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
f : U \to U' is quasi-compact and quasi-separated,
the square
\xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' }
is cartesian, and
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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