Lemma 39.14.4. Let $S$ be a scheme. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid schemes over $S$. 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 39.14.3.

Proof. Since $U \to U'$ is quasi-compact and quasi-separated we see that $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Schemes, Lemma 26.24.1). 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 Schemes, Lemma 30.5.2. 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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