restore page geometry

Author: ecpeterson

cooperations.tex

@@ -160,7 +160,11 @@ \section{Unstable contexts and the Steenrod algebra}\label{UnstableContextsSecti
 
 \begin{corollary}\label{CalculationOfUnstableSteenrodHopfRing}
 There is an isomorphism
-\[\HFtwo_* \OS{\HFtwo}{*} \cong \bigotimes_q \frac{\F_2[\alpha_{(j_1)} \circ \cdots \circ \alpha_{(j_q)} \mid j_1 \le \cdots \le j_q]}{(\alpha_{(j_1)} \circ \cdots \circ \alpha_{(j_q)})^{\ast 2}} \cong \bigotimes_q (\HFtwo_* \RP^\infty)^{\sm q},\] where \((\HFtwo_* \RP^\infty)^{\sm q}\) denotes the \(q\){\th} exterior power of \(\HFtwo_* \RP^\infty\) as a Hopf algebra~\cite[Propposition 5.5]{GoerssDieudonne}. \qed
+\begin{align*}
+\HFtwo_* \OS{\HFtwo}{*} & \cong \bigotimes_q \frac{\F_2[\alpha_{(j_1)} \circ \cdots \circ \alpha_{(j_q)} \mid j_1 \le \cdots \le j_q]}{(\alpha_{(j_1)} \circ \cdots \circ \alpha_{(j_q)})^{\ast 2}} \\
+& \cong \bigotimes_q (\HFtwo_* \RP^\infty)^{\sm q},
+\end{align*}
+where \((\HFtwo_* \RP^\infty)^{\sm q}\) denotes the \(q\){\th} exterior power of \(\HFtwo_* \RP^\infty\) as a Hopf algebra~\cite[Propposition 5.5]{GoerssDieudonne}. \qed
 \end{corollary}
 
 \begin{remark}[{\cite[Theorems 8.5 and 8.11]{Wilson}}]

finite.tex

@@ -107,7 +107,7 @@ \section{Descent and the context of a spectrum}\label{StableContextLecture}
 \end{center}
 A \index{sheaf!simplicial}quasicoherent (and Cartesian~\cite[Tag 09VK]{stacks-project}) sheaf \(\sheaf F\) over a simplicial scheme \(X\) is a sequence of quasicoherent sheaves \(\sheaf F[n]\) on \(X[n]\) as well as, for each map \(\sigma\co [m] \to [n]\) in the simplicial indexing category inducing a map \(X(\sigma)\co X[n] \to X[m]\), a natural choice of isomorphism of sheaves \[\sheaf F(\sigma)^*\co X(\sigma)^* \sheaf F[m] \to \sheaf F[n].\]  In particular, a pullback \(c^* \widetilde{M}\) gives such a quasicoherent sheaf on \(\mathcal D_f\).  By restricting attention to the first three levels we find exactly the structure of the descent datum described before.  Additionally, we have a natural \index{Segal condition}\textit{Segal isomorphism}
 \begin{align*}
-\mathcal D_f[1]^{\times_{\mathcal D_f[0]}(n)} & \xrightarrow{\simeq} \mathcal D_f[n] &
+\mathcal D_f[1]^{\times_{\mathcal D_f[0]}(n)} & \xrightarrow{\simeq} \mathcal D_f[n] \\
 \text{(cf.\ \(S \otimes_R S \otimes_R S\)} & \cong \text{\((S \otimes_R S) \otimes_S (S \otimes_R S)\) at \(n = 2\))},
 \end{align*}
 which shows that any descent datum (including those not arising, a priori, from a pullback) can be naturally extended to a full quasicoherent sheaf on \(\mathcal D_f\).
@@ -449,7 +449,7 @@ \section{The structure of \texorpdfstring{\(\moduli{fg}\)}{Mfg} I: The affine co
 to which we apply \((-) \otimes_{\Q[t]} \Q\) to calculate \[H^* \InternalHom{FormalSchemes}(B\G_a, \G_a)(\Q) = \begin{cases} \Q & \text{when \(* = 0\)}, \\ \Q & \text{when \(* = 1\)}, \\ 0 & \text{otherwise}. \end{cases}\] This means that every \(2\)--cocycle is a coboundary, symmetric or not.
 \item[\(\F_p\):] Now we are computing \(\Tor\) over a free commutative \(\F_p\)--algebra on one generator with divided powers.  Such an algebra splits as a tensor of truncated polynomial algebras, and again computing a minimal free resolution results in the calculation
 \[H^* \InternalHom{FormalSchemes}(B\G_a, \G_a)(\F_p) =\hspace{12em}\]
-\[\hspace{8.5em}=
+\[\hspace{6em}=
 \begin{cases}
 \frac{\F_p[\alpha_k \mid k \ge 0]}{\alpha_k^2 = 0} \otimes \F_p[\beta_k \mid k \ge 1] & \text{when \(p > 2\)}, \\
 \F_2[\alpha_k \mid k \ge 0] & \text{when \(p = 2\)},
@@ -623,7 +623,7 @@ \section{The structure of \texorpdfstring{\(\moduli{fg}\)}{Mfg} II: Large scales
 \begin{proof}
 We begin with the image factorization definition of $\moduli{fgl}^{\ptyp}$:
 \begin{center}
-\begin{tikzcd}[column sep=-1.0em]
+\begin{tikzcd}[column sep=-1.4em]
 & \sheaf O_{\moduli{fgl}, (p)} \arrow["s"]{rdd} \arrow["\eps"]{rrd} \arrow{rrr} & & & \sheaf O_{\moduli{fgl}} \otimes \Q \arrow["\eps"]{rrd} \arrow["s" near end]{rdd} \\
 & & & \sheaf O_{\moduli{fgl}, (p)} \arrow[crossing over]{rrr} & & & \sheaf O_{\moduli{fgl}} \otimes \Q \\
 \Z_{(p)}[v_1, \ldots, v_d, \ldots] \arrow["v"]{rr} & & \sheaf O_{\moduli{fgl}^{\ptyp}} \arrow["i"]{ru} \arrow{rrr} & & & \sheaf O_{\moduli{fgl}^{\ptyp}} \otimes \Q \arrow["i"]{ru}.
@@ -863,7 +863,7 @@ \section{The structure of \texorpdfstring{\(\moduli{fg}\)}{Mfg} III: Small scale
 In the cohomology of the coordinatized deformation complex, classes of degree \(1\) biject with first-order deformations of the identity automorphism, and classes of degree \(2\) biject with first-order deformations of the formal group law \(+_\Gamma\).
 \end{lemma}
 \begin{proof}[Proof sketch]
-Throughout, we consider the square-zero extension \(R = k[\eps] / \eps^2\) with pointing \(\eps = 0\), thought of as a universal tangent vector off of a \(k\)--point.
+Throughout, consider the square-zero extension \(R = k[\eps] / \eps^2\) with pointing \(\eps = 0\), thought of as a universal tangent vector off of a \(k\)--point.
 
 First, we address \(1\)--cocycles.  An automorphism \(\phi\) over \(R\) which reduces to the identity over \(k\) admits a series expansion
 \begin{align*}

preamble.tex

@@ -2,8 +2,8 @@
 
 % \usepackage{showframe}
 % turn these off if you ever resubmit to CUP
-% \usepackage{geometry}
-% \geometry{paperwidth=6in,paperheight=9in}
+\usepackage{geometry}
+\geometry{paperwidth=6in,paperheight=9in}
 
 
 %%% Created by texsupport 20/09/2017
@@ -415,17 +415,18 @@
 
 \usepackage{afterpage}
 \def\afterrectopage#1{{\ifodd\thepage#1\else\afterpage{#1}\fi}}
+% \def\afterrectopage#1{#1}
 
 % from hood chatham: an \intertext command that works in tikzcd
 
 % \usepackage{geometry}
 \makeatletter
-\edef\Gm@rmargin{\the\dimexpr1in + \hoffset + \oddsidemargin}
+\edef\Gm@rmargin{\the\dimexpr1in + \hoffset + \evensidemargin}
 \edef\Gm@lmargin{\the\dimexpr1in + \hoffset + \oddsidemargin}
 \makeatother
 \input{tikzcdintertext}
 
-% from hood chatham: 
+% from hood chatham:
 
 \input{sumfgl}
 
@@ -511,5 +512,3 @@
 \def\footnoterule{\kern-3\p@
   \hrule \@width 4nc \kern 2.6\p@} % the \hrule is .4pt high
 \makeatother
-
-