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Is there a closed immersion $i:Z\to X$ and a finite morphism $f:Z\to Y$ of schemes such that the pushout of the span $Y\stackrel{f}{\leftarrow} Z\stackrel{i}{\rightarrow} X$ does not exist in the category of schemes?

There are several related questions regarding pushouts (and colimits) of schemes on this network here, here, here, here, and there. Some of the answers, comments, and references in these posts provide counterexamples to the existence of the pushout when $f$ is not proper (if I did not miss something).

This proposition asserts that the pushout in question exists when the preimage of every point in $Y$ is contained in the preimage of an open affine subscheme of $X$. However, it does not prove the necessity of this condition. The closest statement, I am aware of, to answering the question is Théorème 7.1 in Conducteur, descente et pincement. It gives a necessary and sufficient condition for the pushout in the category of locally ringed spaces to be a scheme, which is equivalent to the aforementioned condition. But, that is not the same as saying the pushout does not exist in the category of schemes when that condition is violated.

Ultimately, I need to understand if the pushout exists for every proper morphism $f$. So, if you have an answer for the question when $f$ is proper (not necessarily finite), it would be also useful.

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    $\begingroup$ If you work in the category of finite type $k$-schemes, my recollection is that there exists a categorical cofiber coproduct $W$ in some case whose underlying locally ringed space does not equal the categorical cofiber coproduct in the category of locally ringed spaces, yet the morphism $Y\to W$ has positive dimensional fibers. I think if you consider a toric version of Hironaka's example (with a $\mathbb{Z}/n\mathbb{Z}$-action on $Z$ whose quotient equals $Y$), then the categorical cofiber coproduct will factor through a contraction of $Z$ in $X$. $\endgroup$ – Jason Starr Aug 11 '17 at 12:22
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    $\begingroup$ Certainly what I am describing happens if you allow $f$ to be proper. For instance, let $X$ be $\mathbb{P}^2_k$, let $Z$ be a line in $X$, let $f$ be the constant morphism to $\text{Spec}(k)$. The constant morphism from $X$ to $\text{Spec}(k)$ is a cofiber coproduct in the category of schemes. $\endgroup$ – Jason Starr Aug 11 '17 at 12:27
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    $\begingroup$ The Stacks project actually refers to the paper of Ferrand. Also, there is a related paper of Temkin and Tyomkin see arxiv:1305.6014 $\endgroup$ – Count Dracula Aug 11 '17 at 23:18
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    $\begingroup$ @CountDracula: for the convenience of readers, a functioning link is arxiv.org/pdf/1305.6014.pdf $\endgroup$ – nfdc23 Aug 12 '17 at 0:18
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In case you weren't aware, such pushouts with finite $f$ always exist in the wider context of algebraic spaces (with the input objects also permitted to be algebraic spaces) under some mild "finiteness" hypotheses (quasi-compact diagonals, being locally of finite type over some fixed reasonable noetherian base scheme, etc.). More importantly for actually using these things (beyond categorical garbage), their underlying topology is reasonably related back to the given input data and they have good behavior relative to flat base change. This all goes back to Artin (and even further back to Hironaka in some cases when the input objects are schemes, according to the Introduction to Donald Knutson's book Algebraic Spaces); a reference for a write-up of a proof of Artin's result is Theorem 2.2.2 in https://arxiv.org/pdf/0910.5008.pdf

You don't mention your reason for wanting various pushouts to exist as schemes (or if you want such to have some properties such as separatedness, etc.), but would existence as an algebraic space be sufficient for you? Artin's paper Algebraization of formal moduli II in Annals of Math vol. 91 (1970) gives some criteria in section 6 for existence of a pushout with proper $f$ (i.e., "contraction" of a closed subspace along a proper map) as an algebraic space, again with useful topological properties. This was extended in some ways by Joseph Mazur: see his paper Conditions for the Existence of Contractions in the Category of Algebraic Spaces in Transactions of the AMS, vol. 209 (1975) (the paper [10] mentioned in the middle of page 2 of that paper, which from its title must be a version of his PhD work under Artin, does not seem to have been published).

Coming back to schemes, here is an example in dimension 2 where a scheme pushout along a proper $f$ doesn't exist if we make a mild reasonable hypothesis on the pushout, without which it would be totally useless anyway (so avoiding the situation as in Jason Starr's example which exists as a scheme but that I am sure he would agree is so topologically bad as to be basically useless).

In the Introduction to Knutson's book mentioned above (see pp. 21-22) there is an example of a smooth projective surface $X$ over a big-enough algebraically closed field $k$ (he takes $k=\mathbf{C}$, but any field not algebraic over a finite field works there) and a smooth connected curve $Z \subset X$ (in fact an elliptic curve) for which one can make an integral proper algebraic space $P$ of dimension 2 and a (proper) surjection $q:X \rightarrow P$ that carries $Z$ onto a rational point $\xi \in P(k)$ such that $q^{-1}(\xi) = Z$ as closed subsets of $X$, $q$ is an isomorphism over $P - \{\xi\}$, and $P$ is not a scheme. (This construction rests on Cor. 6.10 in Artin's paper mentioned above.) In particular, $P$ is smooth away from $\xi$ (though it has to be non-smooth at $\xi$ because smooth proper algebraic spaces of dimension 2 are necessarily schemes; see the end of section 4 of Chapter V of Knutson's book).

By working over an etale scheme neighborhood of $(P, \xi)$ one sees that $f$ factors uniquely through the $P$-finite normalization of $P$, so we may replace $P$ by its normalization to arrange that $P$ is normal (if it wasn't already normal) without affecting any of the properties we have arranged. We have the closed immersion $i:Z \hookrightarrow X$ and proper surjection $f:Z \rightarrow {\rm{Spec}}(k) =: Y$ induced by $q:X \rightarrow P$. Let's see that the non-scheme algebraic space $P$ is a pushout of $Y \leftarrow Z \hookrightarrow X$ in the category of algebraic spaces and use this to deduce that the pushout does not exist in the category of schemes if we want the pushout to be at all reasonable.

By normality of $P$, one sees that $O_P = f_*(O_X)$. Thus, using that $f$ is a topological quotient map, if $P$ were a scheme then it would be the pushout in the category of locally ringed spaces (in particular, in the category of schemes). This reasoning does not use properness of $P$ over $k$, so it can be applied to the pullback of $f:X \rightarrow P$ over the members of an etale scheme cover of $P$ to deduce (via the reasoning we just did in the scheme setting and via the etale sheaf property of algebraic spaces) that $P$ is necessarily a pushout in the category of algebraic spaces.

Finally, we show there is no "reasonable/useful" pushout in the category of schemes. Suppose that $\pi:X \rightarrow Q$ is a scheme pushout, so $Q$ is canonically a $k$-scheme. By the pushout property of $P$ in the category of algebraic spaces there is a unique $k$-map $h:P \rightarrow Q$ such that $\pi = h \circ q$. Since $P$ is finite type over $k$, clearly $h$ is locally of finite type. We claim that $h$ is separated too. Indeed, in the factorization $$P \rightarrow P \times_Q P \rightarrow P \times_{{\rm{Spec}}(k)} P$$ of the closed immersion $\Delta_{P/k}$, the second map is monic and hence separated, so the first map $\Delta_h$ is also a closed immersion.

We will now make the mild assumption that $\pi$ restricts to an open immersion on $X-Y$ ($Q$ would be useless if that didn't hold). Then $h$ is locally quasi-finite and separated, but an algebraic space that is locally quasi-finite and separated over a scheme is a scheme (see Tag 03XX in the Stacks Project for a proof in the ultimate generality, with no noetherian or other "finiteness" hypotheses on the scheme). Since $P$ is not a scheme, this is a contradiction and so such a $Q$ (with a mild hypothesis on $X \rightarrow Q$) doesn't exist.

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  • $\begingroup$ Thank you very much for your elaborative answer! I might take some time to digest it. For the motivation, the existence of the pushouts in question provides an easy proof for some statements. Yes, I am considering using algebraic spaces (or stacks), if the needed pushouts do not exist in the category of schemes, but that won’t be my first option. $\endgroup$ – user337830 Aug 11 '17 at 15:02
  • $\begingroup$ All schemes I am using are separated and of finite type over the base, and that is the only thing I am looking for in the pushouts. Their mere existence is sufficient to prove some statements, regardless whether they satisfy other mild assumptions or not. $\endgroup$ – user337830 Aug 11 '17 at 15:02

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