Zoran Skoda cleft extension of a space cover

Idea

Recall the category of noncommutative space covers in the sense of Kontsevich-Rosenberg. The objects are called here simply space covers; an object is given by a ring $U$, a $U$-coring $C$ over $U$ and a structure map $s:U\otimes U\to C$ which is a map of corings and epimorphism of $U$-bimodules. (see also: noncommutative space as a cover).

Let $H$ be a Hopf algebra. We are going to generalize the cleft $H$-extensions of rings to cleft $H$-extensions of space covers.

Definition

Let $(U,C,s)$ be a space cover with structure map $s=s_C:U\otimes U \to C$ and $H$ a Hopf algebra. Then a $H$-cleft extension of $(U,C,s)$ consists of the following data:

• a right $H$-comodule algebra $E$ which is an $H$-cleft extension of $U = E^{co H}$

• an $E$-coring $D$ in the category of right $H$-comodules equipped with a monomorphism $C\hookrightarrow D$ of $U$-bimodules such that the image is precisely the subbimodule of $H$-coinvariants in $D$; we identify $C$ with its image and require that $D$ is generated as an $E$-bimodule with $C$.

• a structure map $s_D:E\otimes E\to D$ such that $(E,D,s_D)$ is a space cover, and $s_D:E\otimes E\to D$ is a map in the monoidal category of right $H$-comodules

• extension of $U$-bimodules $C\hookrightarrow D$ is cleft compatibly with cleavage of $U\hookrightarrow E$ in the following sense. Let $\gamma:H\to E$ be the cleavage of $E$. Then it induces an measuring (in smash case an action) $\triangleright:H\otimes U\to U$ as well known, i.e. $h\triangleright u = \gamma(h_{(1)})u \gamma^{-1}(h_{(2)})$. Now there should be left and right cleavage on $D$ in the sense that there are two “extensions” of this measuring to a semimeasuring of $C$ (we will take these semimeasurings as a part of the structure). This means that for all $u\in U$, $c\in C$ and $k = 1,2$ $\triangleright_k:H\otimes C\to C$ is a map such that

  $$h\triangleright_1 (s_C(u\otimes 1)) = s_C((h\triangleright u)\otimes 1)$$
  $$h\triangleright_2 (s_C(1\otimes u)) = s_C(1\otimes (h\triangleright u))$$
  $$h\triangleright_k (s_C(1\otimes 1)) = \epsilon(h) s_C(1\otimes 1)$$
  $$h\triangleright_1 (u c) = \sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright_1 c)$$
  $$h\triangleright_2 (c u) = \sum (h_{(1)}\triangleright_2 c) (h_{(2)}\triangleright u)$$

  In fact, regarding that $s_C$ is a map of $U$-bimodules, the first two properties follow from the other three. Now $\triangleright_1$ will be called the left and $\triangleright_2$ the right semimeasuring of $H$ on $C$.

Constructing smash product case from cocycles

There is a construction which produces some examples from local data. Here I sketch this construction for the simpler case of a smash product rather than a general case of cleft product.

Namely, we start with $(U,C,s)$, Hopf action $\triangleright$ of $H$ on $U$ with semimeasuring extensions $\triangleright_k$ as above, and a transition cocycle $y:H\to C$ in the following sense:

• $y:H\to C$ is convolution invertible in the sense that there is $y^{-1}:H\to C$ such that $\sum y^{-1}(h_{(1)})\otimes_U y(h_{(2)}) = \sum y(h_{(1)})\otimes_U y^{-1}(h_{(2)}) = \epsilon(h) 1\otimes_U 1\in C\otimes_U C$
• $y$ satisfies the cocycle condition:
  $$\sum y(h_{(1)})\otimes_U y(h_{(2)}) = \Delta_C(y(h))$$

  Notice that this equation on the left involves comultiplication in the Hopf algebra, while on the right hand side the comultiplication of the coring.

• $y$ convolution intertwines the action $\triangleright$ of $H$ on $U$ in the sense that for all $u\in U$ and for all $h\in H$ the two conditions
  $$\Delta_C(\sum (h_{(1)}\triangleright u) y(h_{(2)})) = \sum y(h_{(1)}) \otimes_U (h_{(2)}\triangleright_1 s_C(1\otimes u))$$
  $$\sum (h_{(1)}\triangleright_2 s_C(u\otimes 1))\otimes_U y(h_{(2)}) = \Delta_C(\sum y(h_{(1)}) (h_{(2)}\triangleright u))$$

  hold as equalities in $C\otimes_U C$.

One should have a theorem of the form that that $U, C, s_C, \triangleright,y$ suffice to construct $D, s_D$ etc. such that the data form a cleft $H$-extension of $(U,C,s_C)$ and that every cleft $H$-extension, up to a natural isomorphism comes from a cocycle of that form.

Then there is an idea how to compare different cocycles and finally the Čech cohomology related to such cocycles.

Now the main examples should come from covers by localizations and globalized torsors over them (there is a different theory of those). We will here call $\mathcal{E}$ instead of previous $E$ as there is $E$ for another object in this example. If $E_\lambda$ is a family of localizations of a right $H$-comodule algebra $E$, then we say that the family is coaction compatible if in each localization there is a (automatically unique) extension of the coaction $\rho$ to the localization $\rho_\lambda :E_\lambda\to E_\lambda\otimes H$ as an algebra map, and then it is automatically a comodule algebra. This means that $\rho_\lambda\circ i_\lambda = (i_\lambda\otimes H)\otimes \rho$, where $i_\lambda:E\to E_\lambda$ is the localization map (the component of the unit of the adjunction between the localization functor and its adjoint, taken at the object $E$). Then $U_\lambda = (E_\lambda)^{co H}$ for $\rho_\lambda$ coaction). Define $E_{\lambda\mu} = Q_\mu(E_\lambda) = E_\mu\otimes_E E_\lambda$, $U_{\lambda\mu} = (E_{\lambda\mu})^{co H}$, $U = \prod_\lambda U_\lambda$, $C = \prod_{\lambda\mu}U_{\lambda\mu}$, $\mathcal{E} = \prod_\lambda E_\lambda$ and $D = \prod_{\lambda\mu} E_{\lambda\mu}$. The action $\triangleright_1$ is then estricting to $1_\lambda\otimes u_\mu$ as an extension of action $\triangleright_\lambda$ on $u_\mu$ understood as an element in $U_{\mu\lambda}$ (along canonical map from $U_\mu$), and similarly for the other leg. Thus we have, for $u\in U_\mu$,

$$(h_{(1)}\triangleright_\mu u) y_{\mu\lambda}(h_{(2)}) = y_{\mu\lambda}(h_{(1)})(h_{(2)}\triangleright_\lambda u)$$

as an equality in $U_{\lambda\mu\lambda}\subset C\otimes_U C$,
where $y_{\mu\lambda}$ is the projection of $y$ to $U_{\lambda\mu} = s_C(U_\mu\otimes U_\lambda)$. One denotes the restriction $s_{\mu\lambda} = s_C|U_\lambda\otimes U_\mu$. Notice that $\triangleright_\lambda$ is here extended from $U_\lambda$ to $U_{\mu\lambda}$ (one could also extend to $U_{\lambda\mu}$ in another leg). The cocycle condition is for every $\lambda,\mu,\nu,h$ we have $y_{\lambda\mu}(h_{(1)})y_{\mu\nu}(h_{(2)}) = y_{\lambda\nu}(h)$ in $U_{\nu\mu\lambda}$. The convolution invertibility is for every $\mu$, $y_{\mu\nu}(h_{(1)})y_{\nu\mu}(h_{(2)}) = \epsilon(h)1$ in $U_{\mu}$. Though one should check if this fits with the coring version (it looks like there is $U_{\mu\lambda}$-condition there as well, what is strange, but one can not isolate.

Finally at least in a smash product case one expects some sort of $D\cong C\otimes H$ coming from the data. For example, by that isomorphism of $H$-comodules one should have $(c\otimes h)(u\otimes h')= c(h_{(1)}\triangleright u)\otimes h_{(2)} h'$ but $(u\otimes h)(c\otimes h') = u(h_{(1)}\triangleright_1 c) \otimes h_{(2)}h'$; here $\triangleright_1$ is used and not version 2, probably because it is right smash product and left actions. This looks a bit nonsymmetric.