In particular, the $(-1)$-stem in the descent spectral sequence for the Picard sheaf $\pic(\mathcal E)$ gives an upper bound on the size of the relative Brauer group. We will draw consequences from this in the next subsection.

§.§ Brauer groups at height one

§ APPENDIX: DÉCALAGE RESULTS

We make the derivation of the descent spectral sequence a little more explicit, and relate it to the spectral sequence obtained from the covering $\G \twoheadrightarrow *$. For clarity most of this section is written in a general context, but the main result is <ref>, which will be used to relate the descent spectral sequence to the $\K$-local $\E$-Adams spectral sequence through décalage.

Let $\mathcal C$ be a site, and write $
\mathcal A \coloneqq \Sh[\mathcal Sp]{\mathcal C}$. Given any object $\mathcal F \in \mathcal A$, there are two natural filtrations one can consider:

* The usual t-structure on $\mathcal Sp$ induces one on $\mathcal A$, defined by the property that $\mathcal F' \in \mathcal A$ is $t$-truncated if and only if $\mathcal F'(X) \in \mathcal Sp_{\leq t}$ for every $X \in \mathcal C$. One can therefore form the Postnikov tower

\[ \mathcal F \simeq \lim( \cdots \to \tau_{\leq t} \mathcal F \to \cdots) \]

in $\mathcal A$, and obtain a tower of spectra on taking global sections.

* Suppose $U \twoheadrightarrow *$ is a covering of the terminal object. Since $A$ is a sheaf, we can recover $\Gamma \mathcal F$ as the limit of its Čech complex for the covering, and this has an associated tower. Explicitly, write $\operatorname{Tot}_q = \lim_{\bm \Delta_{\leq q}}$ so that

\[ \Gamma \mathcal F \simeq \operatorname{Tot} \mathcal F(U^\bullet) \simeq \lim (\cdots \to \operatorname{Tot}_0 \mathcal F(U^\bullet) ). \]

Any tower of spectra $X = \lim(\cdots \to X_t \to \cdots)$ gives rise to a (conditionally convergent) spectral sequence, as in <ref>. Respectively, in the cases above these read

\begin{equation} \label{eqn:descent spectral sequence for a sheaf}
E^{s,t}_2 = \pi_{t-s} \Gamma \tau_t \mathcal F \implies \pi_{t-s} \Gamma \mathcal F,
\end{equation}

\begin{equation} \label{eqn:Cech spectral sequence for a sheaf}
\check E_2^{p,q} = \pi_{q-p} f_q \mathcal F(U^\bullet) \implies \pi_{q-p} \Gamma \mathcal F,
\end{equation}

where $f_q$ denotes the fibre of the natural transformation $\operatorname{Tot}_q \to \operatorname{Tot}_{q-1}$. In each case, the $d_r$ differential has bidegree $(r+1, r)$ in the displayed grading.

For our purposes, the first spectral sequence, whose $E_2$-page and differentials are both defined at the level of truncations, is useful for interpreting the descent spectral sequence: for example, we use this in <ref>. On the other hand, the second spectral sequence is more easily evaluated once we know the value of a proétale sheaf on the free $\G$-sets. It will therefore be important to be able to compare the two, and this comparison is made using the décalage technique originally due to Deligne in <cit.>. The following material is well known (see for example <cit.>), but we include an exposition for convenience and to fix indexing conventions. The décalage construction of <cit.>, which `turns the page' of a spectral sequence by functorially associating to a filtered spectrum its decalée filtration, is closely related but not immediately equivalent.

Recall that any tower dualises to a filtration (this will be recounted below); the proof is cleanest in the slightly more general context of bifiltered spectra, and so we will make the connection between <ref> and <ref> explicit after proving a slightly more general result.

We suppose therefore that $X$ is a spectrum equipped with a (complete and decreasing) bifiltration. That is, we have a diagram of spectra $X^{t,q}: \Z[]\op \times \Z[]\op \to \mathcal Sp$ such that $X = \colim_{p,q} X^{t,q}$. We will write $ X^{-\infty,q} \coloneqq \colim_t X^{t, q}$ for any fixed $q$, and likewise $ X^{t,-\infty} \coloneqq \colim_q X^{t, q}$ for fixed $t$. Finally, write $X^{t/t', q/q'} \coloneqq \cofib (X^{t',q'} \to X^{t,q})$.

Let $X$ be a bifiltered spectrum, and suppose that for all $t$ and $q$ we have $\pi_s X^{t/t+1, q/q+1} = 0$ unless $s = t - q$. Then there is an isomorphism

\begin{equation} \label{eqn:decalage on first page}
{}^1 E_2^{s,t} \simeq {}^2 E_3^{2s-t,s},
\end{equation}

where the left-hand side is the spectral sequence for the filtration $X = \colim_t X^{t,-\infty}$ and the right-hand for $X = \colim_q X^{-\infty,q}$.

This isomorphism is compatible with differentials, and so extends to isomorphisms

\[ {}^1 E_r^{s,t} \simeq {}^2 E_{r+1}^{2s-t,s} \]

for $1 \leq r \leq \infty$.

We begin with the isomorphism <ref>: it is obtained by considering the following trigraded spectral sequences, which converge to the respective $E_2$-pages.

\begin{align}
E_2^{s,t,q} = \pi_{q-s} X^{t/t+1, q+1/q} &\implies  \pi_{q-s} X^{t/t+1, -\infty} = {}^1 E_2^{s+t-q,t}, \label{eqn:decalage trigraded sseq for first filtration}\\
E_2^{s,t,q} = \pi_{t-s} X^{t+1/t, q+1/q} &\implies \pi_{t-s} X^{-\infty, q+1/q} = {}^2 E_2^{s + q - t,q}. \label{eqn:decalage trigraded sseq for second filtration}
\end{align}

The $d_r$ differentials have $(s,t,q)$-tridegrees $(r+1, 0, r)$ and $(r+1, r, 0)$ respectively. Both spectral sequences therefore take a very simple form, because we have assumed each object $X^{t+1/t, q+1/q}$ is Eilenberg-Mac Lane. They are displayed in <ref>.

[name = firsttrigraded,
Adams grading, scale = .5,
classes = draw = none,
x label = $q-s$,
y label = $s$,
no ticks

in -1,...,10
["*"](, 1 - 2*)

in 0,...,10
[̣red]2(, 1 - 2*)

[name = secondtrigraded,
Adams grading, scale = .5,
classes = draw = none,
x label = $t-s$,
y label = $s$,
no ticks
in 0,...,10
["*"](, 0)

[x range = 05, y range = -22, grid = chess, page = 2, name = firsttrigraded]
Spectral sequence <ref> at fixed $t$. The $y$-intercept is $s = t$.

[x range = 05, y range = -22, grid = chess, page = 3, name = secondtrigraded]
Spectral sequence <ref> at fixed $q$. The $y$-intercept is $s = q$.

The trigraded spectral sequences computing ${}^1 E_2^{*,t}$ and ${}^2 E_2^{*,q}$ respectively.

In particular, the first collapses after the $E_2$-page, so that ${}^1E_2^{*,t}$ is the cohomology of the complex

\begin{equation}
\cdots \to \pi_{t-q} X^{t/t+1, q/q+1} \to \pi_{t-q-1} X^{t/t+1, q+1/q+2} \to \cdots
\label{eqn:complex from decalage trigraded spectral sequence}
\end{equation}

whose differentials are induced by the composites

\[ X^{t+1/t, q/q+1} \to \Sigma X^{t+1/t, q+1} \to \Sigma X^{t+1/t, q+1/q+2} .\]

More precisely, ${}^1 E_2^{s,t} = \pi_{t-s} X^{t/t+1, -\infty} \cong H^{s} \left(\pi_{t-*} X^{t/t+1,*/*+1} \right)$.

The second spectral sequence is even simpler, collapsing immediately to give

\[ {}^2 E_2^{2s-t,s} =\pi_{t-s} X^{-\infty, s/s+1} \cong \pi_{t-s} X^{t/t+1, s/s+1}. \]

We can further identify the first differential on ${}^2E_2^{2s-t,s}$: it is induced by the map $X^{-\infty, 2s-t/2s-t+1} \to \Sigma X^{-\infty, 2s-t+1/2s-t+2}$, and so the identification ${}^1 E_2^{s, t} \cong {}^2 E_3^{2s-t, s}$ follows from the commutative diagram below, in which the top row is part of the complex <ref>.

\[\begin{tikzcd}[column sep = small]
{\pi_{t-s} X^{t/t+1, s/s+1}} & {\pi_{t-s} \Sigma X^{t/t+1, s+1}} & {\pi_{t-s} \Sigma X^{t/t+1, s+1/s+2}} \\
{\pi_{t-s} X^{-\infty, s/s+1}} & {\pi_{t-s} \Sigma X^{-\infty, s+1}} & {\pi_{t-s} \Sigma X^{-\infty, s+1/s+2}}
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-1, to=2-1, "\sim"']
\arrow[from=1-2, to=2-2]
\arrow[from=1-3, to=2-3, "\sim"]
\end{tikzcd}\]

We next argue that this extends to an isomorphism of spectral sequences ${}^1 E_r^{s,t} \cong {}^2 E_{r+1}^{2s-t,s}$. To do so we will give a map of exact couples as below:

\begin{align*}
\begin{array}{ccc}
\left( % https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpX3MgXFxHYW1tYShYXlxcYnVsbGV0LCBcXHRhdV97XFxnZXEgdH1BKSJdLFsyLDAsIlxccGlfcyBcXEdhbW1hKFheXFxidWxsZXQsIFxcdGF1X3tcXGdlcSB0LTF9QSkiXSxbMSwxLCJcXHBpX3MgXFxHYW1tYShYXlxcYnVsbGV0LCBcXHRhdV97dH1BKSJdLFswLDFdLFsxLDJdLFsyLDBdXQ==
\begin{tikzcd}[ampersand replacement = \&, column sep = tiny]
{{}^1 D_2^{s,t}} \&\& {{}^1 D_2^{s,t}} \\
\& {{}^1 E_2^{s,t}}
\arrow[from=1-1, to=1-3]
\arrow[from=1-3, to=2-2]
\arrow[from=2-2, to=1-1]
\end{tikzcd} \right) & \to & \left( % https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpX3MgXFxUb3Ree3Qtc30gIFxcR2FtbWEoWF5cXGJ1bGxldCwgQSkiXSxbMiwwLCJcXHBpX3MgXFxUb3Ree3Qtc30gIFxcR2FtbWEoWF5cXGJ1bGxldCwgQSkiXSxbMSwxLCJcXHBpX3MgRl57dC1zfSAgXFxHYW1tYShYXlxcYnVsbGV0LCBBKSJdLFswLDFdLFsxLDJdLFsyLDBdXQ==
\begin{tikzcd}[ampersand replacement = \&, column sep = tiny]
{{}^2 D_3^{2s-t,s}} \&\& {{}^2 D_3^{2s-t,s}} \\
\& {{}^2 E_3^{2s-t,s}}
\arrow[from=1-1, to=1-3]
\arrow[from=1-3, to=2-2]
\arrow[from=2-2, to=1-1]
\end{tikzcd} \right)
\end{array}
\end{align*}

By definition of the derived couple on the right-hand side, this amounts to giving maps

\[ \pi_{t-s} X^{t, -\infty} \to \im(\pi_{t-s} X^{-\infty, s} \to \pi_{t-s} X^{-\infty, s-1}) \]

for all $s$ and $t$, subject to appropriate naturality conditions. To this end we claim that the first map in each of the spans below induces an isomorphism on $\pi_{t-s}$:

\[ X^{t, -\infty} \leftarrow X^{t, s-1} \to X^{-\infty, s-1}. \]

Indeed, writing $C \coloneqq \cofib(X^{t, s-1} \to X^{t, -\infty})$, we have a filtration $C = \colim C^{t'}$, where $t' \geq t$ and $C^{t'} = \cofib(X^{t', q-1} \to X^{t', -\infty})$. The resulting spectral sequence reads

\begin{equation} \label{eqn:decalage spectral sequence for connectivity argument}
\pi_{t'-s'} C^{t'/t'+1} \implies \pi_{t'-s'} C \qquad (t' \geq t),
\end{equation}

and its $E_2$-page is in turn computed by a trigraded spectral sequence

\begin{equation} \label{eqn:decalage truncated trigraded spectral sequence}
\pi_{q'-s'} X^{t'/t'+1, q'/q'+1} X \implies \pi_{q'-s'} C^{t'/t'+1} \qquad (q' \leq s-2).
\end{equation}

Spectral sequence <ref> can be thought of as a truncation of <ref>; it is in turn displayed in <ref>. the form of its $E_2$-page implies that $C^{t'/t'+1}$ is $(t' - s +2)$-connected, so that <ref> takes the form displayed in <ref>.

[name = truncatedtrigraded,
Adams grading, scale = .5,
classes = draw = none,
x label = $q'-s'$,
y label = $s'$,
no ticks

in 1,...,10
["*"](2 + , 2 - 2*)

in 2,...,10
[̣red]2(2 + , 2 - 2*)

[name = decalage connectivity,
Adams grading, scale = .5,
classes = draw = none,
x label = $t'-s'$,
y label = $s'$,
no ticks

in 0,...,10
in 1,...,10
["*"](1 + + , - )

[x range = 08, y range = -33, grid = chess, page = 2, name = truncatedtrigraded]
Spectral sequence <ref>. The leftmost nonzero entry is $(q'-s', s') = (t' - s + 2 , 2s - t' - 4)$.

[x range = 08, y range = -33, grid = chess, page = 3, name = decalage connectivity]
The $E_2$-page of <ref>. The leftmost nonzero entry is $(t'-s', s') = (t-s+2, s-2)$.

Truncated spectral sequences computing $\pi_{*} C$.

In particular, $C$ is $t-s+1$-connected and so the map $X^{t, s - 1} \to X^{t, -\infty}$ is $(t-s+1)$-connected. Applying an identical analysis to $X^{t, s} \to X^{t, -\infty}$ shows that this map has $(t-s)$-connected cofibre, and so induces a surjection on $\pi_{t-s}$; the diagram below therefore produces the requisite map ${}^1D_2^{s,t} \to {}^2D_3^{2s-t,s}$.

\[\begin{tikzcd}[column sep = small]
{\pi_{t-s} X^{t,s}} & {\pi_{t-s} X^{t,s-1}} & {\pi_{t-s} X^{t, -\infty}} \\
{\pi_{t-s} X^{-\infty, s}} & {\pi_{t-s} X^{-\infty, s - 1}}
\arrow[two heads, from=1-1, to=1-2]
\arrow["\sim", from=1-2, to=1-3]
\arrow[from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow[from=1-3, to=2-2, "\exists"]
\end{tikzcd}\]

We will not show that this indeed defines a map of exact couples, since this result is well-known: for example, see <cit.>.

We now relate this to the context of Postnikov towers and cobar complexes. If $X \simeq \lim_t( \cdots \to X_t \to \cdots)$ is a convergent tower of spectra with $\colim X_t = 0$, we can form a dual filtered spectrum

\[ X^t \coloneqq \fib (X \to X_{t-1}). \]

Then $\colim X^t \xrightarrow{\sim} X$, and we get another spectral sequence

\[ \pi_{t-s} f^t X \implies \pi_{t-s} X, \]

where once again $X^{t/t+1} \coloneqq \cofib(X^{t+1} \to X^t)$. Observe that $X^{t/t+1} \simeq X_{t/t-1} \coloneqq \fib(X_t \to X_{t-1})$, by the octahedral axiom:

\[\begin{tikzcd}[column sep = tiny, row sep = tiny]
& {X^{t+1}} \\
&&& {X^t} \\
{} && X &&& {X_{t/t-1}} \\
&&& {X_t} \\
& {X_{t-1}}
\arrow[from=1-2, to=2-4]
\arrow[from=2-4, to=3-3]
\arrow[from=1-2, to=3-3]
\arrow[from=3-3, to=4-4]
\arrow[from=3-3, to=5-2]
\arrow[from=4-4, to=5-2]
\arrow[from=3-6, to=4-4]
\arrow[from=2-4, to=3-6]
\end{tikzcd}\]

The spectral sequences for $(X_t)$ and $(X^t)$ agree.

The observation above is that the $E_2$-pages agree. To show that the entire spectral sequences match it is enough to show that the differentials $d_2$ do, in other words that the outer diagram below commutes.

\[\begin{tikzcd}[column sep = tiny]
{X_{t/t-1} = X^{t/t+1}} & {\Sigma X^{t+1}} \\
{X_t} & {\Sigma X_{t+1/t} = \Sigma X^{t+1/t+2}}
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow[from=1-2, to=2-2]
\arrow[from=1-1, to=1-2]
\arrow[dashed, from=2-1, to=1-2]
\end{tikzcd}\]

The dashed arrow is given by applying the $3 \times 3$-lemma <cit.> to obtain the diagrams below; note that each triangle above only anticommutes, and so the outer square is commutative.

\begin{align*}
\begin{tikzcd}[ampersand replacement = \&, column sep = tiny]
{\Sigma^{-1}X^{t+1}} \& {\Sigma^{-1}X^{t}} \& {\Sigma^{-1} X^{t/t+1}} \& {X^{t+1}} \\
{\Sigma^{-1}X} \& {\Sigma^{-1}X} \& 0 \& X \\
{\Sigma^{-1} X_t} \& {\Sigma^{-1}X_{t-1}} \& {X_{t/t-1}} \& {X_t} \\
{X^{t+1}} \& {X^t} \& {X^{t/t+1}} \& {\Sigma X^{t+1}}
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=3-1]
\arrow[from=3-1, to=4-1]
\arrow[from=4-1, to=4-2]
\arrow[from=3-1, to=3-2]
\arrow[equals, from=2-1, to=2-2]
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=2-4]
\arrow[from=2-4, to=3-4]
\arrow[from=3-4, to=4-4, dotted]
\arrow[from=4-3, to=4-4]
\arrow[from=4-2, to=4-3]
\arrow[from=3-2, to=3-3]
\arrow[from=3-3, to=3-4]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=1-3, to=2-3]
\arrow[from=2-3, to=3-3]
\arrow[from=3-3, to=4-3]
\arrow[from=1-2, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow[from=3-2, to=4-2]
\arrow["{(-1)}"{description}, draw=none, from=3-3, to=4-4]
\end{tikzcd} & \hskip 20pt &
% https://q.uiver.app/?q=WzAsMTYsWzAsMCwiWF97cC0xfSJdLFsxLDAsIlxcU2lnbWEgZl9wWCJdLFsyLDAsIlxcU2lnbWEgWF9wIl0sWzMsMCwiXFxTaWdtYSBYX3twLTF9Il0sWzAsMSwiXFxTaWdtYSBYXnAiXSxbMSwxLCJcXFNpZ21hIGZecFgiXSxbMiwxLCJcXFNpZ21hXjIgWF57cCsxfSJdLFszLDEsIlxcU2lnbWFeMiBYXnAiXSxbMCwyLCJcXFNpZ21hIFgiXSxbMSwyLCIwIl0sWzIsMiwiXFxTaWdtYV4yIFgiXSxbMywyLCJcXFNpZ21hXjJYIl0sWzAsMywiXFxTaWdtYSBYX3twLTF9Il0sWzEsMywiXFxTaWdtYV4yIGZfcFgiXSxbMiwzLCJcXFNpZ21hXjIgWF9wIl0sWzMsMywiXFxTaWdtYV4yIFhfe3AtMX0iXSxbMCw0XSxbNCw4XSxbOCwxMl0sWzEyLDEzXSxbOCw5XSxbNCw1XSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw3XSxbNywxMV0sWzExLDE1XSxbMTQsMTVdLFsxMywxNF0sWzksMTBdLFsxMCwxMSwiIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzUsNl0sWzYsN10sWzIsNl0sWzYsMTBdLFsxMCwxNF0sWzEsNV0sWzUsOV0sWzksMTNdXQ==
\begin{tikzcd}[ampersand replacement = \&, column sep = tiny]
{X_{t-1}} \& {\Sigma X_{t/t-1}} \& {\Sigma X_t} \& {\Sigma X_{t-1}} \\
{\Sigma X^t} \& {\Sigma X^{t/t+1}} \& {\Sigma^2 X^{t+1}} \& {\Sigma^2 X^t} \\
{\Sigma X} \& 0 \& {\Sigma^2 X} \& {\Sigma^2X} \\
{\Sigma X_{t-1}} \& {\Sigma^2 X_{t/t-1}} \& {\Sigma^2 X_t} \& {\Sigma^2 X_{t-1}}
\arrow[from=1-1, to=2-1, dotted]
\arrow[from=2-1, to=3-1]
\arrow[from=3-1, to=4-1]
\arrow[from=4-1, to=4-2]
\arrow[from=3-1, to=3-2]
\arrow[from=2-1, to=2-2]
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=2-4]
\arrow[from=2-4, to=3-4]
\arrow[from=3-4, to=4-4]
\arrow[from=4-3, to=4-4]
\arrow[from=4-2, to=4-3]
\arrow[from=3-2, to=3-3]
\arrow[equals, from=3-3, to=3-4]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=1-3, to=2-3]
\arrow[from=2-3, to=3-3]
\arrow[from=3-3, to=4-3]
\arrow[from=1-2, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow[from=3-2, to=4-2]
\arrow["{(-1)}"{description}, draw=none, from=1-1, to=2-2]
\end{tikzcd}
\end{align*}

By dualising the Postnikov and $\Tot$-towers, it will therefore suffice to verify that the induced bifiltration satisfies the assumptions of <ref>.

Let $\mathcal F$ be a sheaf of spectra on a site $\mathcal C$, and let $X \twoheadrightarrow *$ be a covering of the terminal object. Suppose that for every $t$ and every $q > 0$ we have $\Gamma(X^q, \tau_t \mathcal F) = \tau_t \Gamma(X^q, \mathcal F).$ Then there is an isomorphism between the descent and Bousfield-Kan spectral sequences, up to reindexing: for all $r$,

\[ E_r^{s,t} \cong \check E_{r+1}^{2s-t,s}. \]

We form the bifiltrations $(\Gamma \mathcal F)_{t,q} = \Tot_q \Gamma(U^\bullet, \tau_{\leq t} \mathcal F)$. Then

\[ (\Gamma \mathcal F)_{t,-\infty} = \lim_q \Tot_q \Gamma(U^\bullet, \tau_{\leq t} \mathcal F) = \Tot \Gamma(U^\bullet, \tau_{\leq t} \mathcal F) = \Gamma \tau_{\leq t} \mathcal F, \]

\[ (\Gamma \mathcal F)_{-\infty, q} = \lim_t \Tot_q \Gamma(U^\bullet, \tau_{\leq t} \mathcal F) = \Tot_q \Gamma(U^\bullet, \lim_t \tau_{\leq t} \mathcal F) = \Tot_q \Gamma(U^\bullet, \mathcal F). \]

Applying <ref> to the dual filtration, we need to verify that

\[ \Tot^{q/q+1} \Gamma(X^\bullet, \tau_t \mathcal F) \]

is Eilenberg-Mac Lane. But recall that for any cosimplicial spectrum $B^\bullet$ we have

\[ \fib(\Tot_q B^\bullet \to \Tot_{q-1} B^\bullet) \simeq \Omega^q N^q B^\bullet, \]

where $N^q$ denotes the fibre of the map from $X^q$ to the $q$-th matching spectrum; $N^q B^\bullet$ is a pointed space with

\[ \pi_j N^q B^\bullet = \pi_j B^q \cap \ker s^0 \cap \ker s^{q-1}. \]

In the case of pointed spaces, this fact is <cit.>; the proof, which appears also as <cit.>, works equally well for a cosimplicial spectrum[Note that the inductive argument there applies in the `cosimplicial' direction, i.e., in the notation of one shows for any fixed $t$ that $N^{n,k} \pi_t X = \ker(\pi_t X \to M^{n,k} \pi_t X)$ for $k, n \leq s$.]. By abuse of notation, we also denote this group by $N^q \pi_j B^\bullet$. Thus

\begin{align*}
\pi_j \Tot^{q/q+1} \Gamma(X^\bullet, \tau_t \mathcal F) &\simeq \pi_j \Omega^q N^q \Gamma(X^\bullet, \tau_t \mathcal F) \\
&\simeq  N^q \pi_{j + q} \Gamma(X^\bullet, \tau_t \mathcal F) \\
&\simeq N^q \pi_{j + q} \tau_t \Gamma(X^\bullet, \mathcal F) \\
& \subset \pi_{j+q} \tau_t \Gamma(X^\bullet, \mathcal F) = \left\{ \begin{array}{ll}
\pi_t \Gamma(X^\bullet, \mathcal F) & j = t - q \\
0 & \text{otherwise.}
\end{array} \right. \qedhere
\end{align*}

On the starting pages, one has

\begin{align*}
E_2^{s,t} &= \pi_{t-s} \Gamma \tau_t \mathcal F \simeq H^{s}(\mathcal C, \pi_t \mathcal F)
\end{align*}

\begin{align*}
\check E_3^{2s-t, s} &= H^{s} (\pi_{t-*} \Omega^* N^* \Gamma(X^\bullet, \mathcal F) ) = H^{s} ( N^* \pi_{t} \Gamma(X^\bullet, \mathcal F) ) \simeq \check H^s(X \twoheadrightarrow *, \pi_t \mathcal F).
\end{align*}

In particular, note that our assumption implies that the Čech-to-derived functor spectral sequence collapses.

§ APPENDIX: THE ADAMS SPECTRAL SEQUENCE AT HEIGHT ONE

In this appendix, we compute the $K(1)$-local $E_1$-Adams spectral sequences at all primes. The material is well-known, and is implicit in the computations of <cit.>. At the prime two we give a different perspective to <cit.> on the computation, which illustrates how one can make use of the Postnikov tower of a sheaf of spectra on $\proet \G$: this approach may be useful in higher height examples that are more computationally challenging. In particular, we use the finite resolution of the $K(1)$-local sphere, and as such our only input is knowledge of the HFPSS for the conjugation action on $KU$, displayed in <ref>.

§.§ Odd primes

At odd primes, the multiplicative lift gives a splitting

\[ \Z[p]^\times \simeq  (1+ p \Z[p]) \times \mu_{p-1} \simeq  \Z[p] \times \mu_{p-1}, \]

with the second isomorphism given by the $p$-adic logarithm. A pair $(a, b) \in \Z[p] \times \mu_{p-1}$ therefore acts on $\pi_t \E$ as $(b\operatorname{exp}_p(a))^{t}$.

The starting page of the $\K$-local $\E$-Adams spectral sequence is

\begin{equation}
E_2^{s,t} = H^{s}(\Z[p]^\times, \pi_{t} \E) = \left\{
\begin{array}{ll}
\Z[p] & t = 0 \text{ and } s = 0, 1 \\
\Z/p^{\nu_p(t') + 1} & t = 2(p-1)t' \neq 0 \text{ and } s = 1
\end{array} \right.
\end{equation}

and zero otherwise. In particular, it collapses immediately to $E_\infty$.

We will use the Lyndon-Hochschild-Serre spectral sequence <cit.> for the inclusion $\Z[p] \simeq 1 + p \Z[p] \hookrightarrow \Z[p]^\times$, which reads

\[ H^i(\mu_{p-1}, H^j(\Z[p], \pi_t KU_p)) \implies  H^{i+j}(\Z[p]^\times, \pi_t KU_p). \]

Since everything is $(p)$-local, taking $\mu_{p-1}$ fixed-points is exact. The spectral sequence therefore collapses, and what remains is to compute $\Z[p]$-cohomology.

By <cit.>, the trivial pro-$p$ module $\Z[p]$ admits a projective resolution

\[ 0 \to \Z[p][[\Z[p]]] \xrightarrow{\zeta - 1} \Z[p][[\Z[p]]] \to \Z[p] \to 0 \]

in the (abelian) category $\mathfrak C_p(\Z[p])$ of pro-$p$ continuous $\Z[p]^\times$-modules; here we write $\zeta$ for a topological generator, and $\Z[p][[G]] \coloneqq \lim_{U <_o G} \Z[p][G/U]$ for the completed group algebra of a profinite group $G$. In particular,

\begin{align*}
H^j(\Z[p], \pi_{2t} KU_p) = H^j \left(\Z[p] \xrightarrow{\exp_p(\zeta)^t - 1} \Z[p] \right) \simeq \left\{ \begin{array}{ll}
\Z[p] & t = 0 \text{ and } j = 0,1 \\
\Z[p]/p^{\nu_p(t)+ 1} & t \neq 0 \text{ and } j = 1
\end{array} \right.
\end{align*}

where for the final isomorphism we have used the isomorphism $\exp_p: p^{j-1} \Z[p]/p^j \Z[p] \simeq 1 + p^j \Z[p]^\times/1 + p^{j+1} \Z[p]^\times$ to obtain $\nu_p(\exp_p(\zeta)^t-1) = \nu_p(t \zeta) + 1 = \nu_p(t) + 1$. The $\mu_{p-1}$-action has fixed points precisely when it is trivial, i.e. when $p-1 \mid t$, which gives the stated form.

§.§ p=2

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

We develop a generalization of the theory of Thom spectra using the
language of ∞-categories. This treatment exposes the conceptual
underpinnings of the Thom spectrum functor: we use a new model of
parameterized spectra, and our deﬁnition is motivated by the geometric
deﬁnition of Thom spectra of May–Sigurdsson. For an A∞-ring spectrum R,
we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid
of a space X to the ∞-category of free rank one R-modules, which we show is
a model for BGL1R; we show that BGL1R classiﬁes homotopy sheaves of
rank one R-modules, which we call R-line bundles. We use our R-module Thom
spectrum to deﬁne the twisted R-homology and cohomology of R-line bundles
over a space classiﬁed by a map X → BGL1R, and we recover the
generalized theory of orientations in this context. In order to compare this
approach to the classical theory, we characterize the Thom spectrum functor
axiomatically, from the perspective of Morita theory.

titleAn $\infty$-categorical approach to $R$-line bundles,
$R$-module Thom spectra, and twisted $R$-homology
journaltitleJournal of Topology

Mathematical Sciences Publishers

titleTopological Hochschild homology and cohomology of $A_\infty$
ring spectra
journaltitleGeometry & Topology

14F22, 18G55, Mathematics - Algebraic Geometry, Mathematics -
Algebraic Topology, Mathematics - Rings and Algebras
In this paper, we study Azumaya algebras and Brauer groups in derived
algebraic geometry. We establish various fundamental facts about Brauer
groups in this setting, and we provide a computational tool, which we use to
compute the Brauer group in several examples. In particular, we show that the
Brauer group of the sphere spectrum vanishes, and we use this to prove two
uniqueness theorems for the stable homotopy category. Our key technical
results include the local geometricity, in the sense of Artin n-stacks, of
the moduli space of perfect modules over a smooth and proper algebra, the
e´tale local triviality of Azumaya algebras over connective derived schemes,
and a local to global principle for the algebraicity of stacks of stable

issn1364-0380, 1465-3060
titleBrauer groups and étale cohomology in derived algebraic
journaltitleGeom. Topol.

14F22, 14H52, 14K10, 55N34, Mathematics - Algebraic Geometry,
Mathematics - Algebraic Topology
We prove that the Brauer group of TMF is isomorphic to the Brauer group
of the derived moduli stack of elliptic curves. Then, we compute the local
Brauer group, i.e., the subgroup of the Brauer group of elements trivialized
by some étale cover of the moduli stack, up to a ﬁnite 2-torsion group.

titlePicard sheaves, local Brauer groups, and topological modular

booktitleSéminaire de Géométrie Algébrique du Bois
Marie-Cohomologie (SGA4)
chapterExposée Vbis
titleThéorie des topos et cohomologie étale des schémas

fibrations, Fundamental group, Homotopy, Homotopy group
isbn978-3-540-04619-6 978-3-540-36142-8
seriesLecture Notes in Mathematics
titleEtale Homotopy
Berlin, Heidelberg

titleThe Brauer group of a commutative ring
journaltitleTransactions of the American Mathematical Society

55N22 (Primary), 55T25 (Secondary), 55P60, 16W30, 13K05, Mathematics
- Algebraic Topology, Mathematics - Commutative Algebra
We investigate Hopf algebroids in the category of L-complete modules over a
commutative Noetherian regular complete local ring. The main examples of
interest in algebraic topology are the Hopf algebroids associated to
Lubin-Tate spectra in the K(n)-local stable homotopy category, and we show
that these have Landweber ﬁltrations for all ﬁnitely generated discrete

title$L$-complete Hopf algebroids and their comodules

The aim of this note is to understand under which conditions invertible
modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell
& May give rise to elements in the algebraic Picard group of invertible
graded modules over the coefﬁcient ring by taking homotopy groups. If a
connective commutative S-algebra R has coherent localizations (R∗)m for
every maximal ideal m R∗, then for every invertible R-module U , U∗ =
π∗U is an invertible graded R∗-module. In some non-connective cases we
can carry the result over under the additional assumption that the
commutative S-algebra has ‘residue ﬁelds’ for all maximal ideals m R∗
if the global dimension of R∗ is small or if R is 2-periodic with
underlying Noetherian complete local regular ring R0. We apply these results
to ﬁnite abelian Galois extensions of Lubin-Tate spectra.

issn0025-2611, 1432-1785
titleInvertible modules for commutative $\mathbb S$-algebras with
residue fields
journaltitlemanuscripta math.

55P43 (Primary) 11R52, 11R32 (Secondary), Mathematics - Algebraic
Geometry, Mathematics - Algebraic Topology, Mathematics - Rings and Algebras
We investigate a notion of Azumaya algebras in the context of structured
ring spectra and give a deﬁnition of Brauer groups. We investigate their
Galois theoretic properties, and discuss examples of Azumaya algebras arising
from Galois descent and cyclic algebras. We construct examples that are
related to topological Hochschild cohomology of group ring spectra and we
present a K(n)-local variant of the notion of Brauer groups.

titleBrauer groups for commutative $\mathbb S$-algebras
journaltitleJournal of Pure and Applied Algebra

Mathematics - Algebraic Topology, Mathematics - Category Theory
The goal of this paper is to prove an equivalence between the model
categorical approach to pro-categories, as studied by Isaksen, Schlank and
the ﬁrst author, and the ∞categorical approach, as developed by Lurie.
Three applications of our main result are described. In the ﬁrst
application we use (a dual version of) our main result to give suﬃcient
conditions on an ω-combinatorial model category, which insure that its
underlying ∞-category is ω-presentable. In the second application we
consider the pro-category of simplicial ´etale sheaves and use it to show
that the topological realization of any Grothendieck topos coincides with the
shape of the hyper-completion of the associated ∞-topos. In the third
application we show that several model categories arising in proﬁnite
homotopy theory are indeed models for the ∞-category of proﬁnite spaces.
As a byproduct we obtain new Quillen equivalences between these models, and
also obtain an example which settles negatively a question raised by G.

issn1472-2739, 1472-2747
titlePro-categories in homotopy theory
journaltitleAlgebr. Geom. Topol.

Mathematics - Algebraic Topology, 55P42, 55P92, 55P91, 55Q51
Following an idea of Hopkins, we construct a model of the determinant
sphere S det in the category of K(n)-local spectra. To do this, we build a
spectrum which we call the Tate sphere S(1). This is a p-complete sphere with
a natural continuous action of Z×p . The Tate sphere inherits an action of
Gn via the determinant and smashing Morava E-theory with S(1) has the eﬀect
of twisting the action of Gn. A large part of this paper consists of
analyzing continuous Gn-actions and their homotopy ﬁxed points in the setup
of Devinatz and Hopkins.

titleConstructing the determinant sphere using a Tate twist

Mathematical Sciences Publishers

titleCompleted power operations for Morava E–theory
journaltitleAlgebraic & Geometric Topology

Mathematics - Algebraic Topology
Let E = En be Morava E-theory of height n. In [DH04] Devinatz and Hopkins
introduced the K(n)-local En-Adams spectral sequence and showed that, under
certain conditions, the E2-term of this spectral sequence can be identiﬁed
with continuous group cohomology. We work with the category of L-complete
E∗∨E-comodules, and show that in a number of cases the E2-term of the
above spectral sequence can be computed by a relative Ext group in this
category. We give suitable conditions for when we can identify this Ext group
with continuous group cohomology.

titleThe $E_2$-term of the $K(n)$-local $E_n$-Adams spectral
journaltitleTopology and its Applications

Mathematics - Algebraic Topology, Mathematics - Algebraic Geometry,
Mathematics - Category Theory, 55P60, 13D45, 14B15, 55U35, Mathematics -
Commutative Algebra
The ﬁrst goal of this paper is to provide an abstract framework in which
to formulate and study local duality in various algebraic and topological
contexts. For any stable ∞-category C together with a collection of compact
objects K ⊂ C we construct local cohomology and local homology functors
satisfying an abstract version of local duality. When specialized to the
derived category of a commutative ring A and a suitable ideal in A, we
recover the classical local duality due to Grothendieck as well as
generalizations by Greenlees and May. More generally, applying our result to
the derived category of quasi-coherent sheaves on a quasi-compact and
separated scheme X implies the local duality theorem of Alonso Tarr´ıo,
Jerem´ıas L´opez, and Lipman.

titleLocal duality in algebra and topology
journaltitleAdvances in Mathematics

Mathematics - Algebraic Topology, Mathematics - Category Theory
Inspired by the Ax–Kochen isomorphism theorem, we develop a notion of
categorical ultraproducts to capture the generic behavior of an inﬁnite
collection of mathematical objects. We employ this theory to give an
asymptotic solution to the approximation problem in chromatic homotopy
theory. More precisely, we show that the ultraproduct of the E(n, p)-local
categories over any non-principal ultraﬁlter on the set of prime numbers is
equivalent to the ultraproduct of certain algebraic categories introduced by
Franke. This shows that chromatic homotopy theory at a ﬁxed height is
asymptotically algebraic.

titleChromatic homotopy theory is asymptotically algebraic

Mathematics - Algebraic Topology, Mathematics - Category Theory
In previous work, we used an ∞-categorical version of ultraproducts to
show that, for a ﬁxed height n, the symmetric monoidal ∞-categories of
En,p-local spectra are asymptotically algebraic in the prime p. In this
paper, we prove the analogous result for the symmetric monoidal
∞-categories of Kp(n)-local spectra, where Kp(n) is Morava K-theory at
height n and the prime p. This requires ∞-categorical tools suitable for
working with compactly generated symmetric monoidal ∞-categories with
non-compact unit. The equivalences that we produce here are compatible with
the equivalences for the En,p-local ∞-categories.

issn0020-9910, 1432-1297
titleMonochromatic homotopy theory is asymptotically algebraic
journaltitleInvent. math.

Mathematics - Algebraic Topology, Mathematics - Algebraic Geometry
Let be a quasicompact quasiseparated scheme. Write Gal( ) for the category
whose objects are geometric points of and whose morphisms are specializations
in the étale topology. We deﬁne a natural proﬁnite topology on the
category Gal( ) that globalizes the topologies of the absolute Galois groups
of the residue ﬁelds of the points of . One of the main results of this
book is that Gal( ) variant of MacPherson’s exit-path category suitable
for the étale topology: we construct an equivalence between representations
of Gal( ) and constructible sheaves on . We show that this exodromy
equivalence holds with nonabelian coeﬃcients and with ﬁnite abelian
coeﬃcients. More generally, by using the pyknotic/condensed formalism, we
extend this equivalence to coeﬃcients in the category of modules over
proﬁnite rings and algebraic extensions of . As an ‘exit-path
category’, the topological category Gal( ) also gives rise to a new,
concrete description of the étale homotopy type of .

Pyknotic objects are (hyper)sheaves on the site of compacta. These provide
a convenient way to do algebra and homotopy theory with additional
topological information present. This appears, for example, when trying to
contemplate the derived category of a local field. In this article, we
present the basic theory of pyknotic objects, with a view to describing a
simple set of everyday examples.

titlePyknotic objects, I. Basic notions

Mathematics - Algebraic Topology
We compute the continuous cohomology of the Morava stabilizer group with
coefficients in Morava $E$-theory,
$Hˆ*(\mathbb{G}_2, E_t)$, at $p=2$,
for $0\leq t  12$, using the Algebraic Duality
Spectral Sequence. Furthermore, in that same range, we compute the
$d_3$-differentials in the homotopy fixed point spectral sequence for the
$K(2)$-local sphere spectrum. These cohomology groups and differentials
play a central role in $K(2)$-local stable homotopy theory, in particular
for the analysis of the $K(2)$-local Picard group.

titleCohomology of the Morava stabilizer group through the duality
resolution at $n=p=2$
mar/Zotero/storage/7QM345NR/Beaudry et a
ogy of the Morava stabilizer group through .pdf:ap
df;arXiv.org Snapshot:/Users/Itamar/Zotero/storage/YETJQYD
sers/Itamar/Zotero/storage/QLL4I7U8/Beaudry et a
Picard Group at the Prime .pdf:
tion/pdf;arXiv.org Snapshot:/Users/Itamar/Zotero/storage/FXMIM
issn1364-0380, 1465-3060
titleChromatic splitting for the $K(2)$-local sphere at $p=2$
journaltitleGeom. Topol.

notePublisher: Mathematical Sciences Publishers
titleThe algebraic duality resolution at p = 2
journaltitleAlgebr. Geom. Topol.

Mathematics - Algebraic Topology, 55P92, 55R25, 55R50, 55U30, 20E18,
The primary goal of this paper is to study Spanier-Whitehead duality in the
K(n)-local category. One of the key players in the K(n)-local category is the
Lubin-Tate spectrum En, whose homotopy groups classify deformations of a
formal group law of height n, in the implicit characteristic p. It is known
that En is self-dual up to a shift; however, that does not fully take into
account the action of the Morava stabilizer group Gn, or even its subgroup of
automorphisms of the formal group in question. In this paper we ﬁnd that
the Gn-equivariant dual of En is in fact En twisted by a sphere with a
non-trivial (when n  1) action by Gn. This sphere is a
dualizing module for the group Gn, and we construct and study such an object
IG for any compact p-adic analytic group G. If we restrict the action of G
on IG to certain type of small subgroups, we identify IG with a speciﬁc
representation sphere coming from the Lie algebra of G. This is done by a
classiﬁcation of p-complete sphere spectra with an action by an elementary
abelian p-group in terms of characteristic classes, and then a speciﬁc
comparison of the characteristic classes in question. The setup makes the
theory quite accessible for computations, as we demonstrate in the later
sections of this paper, determining the K(n)local Spanier-Whitehead duals of
EnhH for select choices of p and n and ﬁnite subgroups H of Gn.

titleDualizing spheres for compact $p$-adic analytic groups and
duality in chromatic homotopy

Let E be a k-local proﬁnite G-Galois extension of an E∞-ring spectrum A
(in the sense of Rognes). We show that E may be regarded as producing a
discrete G-spectrum. Also, we prove that if E is a profaithful k-local
proﬁnite extension which satisﬁes certain extra conditions, then the
forward direction of Rognes’s Galois correspondence extends to the
proﬁnite setting. We show that the function spectrum FA((EhH )k, (EhK
)k) is equivalent to the localized homotopy ﬁxed point spectrum
((E[[G/H]])hK )k where H and K are closed subgroups of G. Applications to
Morava E-theory are given, including showing that the homotopy ﬁxed points
deﬁned by Devinatz and Hopkins for closed subgroups of the extended Morava
stabilizer group agree with those deﬁned with respect to a continuous
action in terms of the derived functor of ﬁxed points.

issn0002-9947, 1088-6850
titleThe homotopy fixed point spectra of profinite Galois
journaltitleTrans. Amer. Math. Soc.

Mathematics - Algebraic Geometry, Mathematics - Commutative Algebra
We give a new deﬁnition of the derived category of constructible
Qℓ-sheaves on a scheme, which is as simple as the geometric intuition
behind them. Moreover, we deﬁne a reﬁned fundamental group of schemes,
which is large enough to see all lisse Qℓ-sheaves, even on non-normal
schemes. To accomplish these tasks, we deﬁne and study the pro-e´tale
topology, which is a Grothendieck topology on schemes that is closely related
to the e´tale topology, and yet better suited for inﬁnite constructions
typically encountered in ℓ-adic cohomology. An essential foundational
result is that this site is locally contractible in a well-deﬁned sense.

titleThe pro-étale topology for schemes

19D10, 19D55, 18D20, 55N15, Mathematics - Algebraic Topology,
Mathematics - K-Theory and Homology
In this paper we establish a universal characterization of higher algebraic
K-theory in the setting of small stable ∞-categories. Speciﬁcally, we
prove that connective algebraic K-theory is the universal additive invariant,
i.e., the universal functor with values in spectra which inverts Morita
equivalences, preserves ﬁltered colimits, and satisﬁes Waldhausen’s
additivity theorem. Similarly, we prove that non-connective algebraic
K-theory is the universal localizing invariant, i.e., the universal functor
that moreover satisﬁes the “Thomason-Trobaugh-Neeman” localization

issn1364-0380, 1465-3060
titleA universal characterization of higher algebraic K-theory
journaltitleGeom. Topol.

American Mathematical Society

Convergence criteria for spectral sequences are developed that apply more
widely than the traditional concepts. In the presence of additional
conditions that depend on data internal to the spectral sequence, they lead
to satisfactory convergence and comparison theorems. The techniques apply to
whole-plane as well as half-plane spectral sequences.

booktitleContemporary Mathematics
isbn978-0-8218-1057-6 978-0-8218-7829-3
titleConditionally convergent spectral sequences
Providence, Rhode Island

We provide a topological duality resolution for the spectrum E2hS12 , which
itself can be used to build the K(2)-local sphere. The resolution is built
from spectra of the form E2hF where E2 is the Morava spectrum for the formal
group of a supersingular curve at the prime 2 and F is a ﬁnite subgroup of
the automorphisms of that formal group. The results are in complete analogy
with the resolutions of Goerss, Henn, Mahowald and Rezk (Ann. of Math. (2)
162 (2005) 777–822) at the prime 3, but the methods are of necessity very
diﬀerent. As in the prime 3 case, the main diﬃculty is in identifying the
top ﬁber; to do this, we make calculations using Henn’s centralizer

issn1753-8416, 1753-8424
titleTopological resolutions in $K(2)$‐local homotopy theory at
the prime $2$
journaltitleJournal of Topology

edition2. corr. pr
isbn978-3-540-06105-2 978-0-387-06105-4
seriesLecture notes in mathematics
titleHomotopy limits, completions and localizations

titleThe Chromatic Nullstellensatz

55P43 (Primary) 55P42, 14C22 (Secondary), Mathematics - Algebraic
We compute the connective spectra of maps from
$\mathbb{Z}$ to the Picard spectra of the spherical Witt
vectors associated with perfect rings of characteristic $p$. As an
application, we determine the connective spectrum of maps from
$\mathbb{Z}$ to the Picard spectrum of the sphere

titleOn the Strict Picard Spectrum of Commutative Ring Spectra
t PDF:/Users/Itamar/Zotero/storage/AVP48URA/Carmeli -
2 - On the Strict Picard Spectrum of Commutative Ring .pdf:applic

titleHigher semiadditive Grothendieck-Witt theory and the
$K(1)$-local sphere

Mathematics - K-Theory and Homology
We study the ´etale sheaﬁﬁcation of algebraic K-theory, called ´etale
K-theory. Our main results show that ´etale K-theory is very close to a
noncommutative invariant called Selmer K-theory, which is deﬁned at the
level of categories. Consequently, we show that ´etale K-theory has
surprisingly well-behaved properties, integrally and without ﬁniteness
assumptions. A key theoretical ingredient is the distinction, which we
investigate in detail, between sheaves and hypersheaves of spectra on ´etale

issn0020-9910, 1432-1297
titleHyperdescent and étale K-theory
journaltitleInvent. math.

titleCohomological Descent
OHOMOLOGICAL DESCENT.pdf:/Users/Itamar/Zotero/storage/THCMI
ixed Points
Northwestern University

typeDoctoral thesis

Let K(n) be the nth Morava K-theory spectrum. Let En be the Lubin-Tate
spectrum, which plays a central role in understanding LK(n)(S0), the
K(n)-local sphere. For any spectrum X, deﬁne E∨(X) to be the spectrum
LK(n)(En ∧ X). Let G be a closed subgroup of the proﬁnite group Gn, the
group of ring spectrum automorphisms of En in the stable homotopy category.
We show that E∨(X) is a continuous G-spectrum, with homotopy ﬁxed point
spectrum (E∨(X))hG. Also, we construct a descent spectral sequence with
abutment π∗((E∨(X))hG).

titleHomotopy fixed points for $L_{K ( n ) }( E_n \wedge X )$
using the continuous action
journaltitleJournal of Pure and Applied Algebra

When G is a proﬁnite group and H and K are closed subgroups, with H
normal in K, it is not known, in general, how to form the iterated homotopy
ﬁxed point spectrum (ZhH )hK/H , where Z is a continuous G-spectrum and
all group actions are to be continuous. However, we show that, if G = Gn, the
extended Morava stabilizer group, and Z = Lb(En∧X), where Lb is Bousﬁeld
localization with respect to Morava K-theory, En is the Lubin-Tate spectrum,
and X is any spectrum with trivial Gn-action, then the iterated homotopy
ﬁxed point spectrum can always be constructed. Also, we show that (EnhH
)hK/H is just EnhK , extending a result of Devinatz and Hopkins.

titleIterated homotopy fixed points for the Lubin–Tate spectrum
journaltitleTopology and its Applications

issn1364-0380, 1465-3060
titleCommutative ring objects in pro-categories and generalized
Moore spectra
journaltitleGeom. Topol.

issn0073-8301, 1618-1913
titleThéorie de Hodge: II
journaltitlePublications Mathématiques de l'IHÉS

Institut des Hautes Études Scientifiques

titleThéorie de Hodge: III
journaltitlePublications Mathématiques de l'IHÉS

issn0073-8301, 1618-1913
titleLa Conjecture de Weil. II
journaltitlePublications Mathématiques de l'IHÉS

titleThe Action of the Morava Stabilizer Group on the Lubin-Tate
Moduli Space of Lifts
journaltitleAmerican Journal of Mathematics

titleThe Action of the Morava Stabilizer Group on the Lubin-Tate
Moduli Space of Lifts
journaltitleAmerican Journal of Mathematics

Let G be a closed subgroup of the semi-direct product of the nth Morava
stabilizer group Sn with the Galois group of the ÿeld extension Fpn =Fp. We
construct a “homotopy ÿxed point spectrum” EnhG whose homotopy ÿxed
point spectral sequence involves the continuous cohomology of G. These
spectra have the expected functorial properties and agree with the
Hopkins-Miller ÿxed point spectra when G is ÿnite.

titleHomotopy fixed point spectra for closed subgroups of the
Morava stabilizer groups

titleNilpotence and Stable Homotopy Theory I
journaltitleThe Annals of Mathematics

We use hypercovers to study the homotopy theory of simplicial presheaves.
The main result says that model structures for simplicial presheaves
involving local weak equivalences can be constructed by localizing at the
hypercovers. One consequence is that the ﬁbrant objects can be explicitly
described in terms of a hypercover descent condition, and the ﬁbrations can
be described by a relative descent condition. We give a few applications for
this new description of the homotopy theory of simplicial presheaves.

issn0305-0041, 1469-8064
titleHypercovers and simplicial presheaves
journaltitleMath. Proc. Camb. Phil. Soc.

We show that if U∗ is a hypercover of a topological space X then the
natural map hocolim U∗→X is a weak equivalence. This fact is used to
construct topological realization functors for the A1-homotopy theory of
schemes over real and complex ﬁelds. In an appendix, we also prove a
theorem about computing homotopy colimits of spaces that are not coﬁbrant.

issn0025-5874, 1432-1823
titleTopological hypercovers and $\mathbb A^1$-realizations
journaltitleMathematische Zeitschrift

given=John P.C.,

Suppose that R is a ring and that A is a chain complex over R. Inside the
derived category of differential graded A-modules there are naturally defined
subcategories of A-torsion objects and of A-complete objects. Under a
finiteness condition on A, we develop a Morita theory for these subcat
egories, find conceptual interpretations for some associated algebraic
functors, and, in appropriate commutative situations, identify the associated
functors as local homology or local cohomology. Some of the results are
suprising even in the case R = Z and A = Z/p.

titleComplete modules and torsion modules
journaltitleAmerican Journal of Mathematics

Springer Berlin Heidelberg

isbn978-3-540-07863-0 978-3-540-38103-7
seriesLecture Notes in Mathematics
titleČech and Steenrod Homotopy Theories with Applications to
Geometric Topology
Berlin, Heidelberg

American Mathematical Soc.

titleRings, modules, and algebras in stable homotopy theory

18D20, 18D50, 55P48, 55U35, Mathematics - Algebraic Topology,
Mathematics - Category Theory
We set up a general theory of weak or homotopy-coherent enrichment in an
arbitrary monoidal ∞-category V. Our theory of enriched ∞-categories has
many desirable properties; for instance, if the enriching ∞-category V is
presentably symmetric monoidal then CatV∞ is as well. These features
render the theory useful even when an ∞-category of enriched ∞-categories
comes from a model category (as is often the case in examples of interest,
e.g. dg-categories, spectral categories, and (∞, n)-categories). This is
analogous to the advantages of ∞-categories over more rigid models such as
simplicial categories — for example, the resulting ∞-categories of
functors between enriched ∞-categories automatically have the correct
homotopy type.

titleEnriched $\infty$-categories via non-symmetric
journaltitleAdvances in Mathematics

Mathematics - Algebraic Topology
In this paper we develop methods for classifying Baker-Richter-Szymik's
Azumaya algebras over a commutative ring spectrum, especially in the largely
inaccessible case where the ring is nonconnective. We give
obstruction-theoretic tools, constructing and classifying these algebras and
their automorphisms with Goerss-Hopkins obstruction theory, and give
descent-theoretic tools, applying Lurie's work on
$\infty$-categories to show that a finite Galois extension
of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on
Brauer spaces. For even-periodic ring spectra $E$, we find that the
"algebraic" Azumaya algebras whose coefficient ring is projective are
governed by the Brauer-Wall group of $\pi_0(E)$, recovering
a result of Baker-Richter-Szymik. This allows us to calculate many examples.
For example, we find that the algebraic Azumaya algebras over Lubin-Tate
spectra have either 4 or 2 Morita equivalence classes depending on whether
the prime is odd or even, that all algebraic Azumaya algebras over the
complex K-theory spectrum $KU$ are Morita trivial, and that the group of
the Morita classes of algebraic Azumaya algebras over the localization
$KU[1/2]$ is $\Bbb Z/8 \times
\Bbb Z/2$. Using our descent results and an obstruction
theory spectral sequence, we also study Azumaya algebras over the real
K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We
show that there exist exactly two Morita equivalence classes of these. The
nontrivial Morita equivalence class is realized by an "exotic"
$KO$-algebra with the same coefficient ring as $End_{KO}(KU)$.
This requires a careful analysis of what happens in the homotopy fixed-point
spectral sequence for the Picard space of $KU$, previously studied by
Mathew and Stojanoska.

issn0010-437X, 1570-5846
titleBrauer groups and Galois cohomology of commutative ring
journaltitleCompositio Math.

Cambridge University Press

titleCentral simple algebras and Galois cohomology

North-Holland Publishing Company

Geometry, Algebraic., Homology theory., Schemes (Algebraic
noteOpen Library ID: OL4617233M
pagetotal[8], 386
titleDix exposés sur la cohomologie des schémas.
Amsterdam, Paris

55P42, Mathematics - Algebraic Topology
We give a calculation of Picard groups P ic2 of K(2)-local invertible
spectra and P ic(L2) of E(2)-local invertible spectra, both at the prime 3.
The main contribution of this paper is to calculation the subgroup κ2 of
invertible spectra X with (E2)∗X ∼= (E2)∗S0 as twisted modules over the
Morava stabilizer group G2.

titleOn Hopkins' Picard groups for the prime 3 and chromatic level
journaltitleJournal of Topology

Cambridge University Press

Let E be a homotopy commutative ring spectrum, and suppose the ring of
cooperations E∗E is ﬂat over E∗. We wish to address the following
question: given a commutative E∗-algebra A in E∗E-comodules, is there an
E∞-ring spectrum X with E∗X ∼= A as comodule algebras? We will
formulate this as a moduli problem, and give a way – suggested by work of
Dwyer, Kan, and Stover – of dissecting the resulting moduli space as a
tower with layers governed by appropriate Andr´e-Quillen cohomology groups.
A special case is A = E∗E itself. The ﬁnal section applies this to
discuss the Lubin-Tate or Morava spectra En.

booktitleStructured Ring Spectra
isbn978-0-521-60305-8 978-0-511-52995-5
titleModuli spaces of commutative ring spectra

algebraic K-theory, Algebraic topology, homological algebra,
homology, Homotopy, homotopy theory, K-theory
isbn978-3-0346-0188-7 978-3-0346-0189-4
titleSimplicial Homotopy Theory

Academic Press

titleDerived functors of $I$-adic completion and local homology
journaltitleJournal of Algebra

Mathematics - Algebraic Topology
These notes explain some descent results for
$\infty$-categories of sheaves on compact Hausdorff spaces
and derive some consequences. Specifically, given a compactly assembled
$\infty$-category $\mathcal{E}$, we show
that the functor sending a locally compact Hausdorff space $X$ to the
of Postnikov complete $\mathcal{E}$-valued sheaves on $ X
$ satisfies descent for proper surjections. This implies proper descent for
left complete derived $\infty$-categories and that the
is a sheaf on the category of compact Hausdorff spaces equipped with the
topology of finite jointly surjective families. Using this, we explain how to
embed Postnikov complete sheaves on a locally compact Hausdorff space into
condensed objects. This implies that the condensed and sheaf cohomologies of
a locally compact Hausdorff space agree.

titleDescent for sheaves on compact Hausdorff spaces
mar/Zotero/storage/YHUD8PFQ/Haine - 2022
local Picard group
University of Melbourne

typeDoctoral thesis

Mathematics - Algebraic Topology
Following a suggestion of Hovey and Strickland, we study the category of
K(k) ∨ K(k +1)∨· · ·∨K(n)-local spectra. When k = 0, this is
equivalent to the category of E(n)-local spectra, while for k = n, this is
the category of K(n)-local spectra, both of which have been studied in detail
by Hovey and Strickland. Based on their ideas, we classify the localizing and
colocalizing subcategories, and give characterizations of compact and
dualizable objects. We construct an Adams type spectral sequence and show
that when p ≫ n it collapses with a horizontal vanishing line above
ﬁltration degree n2 + n − k at the E2-page for the sphere spectrum. We
then study the Picard group of K(k) ∨ K(k + 1) ∨ · · · ∨ K(n)-local
spectra, showing that this group is algebraic, in a suitable sense, when p
≫ n. We also consider a version of Gross–Hopkins duality in this
category. A key concept throughout is the use of descent.

titleThe $\mathop{Sp}_{k,n}$-local stable homotopy category

Mathematics - Algebraic Topology
Using the descent spectral sequence for a Galois extension of ring spectra,
we compute the Picard group of the higher real K-theory spectra of Hopkins
and Miller at height n = p − 1, for p an odd prime. More generally, we
determine the Picard groups of the homotopy ﬁxed points spectra EnhG,
where En is Lubin-Tate E-theory at the prime p and height n = p − 1, and G
is any ﬁnite subgroup of the extended Morava stabilizer group. We ﬁnd
that these Picard groups are always cyclic, generated by the suspension.

issn0010-437X, 1570-5846
titlePicard groups of higher real $K$-theory spectra at height
journaltitleCompositio Math.

Many important results in mathematics deal with the question of figuring
out what objects are ‘the same’. One way of rigorously dealing with this
is through the notion of isomorphic objects. However, it is sometimes better
to consider a weaker form of ‘sameness’, known as homotopy equivalence,
which allow for more flexibility. Although homotopy theory is historically
intertwined with fields of mathematics that appeal to our spatial
imagination, the concept of a homotopy appears under various guises in other
areas as well, and homotopy theoretical generalisations of classical algebra
have recently had an upswing in popularity. The papers included in this
thesis deal with spectral sequences, which can roughly be understood as
computational tools that are able to process large amounts of homotopical
information. In particular, this thesis deals with various constructions of
the Tate spectral sequence, which is a spectral sequence giving us
information on the so-called Tate construction. In my thesis, I construct
multiplicative Tate spectral sequences in a larger generality than what was
known before. This is motivated by the study of an invariant in homotopical
algebra known as topological Hochschild homology, which has been shown to
have important connections to arithmetic questions.

titleMultiplicative Tate Spectral Sequences
typeDoctoral thesis

Cambridge University Press

booktitleElliptic Cohomology
isbn978-0-521-70040-5 978-0-511-72148-9
titleOn finite resolutions of $K(n)$-local spheres
enn - 2007 - On finite resolutions of K ( n

titleA Mini-Course on Morava Stabilizer Groups and Their

titleThe homotopy of the $K(2)$-local Moore spectrum at the prime
$3$ revisited
journaltitleMathematische Zeitschrift

titleOn Brauer Groups of Lubin-Tate Spectra I

American Mathematical Society

booktitleContemporary Mathematics
isbn978-0-8218-5165-4 978-0-8218-7749-4
titleConstructions of elements in Picard groups
Providence, Rhode Island

We show that in a generalized Adams spectral sequence, the presence of a
vanishing line of ﬁxed slope (at some term of the spectral sequence, with
some intercept) is a generic property.

issn1364-0380, 1465-3060
titleVanishing lines in generalized Adams spectral sequences are
journaltitleGeom. Topol.

titleNilpotence and Stable Homotopy Theory II
journaltitleThe Annals of Mathematics

titleOperations and co-operations in Morava $E$-theory
journaltitleHomology, Homotopy and Applications

Morava E-theory En∨∗(−) is a much-studied theory in algebraic
topology, but it is not a homology theory in the usual sense, because it
fails to preserve coproducts (resp. ﬁltered homotopy colimits). The object
of this paper is to construct a spectral sequence to compute the Morava
E-theory of a coproduct (resp. ﬁltered homotopy colimit). The E2-term of
this spectral sequence involves the derived functors of direct sum (resp.
ﬁltered colimit) in an appropriate abelian category. We show that there are
at most n − 1 (resp. n) of these derived functors. When n = 1, we recover
the known result that homotopy commutes with an appropriate version of direct
sum in the K(1)-local stable homotopy category.

titleMorava $E$-theory of filtered colimits
journaltitleTrans. Amer. Math. Soc.

American Mathematical Society

Advancing research. Creating connections.

isbn978-0-8218-1079-8 978-0-8218-6389-3 978-1-4704-0255-6
noteissn: 0065-9266, 1947-6221 Issue: 666
seriesMemoirs of the American Mathematical Society
titleMorava $K$-theories and localisation

titleK-theory of dualizable categories (after A. Efimov)

We discuss the current state of knowledge of stable homotopy groups of
spheres. We describe a computational method using motivic homotopy theory,
viewed as a deformation of classical homotopy theory. This yields a
streamlined computation of the first 61 stable homotopy groups and gives
information about the stable homotopy groups in dimensions 62 through 90. As
an application, we determine the groups of homotopy spheres that classify
smooth structures on spheres through dimension 90, except for dimension 4.
The method relies more heavily on machine computations than previous methods
and is therefore less prone to error. The main mathematical tool is the Adams
spectral sequence.

notePublisher: Proceedings of the National Academy of Sciences
titleStable homotopy groups of spheres
journaltitleProceedings of the National Academy of Sciences

An elementary topological approach to Grothendieck's idea of descent is
given. While being motivated by the idea of localization which is central in
Sheaf Theory, we show how the theory of monads (= triples) provides a direct
categorical approach to Descent Theory. Thanks to an important observation by
B6nabou and Roubaud and by Beck, the monadic description covers descent also
in the abstract context of a bifibred category satisfying the Beck-Chevalley
condition. We present the fundamentals of fibrational descent theory without
requiring any prior knowledge of fibred categories. The paper contains a
number of new topological descent results as well as some new examples in the
context of regular categories which demonstrate the subtlety of the descent
problem in concrete situations.

issn0927-2852, 1572-9095
titleFacets of descent, I
journaltitleAppl Categor Struct

titleContinuous étale cohomology
journaltitleMathematische Annalen: Math. Ann.

Springer Basel

isbn978-3-0348-0065-5 978-3-0348-0066-2
titleGeneralized Étale Cohomology Theories

American Mathematical Soc.

Mathematics / Topology
booktitleHomotopy Theory: Relations with Algebraic Geometry, Group
Cohomology, and Algebraic $K$-Theory
shorttitleHomotopy Theory
titleA relation between the Picard group of the $E(n)$-local
homotopy category and $E(n)$-based Adams spectral sequence

In this paper we calculate the algebraic Hopkins' Picard group
Pic_2ˆ{alg} at the prime p=3, which is a subgroup of
index 2 of the group of isomorphism classes of invertible K(2)-local spectra
i.e. of the Hopkins' Picard group Pic_2.

titleOn Hopkins' Picard Group $\Pic_2$ at the prime $3$
journaltitleAlgebr. Geom. Topol.

titleUne résolution projective pour le second groupe de Morava
pour $p = 5$ et applications
Université de Strasbourg

Mathematics - Algebraic Geometry, Primary 14F42, 55T15, Secondary
We show that the spectral sequence induced by the Betti realization of the
slice tower for the motivic sphere spectrum agrees with the Adams-Novikov
spectral sequence, after a suitable reindexing. The proof relies on a partial
extension of Deligne’s d´ecalage construction to the Tot-tower of a
cosimplicial spectrum.

issn1364-0380, 1465-3060
titleThe Adams-Novikov spectral sequence and Voevodsky's slice
journaltitleGeom. Topol.

titleHigher algebra

titleSpectral algebraic geometry

titleHigher Topos Theory (AM-170)

titleElliptic II: Orientations

titleThe Image of J in the EHP Sequence
journaltitleThe Annals of Mathematics

Mathematics - Algebraic Topology, Mathematics - Category Theory
We classify thick subcategories of the ∞-categories of perfect modules
over ring spectra which arise as functions on even periodic derived stacks
satisfying aﬃneness and regularity conditions. For example, we show that
the thick subcategories of perfect modules over TMF are in natural
bijection with the subsets of the underlying space of the moduli stack of
elliptic curves which are closed under specialization.

issn1364-0380, 1465-3060
titleA thick subcategory theorem for modules over certain ring
journaltitleGeom. Topol.

Mathematics - Algebraic Topology, Mathematics - Category Theory,
Mathematics - Representation Theory
To a “stable homotopy theory” (a presentable, symmetric monoidal stable
∞-category), we naturally associate a category of ﬁnite ´etale algebra
objects and, using Grothendieck’s categorical machine, a proﬁnite group
that we call the Galois group. We then calculate the Galois groups in several
examples. For instance, we show that the Galois group of the periodic
E∞-algebra of topological modular forms is trivial and that the Galois
group of K(n)-local stable homotopy theory is an extended version of the
Morava stabilizer group. We also describe the Galois group of the stable
module category of a ﬁnite group. A fundamental idea throughout is the
purely categorical notion of a “descendable” algebra object and an
associated analog of faithfully ﬂat descent in this context.

titleThe Galois group of a stable homotopy theory

Mathematics - Algebraic Topology, Mathematics - Category Theory
We give a survey of the ideas of descent and nilpotence. We focus on
examples arising from chromatic homotopy theory and from group actions, as
well as a few examples in algebra.

titleExamples of descent up to nilpotence

Mathematics - Algebraic Topology, Mathematics - K-Theory and
We discuss some general properties of $\mathrm{TR}$
and its $K(1)$-localization. We prove that after $K(1)$-localization,
$\mathrm{TR}$ of
$H\mathbb{Z}$-algebras is a truncating invariant in the
sense of Land–Tamme, and deduce $h$-descent results. We show that for
regular rings in mixed characteristic, $\mathrm{TR}$ is
asymptotically $K(1)$-local, extending results of Hesselholt–Madsen. As an
application of these methods and recent advances in the theory of cyclotomic
spectra, we construct an analog of Thomason's spectral sequence relating
$K(1)$-local $K$-theory and \'etale cohomology for
$K(1)$-local $\mathrm{TR}$.

titleOn $K(1)$-local $TR$
mar/Zotero/storage/DYKQMRAM/Mathew - 202
)$-local $mathrm TR $.pdf:application/pdf;arXiv.org Sn

Mathematics - Algebraic Topology, Mathematics - Category Theory
Let G be a ﬁnite group and let F be a family of subgroups of G. We
introduce a class of G-equivariant spectra that we call F -nilpotent. This
deﬁnition ﬁts into the general theory of torsion, complete, and nilpotent
objects in a symmetric monoidal stable ∞-category, with which we begin. We
then develop some of the basic properties of F -nilpotent G-spectra, which
are explored further in the sequel to this paper.

titleNilpotence and descent in equivariant stable homotopy theory
journaltitleAdvances in Mathematics

Mathematics - Algebraic Topology, Mathematics - Category Theory
This paper starts with an exposition of descent-theoretic techniques in the
study of Picard groups of E∞-ring spectra, which naturally lead to the
study of Picard spectra. We then develop tools for the eﬃcient and explicit
determination of diﬀerentials in the associated descent spectral sequences
for the Picard spectra thus obtained. As a major application, we calculate
the Picard groups of the periodic spectrum of topological modular forms T M F
and the nonperiodic and non-connective T mf . We ﬁnd that Pic(T M F ) is
cyclic of order 576, generated by the suspension ΣT M F (a result originally
due to Hopkins), while Pic(T mf ) = Z ⊕ Z/24. In particular, we show that
there exists an invertible T mf -module which is not equivalent to a
suspension of T mf .

364-0380, 1465-3060
titleThe Picard group of topological modular forms via descent
journaltitleGeom. Topol.

We study the categories of modules over real K-theory and TMF. Inspired
by work of Bousfield, we consider TMF-modules at the prime 3 which are
relatively free with respect to TMF(2). We show that a large class of these
can be iteratively built from TMF by coning off torsion elements and
killing generators. This is based on a detailed study of vector bundles on
the moduli stack of elliptic curves. Furthermore, we consider examples of
TMF-modules and show that the categories of TMF-modules and
quasi-coherent sheaves on the derived moduli stack of elliptic curves are
equivalent (at primes bigger than 2).

noteAccepted: 2020-04-18T01:27:04Z
titleUnited Elliptic Homology
Universitäts- und Landesbibliothek Bonn

typeDoctoral thesis

titleOn relations between Adams spectral sequences, with an
application to the stable homotopy of a Moore space
journaltitleJournal of Pure and Applied Algebra

titlePeriodic Phenomena in the Adams-Novikov Spectral Sequence
journaltitleThe Annals of Mathematics

American Mathematical Society

booktitleAlgebraic K-Theory
isbn978-0-8218-0818-4 978-1-4704-2984-3
titleHypercohomology Spectra and Thomason’s Descent Theorem
Providence, Rhode Island

Mathematics - Algebraic Topology, Mathematics - Algebraic Geometry
We show that the hypercomplete $\infty$-topos associated
with any replete topos is Postnikov complete, positively answering a question
of Bhatt and Scholze. Along the way, we construct Milnor sequences for
sheaves of spaces on replete topoi. As a corollary, we generalize a result of
To\"en on affine stacks.

titleOn Postnikov completeness for replete topoi
mar/Zotero/storage/BCW788TY/Mondal and R
cke - 2022 - On Postnikov completeness for replete topoi.pdf:appl

For each prime p, we define a closed model category structure on thé
category of simplicial profinite sets in which thé weak équivalences are
thé maps which induce an isomorphism on continuous modulo p cohomology.
Using thé construction of Bousfield-Kan p-completion of a space, we first
obtain a « rigid » version of thé profinite p-completion of Artin-Mazur
and Sullivan and we deduce then a « concrète » géométrie interprétation
of Lannes' functor T (generalising that of Dror-Smith).

issn0037-9484, 2102-622X
titleEnsembles profinis simpliciaux et interprétation
géométrique du foncteur $T$
journaltitleBul. Soc. Math. France

titleThe cohomology of finite subgroups of Morava stabilizer
groups and Smith-Toda complexes
University of Washington

typeDoctoral thesis

titleTriangulated Categories. (AM-148), Volume 148

Springer Berlin Heidelberg

isbn978-3-540-37888-4 978-3-540-37889-1
seriesGrundlehren der mathematischen Wissenschaften
titleCohomology of number Fields
Berlin, Heidelberg

titleSynthetic spectra and the cellular motivic category

Mathematics - Algebraic Topology
As a consequence of the algebraicity of chromatic homotopy at large primes,
we show that the Hopkins’ Picard group of the K(n)-local category coincides
with the algebraic one when 2p − 2  n2 + n.

titleChromatic Picard groups at large primes

We construct a model structure on simplicial proﬁnite sets such that the
homotopy groups carry a natural proﬁnite structure. This yields a rigid
proﬁnite completion functor for spaces and pro-spaces. One motivation is
the ´etale homotopy theory of schemes in which higher proﬁnite ´etale
homotopy groups ﬁt well with the ´etale fundamental group which is always
proﬁnite. We show that the proﬁnite ´etale topological realization
functor is a good object in several respects.

titleProfinite Homotopy Theory
journaltitleDocumenta Mathematica

Mathematics - Algebraic Topology, Mathematics - Algebraic Geometry,
55P91, 55Q70, 14F35
For a proﬁnite group, we construct a model structure on proﬁnite spaces
and proﬁnite spectra with a continuous action. This yields descent spectral
sequences for the homotopy groups of homotopy ﬁxed point spaces and for
stable homotopy groups of homotopy orbit spaces. Our main example is the
Galois action on proﬁnite ´etale topological types of varieties over a
ﬁeld. One motivation is to understand Grothendieck’s section conjecture
in terms of homotopy ﬁxed points.

titleContinuous group actions on profinite spaces

Mathematics - Algebraic Topology, 55P43, 55T15
We construct a stable model structure on proﬁnite spectra with a
continuous action of an arbitrary proﬁnite group. This provides a natural
framework for a new and conceptually simpliﬁed construction of continuous
homotopy ﬁxed point spectra and of continuous homotopy ﬁxed point
spectral sequences for Lubin-Tate spectra under the action of the extended
Morava stabilizer group.

titleContinuous homotopy fixed points for Lubin-Tate spectra

Exact Sequence, Normal Subgroup, Open Subgroup, Simplicial Group,
Spectral Sequence
titleAn application of simplicial profinite groups
journaltitleCommentarii Mathematici Helvetici

titleLocalization with Respect to Certain Periodic Homology
journaltitleAmerican Journal of Mathematics

titleComplex Cobordism and Stable Homotopy Groups of Spheres

titleNilpotence and Periodicity in Stable Homotopy Theory.
(AM-128), Volume 128

Cambridge University Press

seriesNew Mathematical Monographs
titleCategorical Homotopy Theory

Cambridge University Press

isbn978-1-108-93688-0 978-1-108-83798-9
titleElements of $\infty$-Category Theory

American Mathematical Society

Commutative algebra, Galois theory, Homology theory, Homotopy theory,
Ring extensions (Algebra)
noteOpen Library ID: OL18500212M
pagetotalvii, 137
shorttitleGalois extensions of structured ring spectra
titleGalois extensions of structured ring spectra: stably
dualizable groups
Providence, R.I

titleLectures on Condensed Mathematics

Mathematics - Algebraic Geometry, 14G20, 14G22, 32J27, 14C30,
Mathematics - number Theory
We give proofs of de Rham comparison isomorphisms for rigid-analytic
varieties, with coeﬃcients and in families. This relies on the theory of
perfectoid spaces. Another new ingredient is the pro-´etale site, which
makes all constructions completely functorial.

titlep-adic Hodge theory for rigid-analytic varieties

Algebra, Homological, Algebraic number theory, Galois theory
titleGalois cohomology
Berlin; New York

family=Stacks project authors,

titleThe Stacks project

titleGross–Hopkins duality

Birkhäuser Boston

booktitleNew Horizons in pro-p Groups
isbn978-1-4612-7122-2 978-1-4612-1380-2
titleCohomology of $p$-Adic Analytic Groups
Boston, MA

issn0012-9593, 1873-2151
titleAlgebraic $K$-theory and étale cohomology
journaltitleAnn. Sci. École Norm. Sup.

issn0012-9593, 1873-2151
titleErratum “Algebraic $K$-theory and étale cohomology”
journaltitleAnn. Sci. École Norm. Sup.

Compact Object, Homotopy Category, Left Adjoint, Natural Morphism,
Triangulate Category
We introduce a notion of derived Azumaya algebras over ring and schemes
generalizing the notion of Azumaya algebras of Grothendieck (Le groupe de
Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix Exposés
sur la Cohomologie des Schémas, pp. 46–66, North-Holland, Amsterdam,
1968). We prove that any such algebra B on a scheme X provides a class ϕ(B)
Hˆ{2}_{et}(X,\mathbb{G}_{m})$. We
prove that for X a quasi-compact and quasi-separated scheme ϕ defines a
bijective correspondence, and in particular that any class in
torsion or not, can be represented by a derived Azumaya algebra on X. Our
result is a consequence of a more general theorem about the existence of
compact generators in twisted derived categories, with coefficients in any
local system of reasonable dg-categories, generalizing the well known
existence of compact generators in derived categories of quasi-coherent
sheaves of Bondal and Van Den Bergh (Mosc. Math. J. 3(1):1–36, 2003).
A huge part of this paper concerns the treatment of twisted derived
categories, as well as the proof that the existence of compact generator
locally for the fppf topology implies the existence of a global compact
generator. We present explicit examples of derived Azumaya algebras that are
not represented by classical Azumaya algebras, as well as applications of our
main result to the localization for twisted algebraic K-theory and to the
stability of saturated dg-categories by direct push-forwards along smooth and
proper maps.

titleDerived Azumaya algebras and generators for twisted derived
ën -
- Derived Azumaya algebras and generators for twiste.pdf:applica
itleInvent. math.

Cambridge University Press

The landscape of homological algebra has evolved over the last half-century
into a fundamental tool for the working mathematician. This book provides a
unified account of homological algebra as it exists today. The historical
connection with topology, regular local rings, and semi-simple Lie algebras
are also described. This book is suitable for second or third year graduate
students. The first half of the book takes as its subject the canonical
topics in homological algebra: derived functors, Tor and Ext, projective
dimensions and spectral sequences. Homology of group and Lie algebras
illustrate these topics. Intermingled are less canonical topics, such as the
derived inverse limit functor lim1, local cohomology, Galois cohomology, and
affine Lie algebras. The last part of the book covers less traditional topics
that are a vital part of the modern homological toolkit: simplicial methods,
Hochschild and cyclic homology, derived categories and total derived
functors. By making these tools more accessible, the book helps to break down
the technological barrier between experts and casual users of homological

isbn978-0-521-43500-0 978-0-521-55987-4 978-1-139-64413-6
titleAn Introduction to Homological Algebra

titleClassifying modules over K-theory spectra
journaltitleJournal of Pure and Applied Algebra