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Suppose that $\pi: X \to S$ is a proper morphism of schemes, and $G \to X$ is a flat affine algebraic group scheme over $X$. Is the push-forward $\pi_* G$ also affine?

I'd be happy to assume more -- say that $\pi$ is smooth, and that $G$ is smooth over $X$, or even a torus...

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    $\begingroup$ No, it is not affine. Already for $X\to S$ a smooth, projective morphism whose geometric fibers are connected curves of genus $g>0$, for $G$ the multiplicative group, the pushforward $\pi_*G$ is an extension of an etale group scheme (basically $\mathbb{Z}$) by the relative Jacobian of $X/S,$ and this is an Abelian scheme over $S.$ $\endgroup$
    – Jason Starr
    Commented Nov 5, 2017 at 21:00
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    $\begingroup$ @JasonStarr: did you mean to refer to ${\rm{R}}^1\pi_{\ast}(\mathbf{G}_m)$? For the class of $\pi$ you mention (or more generally smooth proper with geometrically connected fibers of any dimension), $\pi_{\ast}(\mathbf{G}_{m,X})$ is identified with $\mathbf{G}_{m,S}$, as you know. Likewise, $\pi_{\ast}({\rm{GL}}_{n,X}) = {\rm{GL}}_{n,S}$ for such $\pi$ for any $n > 0$. $\endgroup$
    – nfdc23
    Commented Nov 5, 2017 at 21:13
  • $\begingroup$ right -- to clarify, I didn't mean $R^1\pi_*$, but just $\pi_*$. $\endgroup$
    – Danny
    Commented Nov 5, 2017 at 21:16
  • $\begingroup$ @nfdc23. You are completely correct. I was thinking of $\pi_*(B \mathbb{G}_m)$, the pushforward of the classifying stack, not $\pi_* \mathbb{G}_m,$ the pushforward of the group scheme. Sorry about that! $\endgroup$
    – Jason Starr
    Commented Nov 5, 2017 at 21:19
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    $\begingroup$ @R.vanDobbendeBruyn. Representability follows, for instance, from Lemma 2.3.3 of Max Lieblich's article, "Remarks on the stack of coherent algebras": arxiv.org/pdf/math/0603034.pdf I believe that it can also be deduced from Cor. 7.7.8 of EGA $\textrm{III}_2,$ but I cannot quite make that work . . . $\endgroup$
    – Jason Starr
    Commented Nov 5, 2017 at 22:15

2 Answers 2

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Sorry about the mistake in the comment. First of all, for a proper, locally finitely presented morphism of schemes, $\pi:X\to S,$ with $S$ excellent (e.g., a finite type scheme over a field or over $\text{Spec}\ \mathbb{Z}$), for a finitely presented, flat, affine group scheme $\rho:G\to X$, the set-valued functor $\pi_* G$ is representable by a group scheme algebraic space that is locally finitely presented over $S$. (Ed. Thanks to @nfdc23 for pointing out that Lieblich's result only gives an algebraic space.) This follows, for instance, from Lemma 2.3.3 of the following article.

MR2233719 (2008c:14022)
Lieblich, Max(1-PRIN)
Remarks on the stack of coherent algebras.
Int. Math. Res. Not. 2006, Art. ID 75273, 12 pp.
https://arxiv.org/pdf/math/0603034.pdf

However, it is not true that $\pi_*G$ is always affine for $\pi$ a proper, locally finitely presented morphism and $G/X$ a flat, affine group scheme.
Let $S$ be $\mathbb{A}^2_k,$ the affine plane. The open complement, $j:V\hookrightarrow \mathbb{A}^2_k,$ is a flat morphism, but it is not affine.

Let $f:X\to S$ be the blowing up of $\mathbb{A}^2_k$ at the origin. This is a proper morphism. Denote by $E$ the exceptional divisor of $f$. Denote by $U$ the open complement of $E$. As the complement of a Cartier divisor in a smooth scheme, the open immersion $i:U\hookrightarrow X$ is an affine morphism.

Let $\rho:G\to X$ denote an $X$-scheme with two connected components, one of which maps isomorphically to $X$, $$\rho_e:G_e \xrightarrow{\cong} X,$$ and the second of which maps isomorphically to $U$, $$\rho_{\sigma}:G_{\sigma} \xrightarrow{\cong} U.$$ There is a unique structure of $X$-group scheme on $G$: the identity section is the inverse isomorphism of $\rho_e,$ and the multiplication morphism, $$G_\sigma\times_X G_\sigma \to G_e,$$ is the unique open immersion of $X$-schemes.

The $X$-group scheme $G$ is flat and affine. Yet the pushforward $\pi_*G$ is a disjoint union of a copy of $S$ and a copy of the open immersion $j:V\to S.$ This open immersion is not affine.

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  • $\begingroup$ What I was imagining was that if $G$ sits inside of an affine bundle $A \to X$, where $A$ is the relative $Spec$ of the symmetric algebra of the dual to a vector bundle $V/X$, then the pushforward $\pi_* G$ should sit inside the affine bundle associated to $\pi_* V$. I guess the thing that broke in my intuition was that $\pi_* G$ didn't need to sit inside this as a closed subscheme, but just as something constructible...? $\endgroup$
    – Danny
    Commented Nov 6, 2017 at 13:53
  • $\begingroup$ How does Lemma 2.3.3 of the given link imply the representability? (Note that we don't know if $G$ is a closed $X$-subgroup of some ${\rm{GL}}_n$, say even fppf-locally on $S$.) Olsson's paper on Weil restriction for proper flat maps gives that the pushforward is at least a quasi-separated algebraic space locally of finite type. $\endgroup$
    – nfdc23
    Commented Nov 6, 2017 at 15:43
  • $\begingroup$ @nfdc23. Thank you for the correction. I changed "group scheme" to "group algebraic space." $\endgroup$
    – Jason Starr
    Commented Nov 6, 2017 at 15:50
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If $\pi \colon X \to S$ is proper, flat and of finite presentation and $W$ is an affine $X$-scheme, then $\pi_*W \to S$ is affine and of finite presentation. Unless $W$ is etale over $X$, it is difficult to deduce much about the smoothness of $\pi_*W \to S$.

To see this, you can combine an affine representability result for modules, such as https://stacks.math.columbia.edu/tag/08JY or [EGAIII-2, 7.7.8], with the ideas from Proposition 2.5 of:

Lieblich, Max, Remarks on the stack of coherent algebras, Int. Math. Res. Not. 2006, No. 11, Article ID 75273, 12 p. (2006). ZBL1108.14003.

A precise reference is Theorem 2.3 of

Hall, Jack; Rydh, David, General Hilbert stacks and Quot schemes, Mich. Math. J. 64, No. 2, 335-347 (2015). ZBL1349.14013.1434731927. https://projecteuclid.org/euclid.mmj/1434731927

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  • $\begingroup$ If $W$ is smooth over $S$, and if $\pi$ is proper, finitely presented, and flat of relative dimension $\leq 1,$ then it should be possible to prove smoothness of $\pi_*W\to S$ by infinitesimal deformation theory. For a geometric point $\text{Spec}\ k\to S$ and for a section $\sigma:X_k\to W_k$, the obstruction space is $H^2(X_k,\sigma^*(\Omega_{W/K})^\vee).$ This is zero if the relative dimension is $\leq 1.$ $\endgroup$
    – Jason Starr
    Commented Nov 7, 2017 at 11:27
  • $\begingroup$ The previous comment was wrong. Again I was thinking of the pushforward of the classifying stack $BG$, not the pushforward of $G.$ For the pushforward of $G,$ the obstruction space is $H^1(X_k,\sigma^*(\Omega_{W/k})^\vee).$ If $W$ is a smooth, linear group scheme of $X$, then there exists a locally free sheaf $\mathfrak{w}$ on $X$ such that $\Omega_{W/k}$ is the pullback of $\mathfrak{w}^\vee.$ Thus, the obstruction space is $H^1(X_k,\mathfrak{w}^\vee).$ This is typically nonzero. So you were correct: it seems difficult to deduce smoothness (unless $X/S$ is a family of genus $0$ curves). $\endgroup$
    – Jason Starr
    Commented Nov 7, 2017 at 19:48

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