Talbot Talk 1

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Talbot Talk 1: Vesna

  • \(\operatorname{Pic}(R)\): invertible \(R{\hbox{-}}\)modules and equivalences with \(\otimes_R\).
  • \(\pi_0 \operatorname{Pic}(R)\): invertible \(R{\hbox{-}}\)modules modulo equivalence
  • \(\Omega \operatorname{Pic}(R) = \operatorname{GL}_1(R)\)
  • Taking the connected component of \(R\) yields \(\operatorname{Pic}^0(R) = {\mathbf{B}}\operatorname{GL}_1(R)\)
  • \([X, \operatorname{Pic}(R)]\) equals bundles of invertible \(R{\hbox{-}}\)modules in \(X\).
    • Classical example: \([X, \operatorname{Pic}(S^0)]\), stable spherical fibrations on \(X\), motivates most of the development of this theory. Equivalently, what is \(\operatorname{Pic}(S^0)\)?
  • \({\operatorname{ko}}\): connective real \(K\) theory
    • See K-theory
    • The 0th space: \(\Omega^ \infty \cong {\mathbf{Z}}\times{\mathbf{B}}{\operatorname{O}}\), classifies stable real vector bundles.
    • There is a maps \begin{align*} [X, {\mathbf{Z}}\times{\mathbf{B}}{\operatorname{O}}] &\xrightarrow{\sim} [X, \operatorname{Pic}(S^0)] \\ \xi/X &\mapsto \mathop{\mathrm{Th}}(\xi) .\end{align*} Yields an \(\infty{\hbox{-}}\)loop map \({\mathbf{Z}}\times{\mathbf{B}}{\operatorname{O}}\to \operatorname{Pic}(S^0)\) and \({\operatorname{ko}}\to {\operatorname{Pic}}(S^0)\). Yields Adams’ \(J{\hbox{-}}\)homomorphism. See J-homomorphism.
    • Story that develops here: can develop a theory of \(R{\hbox{-}}\)oriented bundles, twisted \(R{\hbox{-}}\)cohomology, twists by ordinary cohomology class, or twist by the space of maps \(\operatorname{Pic}= \mathop{\mathrm{Maps}}({H{\mathbf{Z}}}, {\operatorname{Pic}})\).
  • There is also a Brauer space \(\mathop{\mathrm{Br}}(R)\). See Brauer.

Questions:

  • What are \(\pi_* \operatorname{GL}_1(R)\)?
  • For a space \(X\), show that \([X, \operatorname{GL}(R)] = R^0(X)^X\)
  • What are the invertible \(S^0\) modules?

Computing things

  • \(\pi_0 \mathop{\mathrm{Br}}(R) = ?\)

  • \(\pi_1 \mathop{\mathrm{Br}}(R) = \pi_0 \operatorname{Pic}(R)\)

  • \(\pi_2 \mathop{\mathrm{Br}}(R) = \pi_1 \operatorname{Pic}(R) = \pi_0 \operatorname{GL}_1(R) = (\pi_0 R)^{\times}\)

  • \(\pi_{>2} \mathop{\mathrm{Br}}(R) = \pi_{>1} \operatorname{Pic}(R) = \pi_{>0} \operatorname{GL}_1(R) = \pi_{>0} R\).

  • Can compute low degree \(k\) invariants for \(\operatorname{Pic}(R)\), comes from looking at Steenrod operations.

  • How to compute more:

    • Comparison with algebra (relatively easy, could reduce to open problems)
    • Descent
    • Obstruction theory

Use that there is an injection \(0\to \operatorname{Pic}(\pi_* R) \to \pi_0 {\operatorname{Pic}}(R)\) when \(R\) is connective or \(R\) is weakly even periodic and \(\pi_0 R\) is regular Noetherian. - This is \(\operatorname{Pic}\) over graded rings - But it’s much more complicated to have anything like this for the Brauer group. - Theorem: the functors \(\operatorname{Pic}\) and \(\mathop{\mathrm{Br}}\), \(\mathsf{CAlg}({\mathsf{Sp}}) \to {\Omega}^\infty{\mathsf{Top}}\) satisfy etale descent and Galois descent respectively - \(R\to S\) a map of ring spectra if \(\pi_0 R\to \pi_0 S\) is etale as a map of rings (smooth of dimension zero, or flat + unramified) and there is an equivalence \(\pi_k R \otimes_{\pi_0 R} \pi_0 S \xrightarrow{\sim} \pi_k S\). - \({\operatorname{KO}}\) has no interesting etale extension - \(R\to S^{?}\) is \(G{\hbox{-}}\)Galois if - \(R \xrightarrow{\sim}S^{hG}\), mapping to homotopy fixed points is an equivalence - \(S\otimes_R S \xrightarrow{\sim} \prod_G S\) - \(\pi_* {\operatorname{ku}}= {\mathbf{Z}}[\beta ^{\pm 1}]\) and \(\operatorname{Pic}(\pi_* {\operatorname{ku}}) = {\mathbf{Z}}/2\) where \(\beta\) is the Bott class. In fact \(\operatorname{Pic}({\operatorname{KU}}) = {\mathbf{Z}}/2\), and descent yields \(\operatorname{Pic}({\operatorname{KO}}) = \operatorname{Pic}({\operatorname{KU}})^{hC_2}\) - See descent spectral sequence? - Descent is like a local to global principle.

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