Lemma 21.29.2. With \epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '}) as above. Let A be a set and for \alpha \in A let
\xymatrix{ E_\alpha \ar[d] \ar[r] & Y_\alpha \ar[d] \\ Z_\alpha \ar[r] & X_\alpha }
be a commutative diagram in the category \mathcal{C}. Assume that
a \tau '-sheaf \mathcal{F}' is a \tau -sheaf if \mathcal{F}'(X_\alpha ) = \mathcal{F}'(Z_\alpha ) \times _{\mathcal{F}'(E_\alpha )} \mathcal{F}'(Y_\alpha ) for all \alpha ,
for K' in D(\mathcal{O}_{\tau '}) in the essential image of R\epsilon _* the maps c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha } of Lemma 21.26.1 are isomorphisms for all \alpha .
Then K' \in D^+(\mathcal{O}_{\tau '}) is in the essential image of R\epsilon _* if and only if the maps c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha } are isomorphisms for all \alpha .
Proof.
The “only if” direction is implied by assumption (2). On the other hand, if K' has a unique nonzero cohomology sheaf, then the “if” direction follows from assumption (1). In general we will use an induction argument to prove the “if” direction. Let us say an object K' of D^+(\mathcal{O}_{\tau '}) satisfies (P) if the maps c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha } are isomorphisms for all \alpha \in A.
Namely, let K' be an object of D^+(\mathcal{O}_{\tau '}) satisfying (P). Choose a distinguished triangle
K' \to R\epsilon _*\epsilon ^{-1}K' \to M' \to K'[1]
in D^+(\mathcal{O}_{\tau '}) where the first arrow is the adjuntion map. By (2) and Lemma 21.26.2 we see that M' has (P). On the other hand, applying \epsilon ^{-1} and using that \epsilon ^{-1}R\epsilon _* = \text{id} by Section 21.27 we find that \epsilon ^{-1}M' = 0. In the next paragraph we will show M' = 0 which finishes the proof.
Let K' be an object of D^+(\mathcal{O}_{\tau '}) satisfying (P) with \epsilon ^{-1}K' = 0. We will show K' = 0. Namely, given n \in \mathbf{Z} such that H^ i(K') = 0 for i < n we will show that H^ n(K') = 0. For \alpha \in A we have a distinguished triangle
R\Gamma _{\tau '}(X_\alpha , K') \to R\Gamma _{\tau '}(Z_\alpha , K') \oplus R\Gamma _{\tau '}(Y_\alpha , K') \to R\Gamma _{\tau '}(E_\alpha , K') \to R\Gamma _{\tau '}(X_\alpha , K')[1]
by Lemma 21.26.1. Taking cohomology in degree n and using the assumed vanishing of cohomology sheaves of K' we obtain an exact sequence
0 \to H^ n_{\tau '}(X_\alpha , K') \to H^ n_{\tau '}(Z_\alpha , K') \oplus H^ n_{\tau '}(Y_\alpha , K') \to H^ n_{\tau '}(E_\alpha , K')
which is the same as the exact sequence
0 \to \Gamma (X_\alpha , H^ n(K')) \to \Gamma (Z_\alpha , H^ n(K')) \oplus \Gamma (Y_\alpha , H^ n(K')) \to \Gamma (E_\alpha , H^ n(K'))
We conclude that H^ n(K') is a a \tau -sheaf by assumption (1). However, since the \tau -sheafification \epsilon ^{-1}H^ n(K') = H^ n(\epsilon ^{-1}K') is 0 as \epsilon ^{-1}K' = 0 we conclude that H^ n(K') = 0 as desired.
\square
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