The Stacks project

Lemma 77.12.4. Let $B \to S$ as in Section 77.3. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$. Assume that

  1. $f : U \to U'$ is quasi-compact and quasi-separated,

  2. the square

    \[ \xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' } \]

    is cartesian, and

  3. $s'$ and $t'$ are flat.

Then pushforward $f_*$ given by

\[ (\mathcal{F}, \alpha ) \mapsto (f_*\mathcal{F}, f_*\alpha ) \]

defines a functor from the category of quasi-coherent sheaves on $(U, R, s, t, c)$ to the category of quasi-coherent sheaves on $(U', R', s', t', c')$ which is right adjoint to pullback as defined in Lemma 77.12.3.

Proof. Since $U \to U'$ is quasi-compact and quasi-separated we see that $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma 66.11.2). Moreover, since the squares

\[ \vcenter { \xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' } } \quad \text{and}\quad \vcenter { \xymatrix{ R \ar[d]_ s \ar[r]_ f & R' \ar[d]^{s'} \\ U \ar[r]^ f & U' } } \]

are cartesian we find that $(t')^*f_*\mathcal{F} = f_*t^*\mathcal{F}$ and $(s')^*f_*\mathcal{F} = f_*s^*\mathcal{F}$ , see Cohomology of Spaces, Lemma 68.11.2. Thus it makes sense to think of $f_*\alpha $ as a map $(t')^*f_*\mathcal{F} \to (s')^*f_*\mathcal{F}$. A similar argument shows that $f_*\alpha $ satisfies the cocycle condition. The functor is adjoint to the pullback functor since pullback and pushforward on modules on ringed spaces are adjoint. Some details omitted. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 77.12: Quasi-coherent sheaves on groupoids

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GPM. Beware of the difference between the letter 'O' and the digit '0'.