Finished reading 5.5

sigma.tex

@@ -714,7 +714,7 @@ \section{Unstable additive cooperations for $kU$}
 
 Write $H = H\F_p$.  Today we will study the effect of the map $\hat \Pi_3$ in ordinary homology.  Many parts of the proof we explored for $E$--theory break.  Topologically, the Serre spectral sequence for $H^* BU[6, \infty)$ is not even-concentrated and so is not forced to collapse.  Algebraically, because $\G_a$ is not $p$--divisible the behavior of the model exact sequence is also suspect.  Because the situation has fewer insulating good properties, we are forced to actually consider it carefully.  The upside, however, is that the standard group law on $\G_a$ is simple enough that we can compute the problem to death.
 
-We begin with the topological half of our tasks.  The Serre spectral sequence \[E_2^{*, *} = H\F_p^* BSU \otimes H\F_p^* \OS{H\Z}{3}) \Rightarrow H\F_p^* BU[6, \infty)\] is quite accessible, and we will recount the case of $p = 2$.  In this case, the spectral sequence has $E_2$--page \[E_2^{*, *} = H\F_2^* BSU \otimes H\F_2^* \OS{H\Z}{3} \cong \F_2[c_2, c_3, \ldots] \otimes \F_2\left[\Sq^I \iota_3 \middle| \begin{array}{c} I_j \ge 2I_{j+1}, \\ 2 I_1 - I_+ \end{array}\right].\]  Because the target is $6$--connective, we must have the transgressive differential $d_4 \iota_3 = c_2$, which via the Kudo transgression theorem spurs the much larger family of differentials \[d_{4+I_+} \Sq^I \iota_3 = \Sq^I c_2.\]  This necessitates understanding the action of the Steenrod operations on the cohomology of $BSU$, which is due to Wu\citeme{Wu formulas, maybe May's concise book}:\todo{Can these formulas be read off from the divisorial calculation?  Maybe not, since it's easy to read off the Milnor primitives but harder to see the Steenrod squares.} \[\Sq^{2^j} \cdots \Sq^4 \Sq^2 c_2 = c_{1 + 2^j}.\]  Accounting for the squares of classes left behind, this culminates in the following calculation:\todo{This spectral sequence can be drawn in using Hood's package.}
+We begin with the topological half of our tasks.  The Serre spectral sequence \[E_2^{*, *} = H\F_p^* BSU \otimes H\F_p^* \OS{H\Z}{3} \Rightarrow H\F_p^* BU[6, \infty)\] is quite accessible, and we will recount the case of $p = 2$.  In this case, the spectral sequence has $E_2$--page \[E_2^{*, *} = H\F_2^* BSU \otimes H\F_2^* \OS{H\Z}{3} \cong \F_2[c_2, c_3, \ldots] \otimes \F_2\left[\Sq^I \iota_3 \middle| \begin{array}{c} I_j \ge 2I_{j+1}, \\ 2 I_1 - I_+ \end{array}\right].\] \todo{$2 I_1 - I_+$ doesn't look like a condition.  Also, what is $I_+$?  Somehow this should mean that you start with $\Sq^2$ not $\Sq^1$.}  Because the target is $6$--connective, we must have the transgressive differential $d_4 \iota_3 = c_2$, which via the Kudo transgression theorem spurs the much larger family of differentials \[d_{4+I_+} \Sq^I \iota_3 = \Sq^I c_2.\]  This necessitates understanding the action of the Steenrod operations on the cohomology of $BSU$, which is due to Wu\citeme{Wu formulas, maybe May's concise book}:\todo{Can these formulas be read off from the divisorial calculation?  Maybe not, since it's easy to read off the Milnor primitives but harder to see the Steenrod squares.} \[\Sq^{2^j} \cdots \Sq^4 \Sq^2 c_2 = c_{1 + 2^j}.\]  Accounting for the squares of classes left behind, this culminates in the following calculation:\todo{This spectral sequence can be drawn in using Hood's package.}
 
 \begin{theorem}\label{HF2BU6Calculation}
 There is an isomorphism \[H\F_2^* BU[6, \infty) \cong \frac{H\F_2^* BU}{(c_j \mid j \ne 2^k + 1, j \ge 3)} \otimes F_2[\iota_3^2, (\Sq^2 \iota_3)^2, \ldots]. \qed\]
@@ -739,18 +739,18 @@ \section{Unstable additive cooperations for $kU$}
 
 
 
-Now, we turn to the algebra.  The main idea, as already used in \Cref{CalculationOfLTTangentSpace}, is to first perform a tangent space calculation of \[T_0 C^k(\G_a; \Gm) \cong C^k(\G_a; \G_a),\] then study the behavior of the different tangent directions to determine the full object $C^k(\G_a; \Gm)$.  As a warm-up, we will first consider the case $k = 2$:
+Now, we turn to the algebra.  The main idea, as already used in \Cref{CalculationOfLTTangentSpace}\todo{Is this really the best example to reference?}, is to first perform a tangent space calculation \[T_0 C^k(\G_a; \Gm) \cong C^k(\G_a; \G_a),\] then study the behavior of the different tangent directions to determine the full object $C^k(\G_a; \Gm)$.  As a warm-up, we will first consider the case $k = 2$:
 \begin{corollary}[{cf.\ \Cref{CohomologyOfGa}}]
 The unique symmetric $2$--cocycle of homogeneous degree $n$ has the form \[c_n(x, y) = \begin{cases} (x + y)^n - x^n - y^n & \text{if $n \ne p^j$}, \\ \frac{1}{p} \left( (x + y)^n - x^n - y^n \right) & \text{if $n = p^j$}. \end{cases} \qed\]
 \end{corollary}
 
-Our goal, then, is to select such a $2$--cocycle $f$ and study the minimal conditions needed on a symbol $a$ to produce a multiplicative $2$--cocycle of the form $1 + af + \cdots$.  Since $c_n = \frac{1}{d_n} \delta(x^n)$ is itself produced by an additive formula, life would be easiest if we had access to an exponential, so that we could build \[\text{``$ \delta \exp(a_n x^n)^{1/d_n} = \exp(\delta a_n x^n / d_n) = \exp(a_n c_n). $''}\]  However, the existence of an exponential series is equivalent to requiring that $a_n$ have all fractions, which turns out not to be minimal.  In fact, \emph{no} conditions on $a_n$ are required \emph{at all}, if we tweak the definition of an exponential series:
+Our goal, then, is to select such a $2$--cocycle $f$ and study the minimal conditions needed on a symbol $a$ to produce a multiplicative $2$--cocycle of the form $1 + af + \cdots$.  Since $c_n = \frac{1}{d_n} \delta(x^n)$ \todo{The notation is a little odd, since you're conflating $x^n$ with the function $x \mapsto x^n$.} is itself produced by an additive formula, life would be easiest if we had access to an exponential, so that we could build \[\text{``$ \delta \exp(a_n x^n)^{1/d_n} = \exp(\delta a_n x^n / d_n) = \exp(a_n c_n). $''}\]  However, the existence of an exponential series is equivalent to requiring that $a_n$ have all fractions, which turns out not to be minimal.  In fact, \emph{no} conditions on $a_n$ are required \emph{at all}, if we tweak the definition of an exponential series:
 
 \begin{definition}\todo{Where does this come from? I've never learned a universal property for it. That bothers me. It must have something to do with $p$--typification.}
 The \textit{Artin--Hasse exponential} is the power series \[E_p(t) = \exp\left( \sum_{j=0}^\infty \frac{t^{p^j}}{p^j} \right) \in \Z_{(p)}\ps{t}.\]
 \end{definition}
 
-This series has excellent properties, mimicking those of $\exp(t)$ as closely as possible while keeping coefficients in $\Z_{(p)}$ rather than in $\Q$.  Writing $\delta\co C^1 \to C^2$ and \[d_n = \begin{cases} 1 & \text{if $n = p^j$}, \\ 0 & \text{otherwise}, \end{cases}\] we set \[g_n(x, y) := \delta E_p(a_n x^n)^{1/p^{d_n}} = \exp\left( \sum_{j=0}^\infty \frac{a_n^{p^j} \delta x^{np^j}}{p^{j + d_n}} \right) = \exp\left( \sum_{j=0}^\infty \frac{a_n^{p^j} c_{np^j}(x, y)}{p^j} \right).\]  This gives a point in $C^2(\G_a; \Gm)(\Z_{(p)}[a_n])$, and exhaustion of the tangent space proves the following Lemma:
+This series has excellent properties, mimicking those of $\exp(t)$ as closely as possible while keeping coefficients in $\Z_{(p)}$ rather than in $\Q$.  Writing $\delta\co C^1 \to C^2$ and \[d_n = \begin{cases} 1 & \text{if $n = p^j$}, \\ 0 & \text{otherwise}, \end{cases}\] we set \[g_n(x, y) := \delta E_p(a_n x^n)^{1/p^{d_n}} = \exp\left( \sum_{j=0}^\infty \frac{a_n^{p^j} \delta x^{np^j}}{p^{j + d_n}} \right) = \exp\left( \sum_{j=0}^\infty \frac{a_n^{p^j} c_{np^j}(x, y)}{p^j} \right).\]  This gives a point in $C^2(\G_a; \Gm)(\Z_{(p)}[a_n])$, and exhaustion of the tangent space\todo{Granting that this exhausts the tangent space, how do we recover the global scheme?  Even if we had an inverse function theorem (like Thm.~2.1.4), I would only expect that to be at most a local isomorphism -- unless these are all really formal schemes masquerading as ordinary schemes?} proves the following Lemma:
 
 \begin{lemma}[{\cite[Proposition 3.9]{AHSTheoremOfTheCube}}]
 The map \[\Spec \Z_{(p)}[a_n \mid n \ge 2] \xrightarrow{\prod_{n \ge 2} g_2} C^2(\G_a; \Gm) \times \Spec \Z_{(p)}\] is an isomorphism. \qed