# stacks/stacks-project

Split out arguments in proof smooth base change

I decided to leave in the very long proof because it seems that for some
people this may be easier to read than the version with all the
references to the lemmas building up the proof.

Also added: some more cases where the base change maps are isomorphisms
on sheaves.
aisejohan committed Oct 7, 2018
1 parent 37dae7d commit e241a3c3c2c6086d8d0b243dcc8faaf31c6e7b89
Showing with 1,449 additions and 184 deletions.
1. +1 −1 categories.tex
2. +1 −1 coherent.tex
3. +1,368 −180 etale-cohomology.tex
4. +78 −1 pione.tex
5. +1 −1 sites-cohomology.tex
 @@ -2420,7 +2420,7 @@ \section{Limits and colimits over preordered sets} \item We say $\leq$ is a {\it partial order} if it is a preorder which is antisymmetric (if $i \leq j$ and $j \leq i$, then $i = j$). \item A {\it partially ordered set} is a set endowed with a partial order. \item A {\it directed partially ordered set} is a direct set \item A {\it directed partially ordered set} is a directed set whose ordering is a partial order. \end{enumerate} \end{definition}
 @@ -520,7 +520,7 @@ \section{Quasi-coherence of higher direct images} dimension, because that is defined in terms of vanishing of cohomology of {\it all} $\mathcal{O}_X$-modules. \begin{lemma} \begin{lemma}[Induction Principle] \label{lemma-induction-principle} Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that