Let $f:X\rightarrow Y$ be a morphism of schemes.


1. When $PicY\rightarrow PicX$ is an embedding and $f\_{\*}\mathscr{O}\_{X}$ is invertible, it is the structure sheaf of $Y$.
2. In the proof of Zariski's Main Theorem, we have: If $f$ is birational, finite, integral, and $Y$ is normal, then $f\_{\*}\mathscr{O}\_{X}$ is the structure sheaf of $Y$.


My questions are


1) What exactly prevent $f\_{\*}\mathscr{O}\_{X}$ to be a structure sheaf?


2) Is there any necessary and sufficient condition(s) guarantee that $f\_{\*}\mathscr{O}\_{X}$ is a structure sheaf?



When will the pushforward of a structure sheaf still be a structure sheaf?
**Q**: Exactly what information is contained in $f\_\*\mathscr O\_X$? Look at the
definition. For any $U\subseteq Y$ open, $f\_\*\mathscr O\_X(U) = \mathscr O\_X(f^{-1}(U))$ =
regular functions on $f^{-1}(U)$. So the information in $f\_\*\mathscr O\_X$ is related
to the sets in $X$ of form $f^{-1}(U)$.


Cases where $f\_\*\mathscr O\_X$ contains as little information about $X$ as possible.


If $X$ is irreducible and projective and $f$ is constant, e.g. if $Y$ is affine, then
the only non empty set of form $f^{-1}(U)$ in $X$ is $X$ itself. In this case
$f\_\*\mathscr O\_X$ is a skyscraper sheaf with stalk $k$ supported on the image point
of $f$ in $Y$. There is very little information here about $X$, but perhaps we do
see that $f$ is constant and that $X$ is connected. More generally, if $Z$ is a
projective variety, $Y$ is any variety, and $X = Z\times Y$, and $f:Z\times Y\to Y$
is the projection, then $f^{-1}(U) = Z\times U$, so an element of $f\_\*\mathscr
O\_X(U)$, i.e. a regular function on $f^{-1}(U)$, is determined by its restriction to
$\{p\}\times U$ for any $p\in X$, i.e., a regular function on $U$ in $Y$. Thus in
this case we have $f\_\*\mathscr O\_X = \mathscr O\_Y$. Consequently in this case
$f\_\*\mathscr O\_X$ recovers $Y$, but contains no information at all about $X$.


In general, if $f:X\to Y$ is a projective morphism with every fiber connected, and
$Y$ is any normal variety, then $f\_\*\mathscr O\_X = \mathscr O\_Y$, so again
$f\_\*\mathscr O\_X$ contains little information about $X$. Recall that if $X$ is a
projective variety then every morphism out of $X$ is a projective morphism, and more
generally a projective morphism $X\to Y$ is one that factors via an isomorphism of X
with a closed subvariety of $\mathbb P^n\times Y$, followed by the projection
$\mathbb P^n\times Y\to Y$. Suppose that $f:X\to Y$ is any projective morphism.
Then the fibers $f^{-1}(y)$ over points $y \in Y$ are all finite unions of projective
varieties. Therefore for any open set $U\subseteq Y$ containing the point $y$, the
only regular functions in $\mathscr O\_X(f^{-1}(U)) = f\_\*\mathscr O\_X(U)$ are constant
on every connected component of the fiber $f^{-1}(y)$. Thus $f\_\*\mathscr O\_X$ can
contain little information about $X$ and $f$, other than at most the connected
components of the fibers. We shall see below that it contains exactly this
information.


Cases where $f\_\*\mathscr O\_X$ contains as much information about $X$ as possible.


If $f:X\to Y$ is a map of affine varieties, then the global sections of $f\_\*\mathscr
O\_X$ determine $X$ completely, since then $H^0(Y,f\_\*\mathscr O\_X) = H^0(X,\mathscr
O\_X)$, and then $X = \mathrm{Spec}h^0(X,\mathscr O\_X)$, is the unique affine variety
with coordinate ring $H^0(X,\mathscr O\_X)$. The generalization of this case is that
of any affine map $f:X\to Y$, since then $X$ can be recovered by patching together
the analogous construction from $H^0(U,f\_\*\mathscr O\_X)$ for affine open sets
$U\subseteq Y$. Thus $X$ is completely determined by $f\_\*\mathscr O\_X$ for any
affine map $f:X\to Y$, and this is essentially the only case. I.e. in general
$f\_\*\mathscr O\_X$ is always a quasi coherent $\mathscr O\_Y$ algebra, and if we want
it to determine a variety, as opposed to a "scheme", it is reasonable to assume for
all $U\subseteq Y$ affine open, that $f\_\*\mathscr O\_X(U)$ is a finitely generated k
algebra, as well as an $\mathscr O\_Y(U)$ algebra. We may call temporarily such an
$\mathscr O\_Y$ algebra "of finite type". Thus if $f:X\to Y$ is any morphism such that
$f\_\*\mathscr O\_X$ is of finite type, then the patching construction above yields not
necessarily $X$, but a variety $Z$ and an affine map $h:Z\to Y$ which factors via a
map $g:X\to Z$, where $f = h\circ g$, and where $g\_\*(\mathscr O\_X) = \mathscr O\_Z$.
In particular then, we have $f\_\*\mathscr O\_X = (h\circ g)\_\*(\mathscr O\_X) =
h\_\*(g\_\*(\mathscr O\_X))= h\_\*(\mathscr O\_Z)$. So since $h$ is affine, $f\_\*\mathscr O\_X =
h\_\*(\mathscr O\_Z)$ determines not $X$, but $Z$. (Kempf, section 6.5.)


The case of an arbitrary projective morphism.


Now when $f:X\to Y$ is any projective morphism, then $f\_\*\mathscr O\_X$ is a coherent
$\mathscr O\_Y$-module, hence we get a factorization of $f$ as $h\circ g:X\to Z\to Y$,
where $h:Z\to Y$ is affine, and where also $h\_\*(\mathscr O\_Z) = f\_\*\mathscr O\_X$.
Then $h$ is not only an affine map, but since $h\_\*(\mathscr O\_Z)$ is a coherent $\mathscr
O\_Y$-module, $h$ is also a finite map. Moreover $g:X\to Z$ is also projective and since
$g\_\*(\mathscr O\_X) = \mathscr O\_Z$, it can be shown that the fibers of $g$ are connected.
Hence an arbitrary projective map $f$ factors through a projective map g with connected
fibers, followed by a finite map $h$. Thus in this case, the algebra $f\_\*\mathscr O\_X$
determines exactly the finite part $h:Z\to Y$ of $f$, whose points over $y$ are precisely
the connected components of the fiber $f^{-1}(y)$.


One corollary of this is "Zariski's connectedness theorem". If $f:X\to Y$ is projective
and birational, and $Y$ is normal then $f\_\*\mathscr O\_X= \mathscr O\_Y$, and all fibers
of $f$ are connected, since in this case $Z = Y$ in the Stein factorization described
above. If we assume in addition that $f$ is quasi finite, i.e. has finite fibers, then
$f$ is an isomorphism. More generally, if $Y$ is normal and $f:X\to Y$ is any birational,
quasi - finite, morphism, then $f$ is an embedding onto an open subset of $Y$ ("Zariski's
'main theorem' "). More generally still, any quasi finite morphism factors through
an open embedding and a finite morphism.


