Tags: #web/quick-notes



  • comonads in \(\mathsf{C}\): coalgebra object in \([\mathsf{C}, \mathsf{C}]\).

  • Comodules over a comonad \(T\): an object \(X\), a map \(a^\sharp: X\to TX\), and some coherence conditions. Often called \(T{\hbox{-}}\)algebras, called the category of \(T{\hbox{-}}\)comodules \({\mathsf{T}{\hbox{-}}\mathsf{coMod}}(\mathsf{C})\).

  • A fun but non-obvious consequence of https://stacks.math.columbia.edu/tag/06WS: for $G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}} $ faithfully flat, there is an equivalence of categories \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G) { \, \xrightarrow{\sim}\, }{\mathsf{Rep}}(G) ,\end{align*} the category of regular \(G{\hbox{-}}\)representations, i.e. \({\mathsf{\Gamma(G)}{\hbox{-}}\mathsf{coMod}}\). See regular representation.

  • Why this is true:

    • ${\mathsf{QCoh}}(\operatorname{Spec}R) { , \xrightarrow{\sim}, } {}_{R}{\mathsf{Mod}} $
    • Use \(p: \operatorname{Spec}R\to {\mathbf{B}}G\) induced by \(G\to R\) to induce a pullback functor $p^*: {\mathsf{QCoh}}({\mathbf{B}}G)\to {\mathsf{QCoh}}(\operatorname{Spec}R) \cong {}_{R}{\mathsf{Mod}} $.
    • Set up a adjunction that yields a comonad equivalent to \(F: ({-})\otimes_R \Gamma(G)\).
    • Apply Barr-Beck :
      • Given an adjunction \(\adjunction{L}{R}{\mathsf{D}}{\mathsf{C}}\), get a comonad \(LR\in [\mathsf{C}, \mathsf{C}]\).
      • Then every \(X\in \mathsf{D}\) yields \(L(X) \in {\mathsf{LR}{\hbox{-}}\mathsf{coMod}}\), and \(\mathsf{D} \xrightarrow{L} \mathsf{C}\) factors:

Link to diagram

  • Barr-Beck says \(\tilde L\) is an equivalence under suitable conditions (\(L, R\) conservative with \(L\) preserving equalizers).

  • Set up the Unsorted/adjoint (categorical) \begin{align*} \mathsf{D} \coloneqq\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \coloneqq\mathsf{C} .\end{align*} Then \(LR \coloneqq p^*p_*\), and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{ { \, \xrightarrow{\sim}\, }} {\mathsf{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}

  • Use that if $G\in{\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}{/ {R}} $ then $\Gamma(G) \in \mathsf{Hopf} {}{R} \mathsf{Alg} $. Set $\mathsf{C} \coloneqq {}_{R}{\mathsf{Mod}} $, and \(F\in [\mathsf{C}, \mathsf{C}]\) to be \(F({-}) \coloneqq({-})\otimes_R \Gamma(G)\). Then there is an equivalence of categories \begin{align*} {\mathsf{F}{\hbox{-}}\mathsf{coMod}}(\mathsf{C}) { \, \xrightarrow{\sim}\, }{\mathsf{Rep}}(G) .\end{align*}

  • Then show that \(F\) is equivalent to \(p^*p_*\).


http://individual.utoronto.ca/groechenig/stacks.pdf #resources

Refs: stacks vector bundle Unsorted/descent

  • Vector bundles as descent data: consider describing \(E\to X\); one needs the cocycle condition. This means choosing \({\mathcal{U}}\rightrightarrows X\) and bundle automorphisms \(\phi_{ij}: (U_i \cap U_j)\times {\mathbf{R}}^n {\circlearrowleft}\) of the trivial bundle.
    • We then want to glue up to obtain some \(E\) over \(X\): finding local bundle isomorphisms \(\phi_i: U_i \times {\mathbf{R}}^n { \, \xrightarrow{\sim}\, }{ \left.{{E}} \right|_{{U_i}} }\) with \(\phi_{ij} = \phi_i \circ \phi_j^{-1}\) on \(U_i \cap U_j\). The cocycle condition is necessary, and for topological vector bundles, also sufficient.
    • How to glue: set \(E \coloneqq{\textstyle\coprod}_{i} (U_i \times {\mathbf{R}}^n)/\sim\) where \((x, \mathbf{v})\sim (x, \phi_{ij}(\mathbf{v}))\) with the quotient topology.
  • Alternative formulation:
    • Let \({\mathcal{U}}\rightrightarrows X\) and define \(Y\coloneqq\coprod_i U_i\), which induces \(Y \xrightarrow{\pi} X\) by the inclusions \(U_i \hookrightarrow X\).
    • Then \begin{align*} Y{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{2} } } = \coprod_{(i, j)\in I{ {}^{ \scriptscriptstyle\times^{2} } }} U_i \cap U_j .\end{align*} The cocycle condition becomes the existence of an isomorphism of bundles over \(Y{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{2} } }\):

Link to diagram

  • Note that pullbacks of trivial bundles are trivial, so this is an automorphism of the trivial bundle on \(Y{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{2} } }\)
  • The cocycle condition becomes an identity among bundle isomorphisms on \(Y{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{3} } }\): \begin{align*} p_{12}^* \phi \circ p_{23}^* \phi = p_{13}^*\phi \end{align*} as maps \(p_3^*\tilde E\to p_1^* \tilde E\). Local trivializations translate to \(\pi^* E \cong \tilde E\), the trivial bundle.


  • There is an equivalence of categories ${}{{\mathbf{R}}}{\mathsf{Mod}} { , \xrightarrow{\sim}, }\mathsf{Tw} {}{{\mathbf{C}}}{\mathsf{Mod}} $ where the latter consists of objects which are pairs \((V, f:V\to V)\) where \(f(\lambda \mathbf{v}) = \mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu \mathbf{v}\) is a structure map and \(f^2 = \operatorname{id}_V\) and morphisms \(\phi:V\to W\) that commute with the structure maps.

    • The forward map is \(V\mapsto (V\otimes_{\mathbf{R}}{\mathbf{C}}, f)\) with \(f\) the generator \(f\in { \mathsf{Gal}} ({\mathbf{C}}_{/ {{\mathbf{R}}}} )\), and the inverse is \((V, f)\mapsto V^f\), the \(f{\hbox{-}}\)invariant subspace.
  • For field extensions $L_{/ {k}} $, the ring morphism \(k\hookrightarrow L\) yields \(\operatorname{Spec}L \to \operatorname{Spec}k\), which behaves like a covering space with \(\mathop{\mathrm{Deck}}(\operatorname{Spec}L _{/ {\operatorname{Spec}k}} ) \cong { \mathsf{Gal}} (L_{/ {k}} )\).

  • Vector bundles on \(\operatorname{Spec}k\) correspond to ${}_{k}{\mathsf{Mod}} $, and Galois-equivariant vector bundles on \(\operatorname{Spec}L\) will correspond to vector bundles on the quotient \(\operatorname{Spec}k\).

  • $R\in \mathsf{Alg} _{/ {A}} $: a ring morphism \(A\to R\).

  • Given \(f\in {\mathbf{Z}}[x_1,\cdots, x_n]\), taking the zero locus in a ring \(R\) yields a functor \(\mathsf{CRing}\to {\mathsf{Set}}\). To do this with \(f\in A[x_1,\cdots, x_n]\) for \(A\in \mathsf{CRing}\), one needs $R\in \mathsf{Alg} _{/ {A}} $, so this yields a functor \(\mathsf{Alg} _{/ {A}} \to {\mathsf{Set}}\).

  • Think of spaces as functors \(X\in [\mathsf{CRing}, {\mathsf{Set}}]\), then \(\operatorname{Spec}R \coloneqq\mathsf{CRing}(R, {-})\), so \(R\) corepresents \(\operatorname{Spec}R\) in \(\mathsf{CRing}\).

  • Can represent \(R \left[ { \scriptstyle { {f}^{-1}} } \right] = R[t]/\left\langle{tf-1}\right\rangle\).

  • Standard open subfunctors: \(\operatorname{Spec}R \left[ { \scriptstyle { {f_i}^{-1}} } \right] \to \operatorname{Spec}R\). These form an open cover if \(\left\langle{f_i}\right\rangle = \left\langle{1}\right\rangle\).

  • If \(k\in \mathsf{Field}\), there is an equivalence \(\operatorname{Spec}R(k) \cong Z_f(k)\), the zeros of \(f\) in \(k\). Then \(\operatorname{Spec}R \left[ { \scriptstyle { {h}^{-1}} } \right](k) = Z_f(k)\setminus Z_h(k)\) for \(R = {\mathbf{Z}}[x_1,\cdots, x_n]/\left\langle{f}\right\rangle\).

  • Analog of 2-dimensional \({\mathbf{C}}{\hbox{-}}\)module over a ringer ring: the free \(R{\hbox{-}}\)module \(R{ {}^{ \scriptscriptstyle\times^{2} } }\) of rank 2.

  • \({\mathbf{P}}^1_{{\mathbf{Z}}}: \mathsf{CRing}\to{\mathsf{Set}}\) is the functor sending \(R\) to the set of direct summands \(M \leq R{ {}^{ \scriptscriptstyle\times^{2} } }\) for which there’s an open covering corresponding to \(\left\{{h_i}\right\}\) where \(M \left[ { \scriptstyle { {h_i}^{-1}} } \right] = M\otimes_R R \left[ { \scriptstyle { {h_i}^{-1}} } \right]\) is a free \(R{\hbox{-}}\)module of rank 1 for all \(i\).

    • This recovers ${\mathbf{P}}^1_{{\mathbf{Z}}}({\mathbf{C}}) = {\mathbf{P}}^1_{/ {{\mathbf{C}}}} $ classically, since sub-vector spaces are direct summands.
    • \({\mathbf{P}}^1_{\mathbf{Z}}({\mathbf{Z}}[t])\) induces a continuously varying family of 1-dimensional subspaces of \({\mathbf{C}}^2\)? Somehow, even though \({\mathbf{C}}\) isn’t in the definition..
  • For $S\in {}{R} \mathsf{Alg} $, we have \(\alpha: R\to S\) and for $N\in {}{S}{\mathsf{Mod}} $ we can forget the module structure along this map by defining \begin{align*} R\times N &\to N \\ (r, n) &\mapsto \alpha(r) \cdot n .\end{align*} This induces a restriction functor $\operatorname{res}{\alpha}: {}{S}{\mathsf{Mod}} \to {}_{R}{\mathsf{Mod}} $.

  • Conversely we can tensor \(R{\hbox{-}}\)modules up to \(S{\hbox{-}}\)modules to get a functor \(S\otimes_R({-})\), where the interesting bit is \(s\otimes(rm) \coloneqq\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m\).

  • This yields an adjunction: \begin{align*} \adjunction{({-})\otimes_R S}{\operatorname{res}_{\alpha}}{ {}_{R}{\mathsf{Mod}} }{ {}_{S}{\mathsf{Mod}} } .\end{align*}

  • Any reasonable property of modules should be preserved by base change!

  • Descent for modules: when does \(M\otimes_R S\) having property \(P\) as an \(S{\hbox{-}}\)module descend to \(M\) having property \(P\) has an \(R{\hbox{-}}\)module?

  • Left adjoints are right exact (LARE). In particular, base change is right exact, but not always left exact: take \(\alpha: {\mathbf{Z}}\to {\mathbf{Z}}/2\), take the SES \(0 \to {\mathbf{Z}}\xrightarrow{2} {\mathbf{Z}}\to {\mathbf{Z}}/2\to 0\), and tensor with \({\mathbf{Z}}/2\). So an \(R{\hbox{-}}\)algebra \(S\) is flat precisely when the base change \(S\otimes_R({-})\) is exact.

    • Free implies flat, and every module over a field is free.
  • \(S\) is faithfully flat when \(S\otimes_R M = 0\implies M=0\). Allows checking things after base-changing to \(S\): - Exactness of any sequence, so in particular injectivity/surjectivity - Finite generation (over \(R\) vs \(S\)) - Projectivity, - Flatness - If \(R\to S\) is faithfully flat and \(R\to T\) is an arbitrary ring morphism, the co-base change \(T\to S\otimes_R T\) is faithfully flat.

  • General idea: \(R{\hbox{-}}\)modules \(M\) can be specified by \(S\otimes_R M\) along with Unsorted/descent.

  • faithfully flat descent : there is an equivalence of categories \({}_{R}{\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),

    • descent: pairs \((M, \phi)\) where $M\in {}_{S}{\mathsf{Mod}} $ and \(\phi: M\otimes_RS { \, \xrightarrow{\sim}\, }S\otimes_R M\) is a twist isomorphism.
  • Given \(F\in [\mathsf{A}, \mathsf{B}]\) and \(G\in [\mathsf{A}, \mathsf{C}]\), the left Kan extension of \(G\) along \(F\) is a functor \(L\in [B, C]\) and a sufficiently universal natural transformation \(\alpha\in [G, LF]\).

    • Example: \(G: {}_{A}{\mathsf{Mod}} \to \mathsf{A}\) into some abelian category. Here simplicial resolution by projective objects for projective resolutions, and \({\mathbb{L}}G\) is the left Kan extension of \(G:\mathsf{C} \to K^-(\mathsf{A})\) along the inclusion \(\mathsf{C} \to K^-(\mathsf{A})\), where \(\mathsf{C} \leq K^-(\mathsf{A})\) are complexes of projective modules. So this replaces cofibrant replacement.
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