Merge remote-tracking branch 'refs/remotes/origin/master' into edits-…

…jhfung

quillen.tex

@@ -350,16 +350,15 @@ \section{Projectivization and Thom spaces}\citeme{Section 8 of the $H_\infty$ AH
 \end{remark}
 
 The following example connects this topic with that of \Cref{FormalVarietiesLecture}:
-%<<<<<<< HEAD
-\begin{example}\todo{Make this clearer: point out that the cofiber definition of the Thom space helps you make the first identification (that's what the ``zero section'' refers to), and then point out that $\zeta^*(M) = M / \sheaf I(0) \cdot M$ helps you over the last hump.}\todo{In class you wrote $\L - 1$ instead of $\L$ almost everywhere in this example.} 
-%If $\L$ denotes the canonical line bundle over $\CP^\infty$, then the zero section identifies $E^0 (\CP^\infty)^{\L}$ \footnote{What does this notation mean?  I would guess it's something like maps $\L \to E^0(\CP^\infty)$, but this doesn't seem to make sense.}with the augmentation ideal in $E^0 \CP^\infty$, and so we have an isomorphism $\ThomSheaf{\L} \cong \sheaf I(0)$.  Then, consider the map $\eps: * \to \CP^\infty$, which classifies a line bundle that Thomifies to $\CP^1 \to \CP^\infty$.  Using naturality, we see \todo{I think you should mention that the tilde means the sheaf associated to the module.  Also, in class you wrote $\pi_{-2} E$ instead of $\pi_2 E$}\[\widetilde{\pi_2 E} \cong \ThomSheaf{* \to \CP^\infty} \cong 0^* \sheaf I(0) \cong \omega_{\G_E},\] where $\G_E = \CP^\infty_E$ is the formal group associated to $E$--theory and $\omega_{\G_E}$ is its sheaf of invariant differentials\footnote{The identification of this with the sheaf of invariant differentials is something of a choice.  Certainly it is naturally isomorphic to $T_0^* \CP^\infty_E$, and this in turn is naturally isomorphic to $\omega_{\G_E}$, but deciding which of these two to write is a decision to be borne out as ``correct''.}.  More generally, if $k \eps$ \todo{It might be confusing using $\eps$ for two different things in the same paragraph.} is the trivial bundle of dimension $k$ over a point, then $\ThomSheaf{k \eps} \cong \omega_{\G_E}^{\otimes k}$.  If $f \co E \to F$ is an $E$--algebra (e.g., $F = E^{X_+}$), then this gives an interpretation of $\pi_{2k} F$ as $f_E^* \omega_{\G_E}^{\otimes k}$.
-If $\L$ denotes the canonical line bundle over $\C P^\infty$, then recall that its Thom complex is $T(\L) \simeq \C P^\infty$.  This is because the total space of the sphere subbundle of $\L$ is contractible: \[ S^1 \rightarrow EU(1) \rightarrow \C P^\infty \xrightarrow{cofib} T(\L). \]  Applying $E^*$ to the same cofiber sequence realizes $E^* T(\L)$ as the kernel of the augmentation map $E^* \C P^\infty \to E^* \S^0$, i.e., $E^* T(\L)$ is the augmentation ideal in $E^* \C P^\infty$.  So we have an isomorphism $\ThomSheaf{\L} \cong \sheaf I (0)$, the ideal sheaf of functions on $\C P^\infty_E$ vanishing at a point.  Then consider the map $\eps: * = \C P^0 \to \C P^\infty$, which classifies a line bundle that Thomifies to $\C P^1 \to \C P^\infty$.  This gives us the identification $\pi_{-*+2} E \cong E^* S^2 \cong E^* T(\epsilon^* L)$, which is a module over $E^* \C P^\infty$.  Using naturality, the associated line bundle over $\C P^\infty$ is \[\widetilde{\pi_{-*+2} E} = \ThomSheaf{\eps^* \L} \cong \eps_E^* \ThomSheaf{\L} \cong \eps_E^* \sheaf I (0).\] Picking a coordinate\todo{Can you do this without coordinates? Yes, sort of! To show that $\eps^* \sheaf I (0) \cong \eps^* \sheaf I (0)/\sheaf I (0)$, consider the SES $0 \to \sheaf I (0)^2 \to \sheaf I (0) \to \sheaf I (0) / \sheaf I (0)^2 \to 0$.  Applying $\eps^*$ and noting that it is right exact, we get a shorter exact sequence $\eps^* \sheaf I (0)^2 \to \eps^* \sheaf I (0) \to \eps* \sheaf I (0)/\sheaf I (0) \to 0$.  We will obtain our desired isomorphism once we show that $\eps^* \sheaf I (0)^2 \to \eps^* \sheaf I (0)$ is the zero map.  This is easily shown in coordinates, but all this is really saying is that a function that vanishes to second order has zero first derivative, once we observe that $\eps^* \sheaf I (0)^n$, i.e., pulling back to a point, only remembers the value of the $n$th derivative $f^{(n)}(0)$ at that point.} $x$ on $\C P^\infty$, we see that $\sheaf I(0)$ is the sheaf of sections of the $E_* \llbracket x \rrbracket$-module $x E_* \llbracket x \rrbracket$, so $\eps_E^* \sheaf I(0)$ is the sheaf of sections of the $E_*$-module $E_* \otimes_{E_* \llbracket x \rrbracket} x E_* \llbracket x \rrbracket \cong E_* \cdot x$, so on the level of sheaves we have \[\eps^* \sheaf I(0) \cong \eps^* \sheaf I(0)/\sheaf I(0)^2 = \eps^* T_0^* \G_E \cong \eps^* \omega_{\G_E}, \] where $\G_E = \CP^\infty_E$ is the formal group associated to $E$--theory and $\omega_{\G_E}$ is its sheaf of invariant differentials\footnote{The identification of this with the sheaf of invariant differentials is something of a choice.  Certainly it is naturally isomorphic to $T_0^* \CP^\infty_E$, and this in turn is naturally isomorphic to $\omega_{\G_E}$, but deciding which of these two to write is a decision to be borne out as ``correct''.}.  More generally, if $k \eps$ \todo{It might be confusing using $\eps$ for two different things in the same paragraph.} is the trivial bundle of dimension $k$ over a point, then $\ThomSheaf{k \eps} \cong \omega_{\G_E}^{\otimes k}$.  If $f \co E \to F$ is an $E$--algebra (e.g., $F = E^{X_+}$), then this gives an interpretation of $\pi_{2k} F$ as $f_E^* \omega_{\G_E}^{\otimes k}$.
-\todo{There is an issue with grading here.  Really when I wrote $E^*$ I meant $EP^0$ instead, and then I would have to consider $\L - 1$ instead of $\L$, where $T(\L - 1) \cong \Sigma^{-2} \C P^\infty$ to get the rest of the grading right, and also $\pi_* E$ instead of $\pi_0 E$. (I think.) I've noticed this problem elsewhere in the notes, and I think Eric needs to make a choice one and for all either way.}
-%=======
-%\begin{example}\label{Pi2AndInvariantDiffls}
-%\todo{Make this clearer: point out that the cofiber definition of the Thom space helps you make the first identification (that's what the ``zero section'' refers to), and then point out that $\zeta^*(M) = M / \sheaf I(0) \cdot M$ helps you over the last hump.}\todo{In class you wrote $\L - 1$ instead of $\L$ almost everywhere in this example.} 
-%If $\L$ denotes the canonical line bundle over $\CP^\infty$, then the zero section identifies $E^0 (\CP^\infty)^{\L}$ with the augmentation ideal in $E^0 \CP^\infty$, and so we have an isomorphism $\ThomSheaf{\L} \cong \sheaf I(0)$.  Then, consider the map $\eps: * \to \CP^\infty$, which classifies a line bundle that Thomifies to $\CP^1 \to \CP^\infty$.  Using naturality, we see \todo{I think you should mention that the tilde means the sheaf associated to the module.  Also, in class you wrote $\pi_{-2} E$ instead of $\pi_2 E$}\[\widetilde{\pi_2 E} \cong \ThomSheaf{* \to \CP^\infty} \cong 0^* \sheaf I(0) \cong \omega_{\G_E},\] where $\G_E = \CP^\infty_E$ is the formal group associated to $E$--theory and $\omega_{\G_E}$ is its sheaf of invariant differentials\footnote{The identification of this with the sheaf of invariant differentials is something of a choice.  Certainly it is naturally isomorphic to $T_0^* \CP^\infty_E$, and this in turn is naturally isomorphic to $\omega_{\G_E}$, but deciding which of these two to write is a decision to be borne out as ``correct''.}.  More generally, if $k \eps$ \todo{It might be confusing using $\eps$ for two different things in the same paragraph.} is the trivial bundle of dimension $k$ over a point, then $\ThomSheaf{k \eps} \cong \omega_{\G_E}^{\otimes k}$.  If $f \co E \to F$ is an $E$--algebra (e.g., $F = E^{X_+}$), then this gives an interpretation of $\pi_{2k} F$ as $f_E^* \omega_{\G_E}^{\otimes k}$.
-%>>>>>>> refs/remotes/origin/master
+
+% Eric, I wrote up this blurb following our discussion that one afternoon.  You've been changing things around in this section, so I don't know if it's still useful.
+
+%If $\L$ denotes the canonical line bundle over $\C P^\infty$, then recall that its Thom complex is $T(\L) \simeq \C P^\infty$.  This is because the total space of the sphere subbundle of $\L$ is contractible: \[ S^1 \rightarrow EU(1) \rightarrow \C P^\infty \xrightarrow{cofib} T(\L). \]  Applying $E^*$ to the same cofiber sequence realizes $E^* T(\L)$ as the kernel of the augmentation map $E^* \C P^\infty \to E^* \S^0$, i.e., $E^* T(\L)$ is the augmentation ideal in $E^* \C P^\infty$.  So we have an isomorphism $\ThomSheaf{\L} \cong \sheaf I (0)$, the ideal sheaf of functions on $\C P^\infty_E$ vanishing at a point.  Then consider the map $\eps: * = \C P^0 \to \C P^\infty$, which classifies a line bundle that Thomifies to $\C P^1 \to \C P^\infty$.  This gives us the identification $\pi_{-*+2} E \cong E^* S^2 \cong E^* T(\epsilon^* L)$, which is a module over $E^* \C P^\infty$.  Using naturality, the associated line bundle over $\C P^\infty$ is \[\widetilde{\pi_{-*+2} E} = \ThomSheaf{\eps^* \L} \cong \eps_E^* \ThomSheaf{\L} \cong \eps_E^* \sheaf I (0).\] Picking a coordinate\todo{Can you do this without coordinates? Yes, sort of! To show that $\eps^* \sheaf I (0) \cong \eps^* \sheaf I (0)/\sheaf I (0)$, consider the SES $0 \to \sheaf I (0)^2 \to \sheaf I (0) \to \sheaf I (0) / \sheaf I (0)^2 \to 0$.  Applying $\eps^*$ and noting that it is right exact, we get a shorter exact sequence $\eps^* \sheaf I (0)^2 \to \eps^* \sheaf I (0) \to \eps* \sheaf I (0)/\sheaf I (0) \to 0$.  We will obtain our desired isomorphism once we show that $\eps^* \sheaf I (0)^2 \to \eps^* \sheaf I (0)$ is the zero map.  This is easily shown in coordinates, but all this is really saying is that a function that vanishes to second order has zero first derivative, once we observe that $\eps^* \sheaf I (0)^n$, i.e., pulling back to a point, only remembers the value of the $n$th derivative $f^{(n)}(0)$ at that point.} $x$ on $\C P^\infty$, we see that $\sheaf I(0)$ is the sheaf of sections of the $E_* \llbracket x \rrbracket$-module $x E_* \llbracket x \rrbracket$, so $\eps_E^* \sheaf I(0)$ is the sheaf of sections of the $E_*$-module $E_* \otimes_{E_* \llbracket x \rrbracket} x E_* \llbracket x \rrbracket \cong E_* \cdot x$, so on the level of sheaves we have \[\eps^* \sheaf I(0) \cong \eps^* \sheaf I(0)/\sheaf I(0)^2 = \eps^* T_0^* \G_E \cong \eps^* \omega_{\G_E}, \] where $\G_E = \CP^\infty_E$ is the formal group associated to $E$--theory and $\omega_{\G_E}$ is its sheaf of invariant differentials\footnote{The identification of this with the sheaf of invariant differentials is something of a choice.  Certainly it is naturally isomorphic to $T_0^* \CP^\infty_E$, and this in turn is naturally isomorphic to $\omega_{\G_E}$, but deciding which of these two to write is a decision to be borne out as ``correct''.}.  More generally, if $k \eps$ \todo{It might be confusing using $\eps$ for two different things in the same paragraph.} is the trivial bundle of dimension $k$ over a point, then $\ThomSheaf{k \eps} \cong \omega_{\G_E}^{\otimes k}$.  If $f \co E \to F$ is an $E$--algebra (e.g., $F = E^{X_+}$), then this gives an interpretation of $\pi_{2k} F$ as $f_E^* \omega_{\G_E}^{\otimes k}$.
+%\todo{There is an issue with grading here.  Really when I wrote $E^*$ I meant $EP^0$ instead, and then I would have to consider $\L - 1$ instead of $\L$, where $T(\L - 1) \cong \Sigma^{-2} \C P^\infty$ to get the rest of the grading right, and also $\pi_* E$ instead of $\pi_0 E$. (I think.) I've noticed this problem elsewhere in the notes, and I think Eric needs to make a choice one and for all either way.}
+
+\begin{example}\label{Pi2AndInvariantDiffls}
+\todo{Make this clearer: point out that the cofiber definition of the Thom space helps you make the first identification (that's what the ``zero section'' refers to), and then point out that $\zeta^*(M) = M / \sheaf I(0) \cdot M$ helps you over the last hump.}\todo{In class you wrote $\L - 1$ instead of $\L$ almost everywhere in this example.}
+If $\L$ denotes the canonical line bundle over $\CP^\infty$, then the zero section identifies $E^0 (\CP^\infty)^{\L}$ \footnote{What does this notation mean?  I would guess it's something like maps $\L \to E^0(\CP^\infty)$, but this doesn't seem to make sense.}with the augmentation ideal in $E^0 \CP^\infty$, and so we have an isomorphism $\ThomSheaf{\L} \cong \sheaf I(0)$.  Then, consider the map $\eps: * \to \CP^\infty$, which classifies a line bundle that Thomifies to $\CP^1 \to \CP^\infty$.  Using naturality, we see \todo{I think you should mention that the tilde means the sheaf associated to the module.  Also, in class you wrote $\pi_{-2} E$ instead of $\pi_2 E$}\[\widetilde{\pi_2 E} \cong \ThomSheaf{* \to \CP^\infty} \cong 0^* \sheaf I(0) \cong \omega_{\G_E},\] where $\G_E = \CP^\infty_E$ is the formal group associated to $E$--theory and $\omega_{\G_E}$ is its sheaf of invariant differentials\footnote{The identification of this with the sheaf of invariant differentials is something of a choice.  Certainly it is naturally isomorphic to $T_0^* \CP^\infty_E$, and this in turn is naturally isomorphic to $\omega_{\G_E}$, but deciding which of these two to write is a decision to be borne out as ``correct''.}.  More generally, if $k \eps$ \todo{It might be confusing using $\eps$ for two different things in the same paragraph.} is the trivial bundle of dimension $k$ over a point, then $\ThomSheaf{k \eps} \cong \omega_{\G_E}^{\otimes k}$.  If $f \co E \to F$ is an $E$--algebra (e.g., $F = E^{X_+}$), then this gives an interpretation of $\pi_{2k} F$ as $f_E^* \omega_{\G_E}^{\otimes k}$.
 \end{example}
 
 Aside from this example, though, this construction on its own does not allow for the ready manipulation of line bundles.  However, our discussion yesterday centered on an equivalent presentation of line bundles on a formal curve: their corresponding divisors.  Following that cue, we now seek out a topological construction on vector bundles $V \to X$ which produces finite schemes over $X_E$.  A quick browse through the literature will lead one to the following:

sigma.tex

@@ -198,7 +198,7 @@ \section{Special divisors and the special splitting principle}\label{MSUDay}
 However, the use of \Cref{BSUtoBUtoCPinftyIsSexseq} inspires us to spend a moment longer with the associated formal schemes.  An equivalent statement is that there is a short exact sequence of formal group schemes
 \begin{center}
 \begin{tikzcd}
-BSU_E \arrow{r} \arrow[-,double]{d} & BU_E \arrow{r}{B\det} \arrow[-,double]{d} & BU(1)_E \arrow[-,double]{d} \\
+BSU_E \arrow{r} \arrow[equal]{d} & BU_E \arrow{r}{B\det} \arrow[equal]{d} & BU(1) \arrow[equal]{d} \\
 \SDiv_0 \CP^\infty_E \arrow{r} & \Div_0 \CP^\infty_E \arrow{r}{\mathrm{sum}} & \CP^\infty_E,
 \end{tikzcd}
 \end{center}