Mixing
Power operations from a Tate construction
$begingroup$
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $mathbb{E}_{infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $hat{T}_p: X mapsto ( X^{otimes p})^{tC_p}$ and its best colimit-preserving approximation, $T_p.$ For any $mathbb{E}_infty$ $k$-algebra $X,$ one obtains a map $T_p(X)[-1] to X.$ $T_p(X)$ is given by tensoring with a $k$-bimodule which is equivalent on one side to $k^{tC_p}$ and this allows us to obtain operations (not $k$-linear) from elements of Tate cohomology, $ pi_* k^{tC_p} simeq hat{H}^{-*}(C_p, k).$
The precise statement is in construction 2.2.6, which applies the observation that for $k$ a discrete ring of characteristic $p$, $1 in k$ determines a canonical element of $hat{H}^{-1}(C_p, k),$ precisely because that group is given as the kernel of the norm. This defines a map $k to k^{tC_p}[-1]$ which upon composition with the map in the previous paragraph gives a map $X to X.$ This map is supposed to be the derived witness to $P^0.$
Here is remark 2.2.9
Construction 2.2.6 can be generalized: given any class $x in hat{H}^{n-1}(mathbb{Z} / pmathbb{Z}; k)$, we obtain an associated
map $P(x) : A to A[n]$, which induces
group homomorphisms $pi_m(A) to pi_{m-n}(A)$. These operations depend functorially
on $A$ and generate an algebra (the extended Steenrod algebra) of “power
operations” which act on the homotopy groups of every $mathbb{E}_infty$-algebra over $k.$
My questions: is there any reference where this construction of the extended powers is fully elaborated? How much of the elementary structure of the Steenrod algebra (e.g. Adem relations, structure of the dual Steenrod algebra, etc.) can be translated to this point of view?
Just to fill in a few details, Lurie constructs these power operations by first rotating the fiber sequence defining the Tate construction to yield
$hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to (X^{otimes p})^{hC_p}$
If $X$ is an $mathbb{E}_infty$ $k$-algebra, one then computes the composition
$T_p(X)[-1] to hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to X^{otimes p}_{hSigma_p} to X$
where the maps are given by approximation, the first map in the rotated fiber sequence, a tautological map between colimits, and the $mathbb{E}_infty$ multiplication respectively.
at.algebraic-topology homotopy-theory derived-algebraic-geometry steenrod-algebra
asked Jan 7 at 19:26
pupshawpupshaw
1256
$endgroup$
add a comment |
$begingroup$
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $mathbb{E}_{infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $hat{T}_p: X mapsto ( X^{otimes p})^{tC_p}$ and its best colimit-preserving approximation, $T_p.$ For any $mathbb{E}_infty$ $k$-algebra $X,$ one obtains a map $T_p(X)[-1] to X.$ $T_p(X)$ is given by tensoring with a $k$-bimodule which is equivalent on one side to $k^{tC_p}$ and this allows us to obtain operations (not $k$-linear) from elements of Tate cohomology, $ pi_* k^{tC_p} simeq hat{H}^{-*}(C_p, k).$
The precise statement is in construction 2.2.6, which applies the observation that for $k$ a discrete ring of characteristic $p$, $1 in k$ determines a canonical element of $hat{H}^{-1}(C_p, k),$ precisely because that group is given as the kernel of the norm. This defines a map $k to k^{tC_p}[-1]$ which upon composition with the map in the previous paragraph gives a map $X to X.$ This map is supposed to be the derived witness to $P^0.$
Here is remark 2.2.9
Construction 2.2.6 can be generalized: given any class $x in hat{H}^{n-1}(mathbb{Z} / pmathbb{Z}; k)$, we obtain an associated
map $P(x) : A to A[n]$, which induces
group homomorphisms $pi_m(A) to pi_{m-n}(A)$. These operations depend functorially
on $A$ and generate an algebra (the extended Steenrod algebra) of “power
operations” which act on the homotopy groups of every $mathbb{E}_infty$-algebra over $k.$
My questions: is there any reference where this construction of the extended powers is fully elaborated? How much of the elementary structure of the Steenrod algebra (e.g. Adem relations, structure of the dual Steenrod algebra, etc.) can be translated to this point of view?
Just to fill in a few details, Lurie constructs these power operations by first rotating the fiber sequence defining the Tate construction to yield
$hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to (X^{otimes p})^{hC_p}$
If $X$ is an $mathbb{E}_infty$ $k$-algebra, one then computes the composition
$T_p(X)[-1] to hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to X^{otimes p}_{hSigma_p} to X$
where the maps are given by approximation, the first map in the rotated fiber sequence, a tautological map between colimits, and the $mathbb{E}_infty$ multiplication respectively.
at.algebraic-topology homotopy-theory derived-algebraic-geometry steenrod-algebra
asked Jan 7 at 19:26
pupshawpupshaw
1256
$endgroup$
add a comment |
$begingroup$
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $mathbb{E}_{infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $hat{T}_p: X mapsto ( X^{otimes p})^{tC_p}$ and its best colimit-preserving approximation, $T_p.$ For any $mathbb{E}_infty$ $k$-algebra $X,$ one obtains a map $T_p(X)[-1] to X.$ $T_p(X)$ is given by tensoring with a $k$-bimodule which is equivalent on one side to $k^{tC_p}$ and this allows us to obtain operations (not $k$-linear) from elements of Tate cohomology, $ pi_* k^{tC_p} simeq hat{H}^{-*}(C_p, k).$
The precise statement is in construction 2.2.6, which applies the observation that for $k$ a discrete ring of characteristic $p$, $1 in k$ determines a canonical element of $hat{H}^{-1}(C_p, k),$ precisely because that group is given as the kernel of the norm. This defines a map $k to k^{tC_p}[-1]$ which upon composition with the map in the previous paragraph gives a map $X to X.$ This map is supposed to be the derived witness to $P^0.$
Here is remark 2.2.9
Construction 2.2.6 can be generalized: given any class $x in hat{H}^{n-1}(mathbb{Z} / pmathbb{Z}; k)$, we obtain an associated
map $P(x) : A to A[n]$, which induces
group homomorphisms $pi_m(A) to pi_{m-n}(A)$. These operations depend functorially
on $A$ and generate an algebra (the extended Steenrod algebra) of “power
operations” which act on the homotopy groups of every $mathbb{E}_infty$-algebra over $k.$
My questions: is there any reference where this construction of the extended powers is fully elaborated? How much of the elementary structure of the Steenrod algebra (e.g. Adem relations, structure of the dual Steenrod algebra, etc.) can be translated to this point of view?
Just to fill in a few details, Lurie constructs these power operations by first rotating the fiber sequence defining the Tate construction to yield
$hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to (X^{otimes p})^{hC_p}$
If $X$ is an $mathbb{E}_infty$ $k$-algebra, one then computes the composition
$T_p(X)[-1] to hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to X^{otimes p}_{hSigma_p} to X$
where the maps are given by approximation, the first map in the rotated fiber sequence, a tautological map between colimits, and the $mathbb{E}_infty$ multiplication respectively.
at.algebraic-topology homotopy-theory derived-algebraic-geometry steenrod-algebra
asked Jan 7 at 19:26
pupshawpupshaw
1256
$endgroup$
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $mathbb{E}_{infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $hat{T}_p: X mapsto ( X^{otimes p})^{tC_p}$ and its best colimit-preserving approximation, $T_p.$ For any $mathbb{E}_infty$ $k$-algebra $X,$ one obtains a map $T_p(X)[-1] to X.$ $T_p(X)$ is given by tensoring with a $k$-bimodule which is equivalent on one side to $k^{tC_p}$ and this allows us to obtain operations (not $k$-linear) from elements of Tate cohomology, $ pi_* k^{tC_p} simeq hat{H}^{-*}(C_p, k).$
The precise statement is in construction 2.2.6, which applies the observation that for $k$ a discrete ring of characteristic $p$, $1 in k$ determines a canonical element of $hat{H}^{-1}(C_p, k),$ precisely because that group is given as the kernel of the norm. This defines a map $k to k^{tC_p}[-1]$ which upon composition with the map in the previous paragraph gives a map $X to X.$ This map is supposed to be the derived witness to $P^0.$
Here is remark 2.2.9
Construction 2.2.6 can be generalized: given any class $x in hat{H}^{n-1}(mathbb{Z} / pmathbb{Z}; k)$, we obtain an associated
map $P(x) : A to A[n]$, which induces
group homomorphisms $pi_m(A) to pi_{m-n}(A)$. These operations depend functorially
on $A$ and generate an algebra (the extended Steenrod algebra) of “power
operations” which act on the homotopy groups of every $mathbb{E}_infty$-algebra over $k.$
My questions: is there any reference where this construction of the extended powers is fully elaborated? How much of the elementary structure of the Steenrod algebra (e.g. Adem relations, structure of the dual Steenrod algebra, etc.) can be translated to this point of view?
Just to fill in a few details, Lurie constructs these power operations by first rotating the fiber sequence defining the Tate construction to yield
$hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to (X^{otimes p})^{hC_p}$
If $X$ is an $mathbb{E}_infty$ $k$-algebra, one then computes the composition
$T_p(X)[-1] to hat{T}_p(X)[-1] to (X^{otimes p})_{hC_p} to X^{otimes p}_{hSigma_p} to X$
where the maps are given by approximation, the first map in the rotated fiber sequence, a tautological map between colimits, and the $mathbb{E}_infty$ multiplication respectively.
at.algebraic-topology homotopy-theory derived-algebraic-geometry steenrod-algebra
at.algebraic-topology homotopy-theory derived-algebraic-geometry steenrod-algebra
asked Jan 7 at 19:26
pupshawpupshaw
1256
asked Jan 7 at 19:26
pupshawpupshaw
1256
asked Jan 7 at 19:26
pupshawpupshaw
1256
asked Jan 7 at 19:26
pupshawpupshaw
1256
asked Jan 7 at 19:26
pupshawpupshaw
1256
1256
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $Sigma_2$ and operations, or the relation between $Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $Hunderline{Bbb{F}}_2$ and a cellular construction of $BP{Bbb R}$.
There is probably more to say but you should feel free to contact me by email if you want further elaboration.
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320321%2fpower-operations-from-a-tate-construction%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $Sigma_2$ and operations, or the relation between $Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $Hunderline{Bbb{F}}_2$ and a cellular construction of $BP{Bbb R}$.
There is probably more to say but you should feel free to contact me by email if you want further elaboration.
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
$endgroup$
add a comment |
$begingroup$
What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $Sigma_2$ and operations, or the relation between $Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $Hunderline{Bbb{F}}_2$ and a cellular construction of $BP{Bbb R}$.
There is probably more to say but you should feel free to contact me by email if you want further elaboration.
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
$endgroup$
add a comment |
$begingroup$
What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $Sigma_2$ and operations, or the relation between $Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $Hunderline{Bbb{F}}_2$ and a cellular construction of $BP{Bbb R}$.
There is probably more to say but you should feel free to contact me by email if you want further elaboration.
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
$endgroup$
What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $Sigma_2$ and operations, or the relation between $Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $Hunderline{Bbb{F}}_2$ and a cellular construction of $BP{Bbb R}$.
There is probably more to say but you should feel free to contact me by email if you want further elaboration.
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
answered Jan 7 at 20:28
Tyler LawsonTyler Lawson
39.1k8136199
39.1k8136199
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320321%2fpower-operations-from-a-tate-construction%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
