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Sam
Sam
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Recently, I have been studying about quasi-coherator and I have some doubts.

  1. I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you please justify whether or not it exists for any arbitrary scheme ?

  2. What is the quasi-coherator in case we consider sheaves on a ringed site ? At least for the simplest case if we consider the site $X_{etale}$ where $X$ is a scheme, what is the exact construction of the quasi-coherator ? My guess is that in this case it is the same construction as in case of schemes, since here the site under consideration is the one obtained from a scheme. However, what if we are working with an arbitrary ringed site ?

  3. Now, suppose we understand the meaning of quasi-coherator on a ringed site, I would like to ask what is the quasi-coherator on a fibred category (say, a gerb) on a site ?

Consider, $$\chi \xrightarrow{p} X_{etale}$$ be a firedfibred category. Now, $\chi$ has a site structure inherited from $X_{etale}$ and also has a structure sheaf $O_\chi = p^{-1}(O_X)$ inherited from that of $X$. Now, what is the quasi-coherator of a sheaf on this fibred category?

Thanks in advance !

Recently, I have been studying about quasi-coherator and I have some doubts.

  1. I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you please justify whether or not it exists for any arbitrary scheme ?

  2. What is the quasi-coherator in case we consider sheaves on a ringed site ? At least for the simplest case if we consider the site $X_{etale}$ where $X$ is a scheme, what is the exact construction of the quasi-coherator ? My guess is that in this case it is the same construction as in case of schemes, since here the site under consideration is the one obtained from a scheme. However, what if we are working with an arbitrary ringed site ?

  3. Now, suppose we understand the meaning of quasi-coherator on a ringed site, I would like to ask what is the quasi-coherator on a fibred category (say, a gerb) on a site ?

Consider, $$\chi \xrightarrow{p} X_{etale}$$ be a fired category. Now, $\chi$ has a site structure inherited from $X_{etale}$ and also has a structure sheaf $O_\chi = p^{-1}(O_X)$ inherited from that of $X$. Now, what is the quasi-coherator of a sheaf on this fibred category?

Thanks in advance !

Recently, I have been studying about quasi-coherator and I have some doubts.

  1. I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you please justify whether or not it exists for any arbitrary scheme ?

  2. What is the quasi-coherator in case we consider sheaves on a ringed site ? At least for the simplest case if we consider the site $X_{etale}$ where $X$ is a scheme, what is the exact construction of the quasi-coherator ? My guess is that in this case it is the same construction as in case of schemes, since here the site under consideration is the one obtained from a scheme. However, what if we are working with an arbitrary ringed site ?

  3. Now, suppose we understand the meaning of quasi-coherator on a ringed site, I would like to ask what is the quasi-coherator on a fibred category (say, a gerb) on a site ?

Consider, $$\chi \xrightarrow{p} X_{etale}$$ be a fibred category. Now, $\chi$ has a site structure inherited from $X_{etale}$ and also has a structure sheaf $O_\chi = p^{-1}(O_X)$ inherited from that of $X$. Now, what is the quasi-coherator of a sheaf on this fibred category?

Thanks in advance !

Source Link
Sam
Sam
  • 383
  • 1
  • 6

Quasi-coherator

Recently, I have been studying about quasi-coherator and I have some doubts.

  1. I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you please justify whether or not it exists for any arbitrary scheme ?

  2. What is the quasi-coherator in case we consider sheaves on a ringed site ? At least for the simplest case if we consider the site $X_{etale}$ where $X$ is a scheme, what is the exact construction of the quasi-coherator ? My guess is that in this case it is the same construction as in case of schemes, since here the site under consideration is the one obtained from a scheme. However, what if we are working with an arbitrary ringed site ?

  3. Now, suppose we understand the meaning of quasi-coherator on a ringed site, I would like to ask what is the quasi-coherator on a fibred category (say, a gerb) on a site ?

Consider, $$\chi \xrightarrow{p} X_{etale}$$ be a fired category. Now, $\chi$ has a site structure inherited from $X_{etale}$ and also has a structure sheaf $O_\chi = p^{-1}(O_X)$ inherited from that of $X$. Now, what is the quasi-coherator of a sheaf on this fibred category?

Thanks in advance !