Omega Y$ is the kernel of the 0-map,while the suspension $\Sigma X$ is the cokernel \begin{displaymath} \itexarray{ X &\stackrel{f}{\to}& 0 \\ \downarrow &\Downarrow& \downarrow \\ 0 &\to& \Sigma X } \;\;\;\; \itexarray{ \Omega Y &\stackrel{}{\to}& 0 \\ \downarrow &\Downarrow& \downarrow^g \\ 0 &\to& Y } \end{displaymath} One example of this is the [[(infinity,1)-category]] of [[pointed object|pointed]] [[topological space]]s. \textbf{definition} a [[prespectrum object]] in an [[(infinity,1)-category]] $C$ with the properties as above is a [[(infinity,1)-functor]] \begin{displaymath} X : N(\mathbb{Z} \times \mathbb{Z}) \to C \end{displaymath} such that $X(i,j)$ for $i \neq j$ is [[zero object]] 0. \begin{displaymath} \itexarray{ X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0 \\ \downarrow &\searrow& \downarrow^g \\ X(n+1,n) \simeq 0 &\to& X(n+1,n+1) } \end{displaymath} (everything filled with 2-cells aka homotopies) since we have cokernels we get maps from the universal property \begin{displaymath} \itexarray{ X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0 \\ \downarrow &\searrow& \downarrow^g \\ 0 &\to& \Sigma X(n,n) \\ &&& \searrow^{\alpha_n} \\ &&&& X(n+1,n+1) } \end{displaymath} and analogously maps $\beta_n : X(n,n) \to \Omega X(n+1, n+1)$ now $X$ is a \textbf{spectrum object} if the $\beta_n$ are equivalences, for all $n$. (We don't require $\alpha_n$ to be equivalences.) so to each [[(infinity,1)-category]] $C$ we get another [[(infinity,1)-category]] $Sp(C)$, the full subcategory $Fun(N(\mathbb{Z}\times \mathbb{Z}), C)$ on the [[spectrum object]]s. In particular, we set \begin{displaymath} Sp := Sp(Top) \end{displaymath} the [[stable (infinity,1)-category of spectra]] is the stabilization of the [[(infinity,1)-category]] [[Top]] of [[topological space]]s. \begin{quote}% I think we need pointed topological spaces here? \end{quote} \textbf{Fact}: $Sp$ has an essentially unique structure of a [[symmetric monoidal (infinity,1)-category]]. This monoidal structure $\otimes$ is uniquely characterized by the following two properties: \begin{enumerate}% \item $\otimes$ preserves limits and colimits. \item the [[sphere spectrum]] is the [[monoidal unit]]/[[tensor unit]] wrt $\otimes$. \end{enumerate} \textbf{definition} A [[symmetric monoidal (infinity,1)-category]] structure on an [[(infinity,1)-category]] $C$ is given by the following data: \begin{enumerate}% \item another [[(infinity,1)-category]] $C^\otimes$ with an [[(infinity,1)-functor]] $C^\otimes \to N(\Gamma)$ that is a [[coCartesian fibration]] \end{enumerate} where $\Gamma$ is [[Segal's category]] with objects finite [[pointed object|pointed]] [[set]]s and morphisms basepoint preserving [[function]]s between sets. such that $C^\otimes_{\langle 1\rangle} \simeq C$ where $C^\otimes_{\langle 1\rangle}$ is the fiber over $\langle 1\rangle = \{*,1\}$, i.e. the pullback \begin{displaymath} \itexarray{ C^\otimes_{\langle 1\rangle} &\to& C^\otimes \\ \downarrow &pullback& \downarrow \\ \{\langle 1\rangle\} &\to& N(\Gamma) } \end{displaymath} \begin{quote}% here should go some pictures that illustarte this. But see the first few pages of [[Noncommutative Algebra]] for the intuition and motivation. \end{quote} so let $C$ now be a [[symmetric monoidal (infinity,1)-category]]. \textbf{definition} A [[commutative monoid in an (infinity,1)-category|commutative monoid in]] $C$ is a [[section]] $s$ of the structure map mentioned above $C^\otimes \to N(\Gamma) \stackrel{s}{\to} C^\otimes$. The monoid object itself is the image of $\langle 1 \rangle$ under $s$, $A = s(\langle 1 \rangle)$. (Sort of. I think the whole point is that we don't ever say something like ``this \emph{particular} $A$ is \emph{the} monoid object''. Rather, the picture should roughly be that we have all of the standard diagrams describing a commutative monoid object, except that the various objects in the diagrams are \emph{not necessarily the same object}. However, these \emph{a priori} different objects will be \emph{a fortiori} homotopy equivalent, so that in particular the usual picture will reappear in the homotopy category. Moreover, of course, these diagrams will not be strictly commutative, but commutative up to coherent homotopy, so that in particular the usual strict commutativity reappears after passage to the homotopy category.) \begin{quote}% There is one more condition on $s$, though. \end{quote} \textbf{definition} an [[E-infinity ring]] spectrum is a [[commutative monoid in an (infinity,1)-category]] in the [[stable (infinity,1)-category of spectra]] $Sp$. $E_\infty$-rings themselves form an [[(infinity,1)-category]]. And this has all [[limit]]s and [[colimit]]s (see DAG III 2.1, 2.7), so we can talk about sheaves of $E_\infty$ rings! \hypertarget{part_3__brave_new_schemes}{}\section*{{part 3 - brave new schemes}}\label{part_3__brave_new_schemes} Now the theory of [[scheme]]s and [[derived scheme]]s, but not over [[simplicial commutative ring]]s, but over [[E-infinity ring]]s. So we are trying to guess the content of the not-yet-existsting \begin{itemize}% \item [[Jacob Lurie]], [[Spectral Schemes]]. \end{itemize} Let $A$ be an [[E-infinity ring]]. Define its [[spectrum of an E-infinity ring]] $Spec A$ as the [[ringed space]] $(|Spec A|, \mathcal{O}_{Spec A})$ whose underlying [[topological space]] is the ordinary spectrum of the degree-0 ring \begin{displaymath} |Spec A| := Spec \pi_0 A \end{displaymath} and where $\mathcal{O}_{Spec A}$ is given on Zariski-opens $D(f)$ for any $f \in \pi_0 A$ by \begin{displaymath} \mathcal{O}_{Spec A}(D(f)) := A[f^{-1}] \,. \end{displaymath} Here $A \to A[f^{-1}]$ is characterized by the following equivalent ways: \begin{enumerate}% \item $\pi_\bullet A \to \pi_*(A[f^{-1}])$ identify $\pi_{\bullet}(A[f^{-1}])$ with $\pi_\bullet$ \item $\forall$ $E_\infty$-rings the induced map $Hom(A[f^{-1}],B) \to Hom(A,B)$ is a [[homotopy equivalence]] of the left hand side with the subspace of the right hand side which takes $f \in \pi_0 A$ to an invertible element of $\pi_0 B$. \end{enumerate} This geometry over [[E-infinity ring]]s is in [[spectral algebraic geometry]]/[[brave new algebra|brave new algebraic geometry]]. The analog for [[simplicial commutative ring]]s instead of is what is discussed at [[derived scheme]]. \textbf{theorem} ([[Jacob Lurie]]) If $X$ s a space and $\mathcal{O}$ a sheaf of [[E-infinity ring]]s then $(X,\pi_0 \mathcal{O}_X)$ is a classical [[scheme]] and $\pi_n \mathcal{O}_X$ is a quasicoherent $\pi_0 \mathcal{O}_X$-[[module]]. \textbf{theorem} there exists a [[derived Deligne-Mumford stack]] $(M_{1,1}, \mathcal{O}^{der}_{M_{1,1}})$ such that $(M_{1,1}, \pi_0 \mathcal{O}^{der}_{M_{1,1}})$ is the ordinary [[Deligne-Mumford stack|DM-]] [[moduli stack]] of [[elliptic curve]]s. \hypertarget{references}{}\section*{{References}}\label{references} \begin{itemize}% \item [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]] \end{itemize} \end{document}