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. $\square$

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