How are multiplication maps of spectra defined?












1












$begingroup$


In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition.




For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} mathrm{MU}(p) to mathrm{MU}(p)$ given by multiplication
by $t_k$.




Here, $mathrm{MU}(p)$ is mod-$p$ complex bordism, which I presume means the localization of $mathrm{MU}$ at $mathrm{H}mathbb{F}_p$, and $t_k$ is a generator of $pi_{2k}(mathrm{MU}(p))$.




Question. What does this mean?




Some thoughts: I can interpret $Sigma^{2k} mathrm{MU}(p)$ as $Sigma^{infty}S^{2k} wedge mathrm{MU}(p)$, so we can consider a map $Sigma^{infty} S^{2k} wedge mathrm{MU}(p) to mathrm{MU}(p) wedge mathrm{MU}(p)$ defined by, say, $t_k wedge operatorname{Id}$, followed by some kind of multiplication map on $operatorname{MU}(p)$. I believe $operatorname{MU}$ is a ring spectrum, which I think implies that $mathrm{MU}$ admits a multiplication map of some sort. Maybe that's what's being implied here.




Followup question / rant. Learning about homotopy theory has been a pain to me, much more than learning any other subject, due to the sheer number of insider's jargon, and the lack of introductions explaining them. If possible, any advice on how to deal with this issue, including references that I may not know of, would be appreciated.











share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition.




    For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} mathrm{MU}(p) to mathrm{MU}(p)$ given by multiplication
    by $t_k$.




    Here, $mathrm{MU}(p)$ is mod-$p$ complex bordism, which I presume means the localization of $mathrm{MU}$ at $mathrm{H}mathbb{F}_p$, and $t_k$ is a generator of $pi_{2k}(mathrm{MU}(p))$.




    Question. What does this mean?




    Some thoughts: I can interpret $Sigma^{2k} mathrm{MU}(p)$ as $Sigma^{infty}S^{2k} wedge mathrm{MU}(p)$, so we can consider a map $Sigma^{infty} S^{2k} wedge mathrm{MU}(p) to mathrm{MU}(p) wedge mathrm{MU}(p)$ defined by, say, $t_k wedge operatorname{Id}$, followed by some kind of multiplication map on $operatorname{MU}(p)$. I believe $operatorname{MU}$ is a ring spectrum, which I think implies that $mathrm{MU}$ admits a multiplication map of some sort. Maybe that's what's being implied here.




    Followup question / rant. Learning about homotopy theory has been a pain to me, much more than learning any other subject, due to the sheer number of insider's jargon, and the lack of introductions explaining them. If possible, any advice on how to deal with this issue, including references that I may not know of, would be appreciated.











    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition.




      For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} mathrm{MU}(p) to mathrm{MU}(p)$ given by multiplication
      by $t_k$.




      Here, $mathrm{MU}(p)$ is mod-$p$ complex bordism, which I presume means the localization of $mathrm{MU}$ at $mathrm{H}mathbb{F}_p$, and $t_k$ is a generator of $pi_{2k}(mathrm{MU}(p))$.




      Question. What does this mean?




      Some thoughts: I can interpret $Sigma^{2k} mathrm{MU}(p)$ as $Sigma^{infty}S^{2k} wedge mathrm{MU}(p)$, so we can consider a map $Sigma^{infty} S^{2k} wedge mathrm{MU}(p) to mathrm{MU}(p) wedge mathrm{MU}(p)$ defined by, say, $t_k wedge operatorname{Id}$, followed by some kind of multiplication map on $operatorname{MU}(p)$. I believe $operatorname{MU}$ is a ring spectrum, which I think implies that $mathrm{MU}$ admits a multiplication map of some sort. Maybe that's what's being implied here.




      Followup question / rant. Learning about homotopy theory has been a pain to me, much more than learning any other subject, due to the sheer number of insider's jargon, and the lack of introductions explaining them. If possible, any advice on how to deal with this issue, including references that I may not know of, would be appreciated.











      share|cite|improve this question









      $endgroup$




      In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition.




      For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} mathrm{MU}(p) to mathrm{MU}(p)$ given by multiplication
      by $t_k$.




      Here, $mathrm{MU}(p)$ is mod-$p$ complex bordism, which I presume means the localization of $mathrm{MU}$ at $mathrm{H}mathbb{F}_p$, and $t_k$ is a generator of $pi_{2k}(mathrm{MU}(p))$.




      Question. What does this mean?




      Some thoughts: I can interpret $Sigma^{2k} mathrm{MU}(p)$ as $Sigma^{infty}S^{2k} wedge mathrm{MU}(p)$, so we can consider a map $Sigma^{infty} S^{2k} wedge mathrm{MU}(p) to mathrm{MU}(p) wedge mathrm{MU}(p)$ defined by, say, $t_k wedge operatorname{Id}$, followed by some kind of multiplication map on $operatorname{MU}(p)$. I believe $operatorname{MU}$ is a ring spectrum, which I think implies that $mathrm{MU}$ admits a multiplication map of some sort. Maybe that's what's being implied here.




      Followup question / rant. Learning about homotopy theory has been a pain to me, much more than learning any other subject, due to the sheer number of insider's jargon, and the lack of introductions explaining them. If possible, any advice on how to deal with this issue, including references that I may not know of, would be appreciated.








      reference-request algebraic-topology homotopy-theory stable-homotopy-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 18 '18 at 9:58









      guestguest

      1005




      1005






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $mu:MUwedge MUrightarrow MU$ as well as a unit $eta:Srightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $xinpi_rMU$, multiplication by $x$ is the composite



          $$hat x:S^rwedge MUxrightarrow{xwedge 1}MUwedge MUxrightarrow{mu}MU.$$



          It is standard to drop the hat and denote the induced map by the same symbol as the element $xinpi_rMU$. Thus $t_kinpi_{2k}MU$ is a generator and the map you are interested in is what I have called $hat{t}^k$ in the previous notation.



          In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 11:12






          • 1




            $begingroup$
            I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:21






          • 1




            $begingroup$
            I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:26






          • 1




            $begingroup$
            For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:31






          • 1




            $begingroup$
            You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
            $endgroup$
            – guest
            Dec 18 '18 at 16:42











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          3












          $begingroup$

          The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $mu:MUwedge MUrightarrow MU$ as well as a unit $eta:Srightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $xinpi_rMU$, multiplication by $x$ is the composite



          $$hat x:S^rwedge MUxrightarrow{xwedge 1}MUwedge MUxrightarrow{mu}MU.$$



          It is standard to drop the hat and denote the induced map by the same symbol as the element $xinpi_rMU$. Thus $t_kinpi_{2k}MU$ is a generator and the map you are interested in is what I have called $hat{t}^k$ in the previous notation.



          In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 11:12






          • 1




            $begingroup$
            I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:21






          • 1




            $begingroup$
            I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:26






          • 1




            $begingroup$
            For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:31






          • 1




            $begingroup$
            You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
            $endgroup$
            – guest
            Dec 18 '18 at 16:42
















          3












          $begingroup$

          The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $mu:MUwedge MUrightarrow MU$ as well as a unit $eta:Srightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $xinpi_rMU$, multiplication by $x$ is the composite



          $$hat x:S^rwedge MUxrightarrow{xwedge 1}MUwedge MUxrightarrow{mu}MU.$$



          It is standard to drop the hat and denote the induced map by the same symbol as the element $xinpi_rMU$. Thus $t_kinpi_{2k}MU$ is a generator and the map you are interested in is what I have called $hat{t}^k$ in the previous notation.



          In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 11:12






          • 1




            $begingroup$
            I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:21






          • 1




            $begingroup$
            I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:26






          • 1




            $begingroup$
            For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:31






          • 1




            $begingroup$
            You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
            $endgroup$
            – guest
            Dec 18 '18 at 16:42














          3












          3








          3





          $begingroup$

          The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $mu:MUwedge MUrightarrow MU$ as well as a unit $eta:Srightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $xinpi_rMU$, multiplication by $x$ is the composite



          $$hat x:S^rwedge MUxrightarrow{xwedge 1}MUwedge MUxrightarrow{mu}MU.$$



          It is standard to drop the hat and denote the induced map by the same symbol as the element $xinpi_rMU$. Thus $t_kinpi_{2k}MU$ is a generator and the map you are interested in is what I have called $hat{t}^k$ in the previous notation.



          In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.






          share|cite|improve this answer









          $endgroup$



          The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $mu:MUwedge MUrightarrow MU$ as well as a unit $eta:Srightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $xinpi_rMU$, multiplication by $x$ is the composite



          $$hat x:S^rwedge MUxrightarrow{xwedge 1}MUwedge MUxrightarrow{mu}MU.$$



          It is standard to drop the hat and denote the induced map by the same symbol as the element $xinpi_rMU$. Thus $t_kinpi_{2k}MU$ is a generator and the map you are interested in is what I have called $hat{t}^k$ in the previous notation.



          In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 18 '18 at 11:09









          TyroneTyrone

          4,66511225




          4,66511225












          • $begingroup$
            p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 11:12






          • 1




            $begingroup$
            I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:21






          • 1




            $begingroup$
            I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:26






          • 1




            $begingroup$
            For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:31






          • 1




            $begingroup$
            You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
            $endgroup$
            – guest
            Dec 18 '18 at 16:42


















          • $begingroup$
            p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 11:12






          • 1




            $begingroup$
            I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:21






          • 1




            $begingroup$
            I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:26






          • 1




            $begingroup$
            For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
            $endgroup$
            – Tyrone
            Dec 18 '18 at 13:31






          • 1




            $begingroup$
            You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
            $endgroup$
            – guest
            Dec 18 '18 at 16:42
















          $begingroup$
          p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
          $endgroup$
          – Tyrone
          Dec 18 '18 at 11:12




          $begingroup$
          p.s. the 'obvious' diagrams correspond to the associativity $mu(1wedgemu)simeq mu(muwedge1):MUwedge MUwedge MUrightarrow MU$ and unit $mu(1wedgeeta)simeq mu(etawedge 1)simeq id_{MU}:Swedge MUcong MUwedge Scong MUrightarrow MU$.
          $endgroup$
          – Tyrone
          Dec 18 '18 at 11:12




          1




          1




          $begingroup$
          I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:21




          $begingroup$
          I believe that $MU_{(p)}$ does inherit a ring structure. In fact Bousfield discusses this himself in his paper "The Localization of Spectra with Respect to Homology". As i understand it, $p$-localisation is a so-called 'smashing' localisation, so $MU_{p}simeq MUwedge S_{(p)}$, where $S_{(p)}$ is the $p$-local sphere, so things work out fairly easily. (Although I'd recommend checking these statements)
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:21




          1




          1




          $begingroup$
          I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:26




          $begingroup$
          I do think Lurie's notes are very good, but limited in the sense I wrote above. I'm not sure I can make a good recommendation in general for a set of notes. I know Morava K-theory is discussed in Ravenel's classics "Complex Cobordism and Stable Homotopy Groups of Spheres" and "Nilpotence and Periodicity in Stable Homotopy Theory". The presentation there is very low-tech compared to Lurie's, however.
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:26




          1




          1




          $begingroup$
          For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:31




          $begingroup$
          For some background you could read Würgler's, "Morava K-Theories: A Survey" and its references. This paper is now quite old, and you are probably already above most of it, but you might be interested to see the material presented in a more standard way. Alternately, Mike Hopkins is the deserved golden boy of modern stable homotopy theory, and I believe he together with Ravenel and others has written a fair bit about chromatic homotopy theory, elliptic cohomology, etc...
          $endgroup$
          – Tyrone
          Dec 18 '18 at 13:31




          1




          1




          $begingroup$
          You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
          $endgroup$
          – guest
          Dec 18 '18 at 16:42




          $begingroup$
          You're a saint, Tyrone. I followed your suggestion that smashing localizations send ring spectra to ring spectra, and I'm convinced it works out (though I'm stuck proving a diagram that looks seriously commutative really is commutative). I'm going to look up the references you mentioned and see if it helps me out. Thanks again.
          $endgroup$
          – guest
          Dec 18 '18 at 16:42


















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