---
abstract: 'Recently N. Nitsure showed that for a coherent sheaf ${{\mathcal F}}$ on a noetherian scheme the automorphism functor ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable if and only if ${{\mathcal F}}$ is locally free. Here we remove the noetherian hypothesis and show that the same result holds for the endomorphism functor ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ even if one asks for representability by an algebraic space.'
author:
- Niko Naumann
title: 'Representability of ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$'
---

[MSC2000: 14A25]{}

Statement of results {#s1}
====================

 {#s11}

Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. We are interested in the representability of the following two functors on the category of $X$-schemes: $$\begin{aligned}
{\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{Aut}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}}) \\
{\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{End}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}})\end{aligned}$$ for $f : X' \to X$ an $X$-scheme.

The result is as follows:

\[thm11\] Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. Then the following are equivalent:

1.  ${{\mathcal F}}$ is locally free.

2.  ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme.

3.  ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme.

If $X$ is locally noetherian, these conditions are also equivalent to the following:

1.  ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space.

2.  ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space.

 {#s12}

The equivalence of 1) and 2) in theorem \[thm11\] in case $X$ is noetherian is the main result of [@N]. Our proof follows the ideas of [*loc.cit.*]{} closely. The main steps are contained in the following two lemmas:

\[t1\] Let $A$ be a local ring and $M$ a finitely presented $A$-module which is [*not*]{} free. Then there is a local homomorphism $A \to B$ such that $$M \otimes_A B \cong B^n \oplus (B/b)^m \; ,$$ for some $0 \neq b \in B , b^2 = 0$ and $m \ge 1,n \ge 0$.\
If $A$ is noetherian, $B$ can be chosen to be artin.

We observe that in the last statement of the lemma the noetherian hypothesis is indispensable: let $(B , {{\mathfrak{m}}})$ be a local ring such that there is $0 \neq b \in \bigcap_{n \ge 1} {{\mathfrak{m}}}^n$. Clearly $(b^2) \subsetneq (b)$, so after dividing out $(b^2)$ one gets a ring $B$ as in the lemma but for any local homomorphism $f : B \to C$ with $C$ [*noetherian*]{} one clearly has $f (b) = 0$.

\[t2\] Let $S$ be a scheme and $S_0 \subseteq S$ a closed subscheme defined by a nilpotent ideal sheaf. Assume $X$ is a flat $S$-scheme and $f : X \to Y$ is an $S$-morphism such that $f \times {\mathrm{id}}_{S_0}$ is an isomorphism. Then $f$ is an isomorphism.

 {#s13}

In order to treat the representability of ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ we will use the following observation:

\[t3\] Under the assumptions of 1.1 the obvious natural transformation of (set-valued) functors ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} \to {\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is relatively representable by an open immersion.

For completeness we also include a proof of the next lemma which is essentially lemma 5 of [@N] and shows the relative representability of a “parabolic” sub-group functor:\
Let $X$ be a scheme and $$\label{eq:1}
  0 \longrightarrow {{\mathcal F}}' \longrightarrow {{\mathcal F}}\longrightarrow {{\mathcal F}}'' \longrightarrow 0$$ a short exact sequence of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free. For any morphism $f : Y \to X$, the sequence $f^* ((\ref{eq:1}))$ is exact because ${{\mathcal F}}''$ is in particular ${{\mathcal O}}_X$-flat and it makes sense to consider $$P (Y) := \{ \alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) {\, | \,}\alpha (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \} \subseteq {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (Y) \; .$$

\[t4\] In the above situation, the natural transformation $P \hookrightarrow {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is relatively representable by a closed immersion.

For basic facts about (relative) representability we refer to [@BLR], 7.6.

Proofs
======

 {#s21}

In this subsection we dispense with the easy implications of theorem \[thm11\], the assumptions and notations of which we now assume:\
As ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ are clearly Zariski sheaves the problem of representing them is Zariski local on $X$, i.e. we can assume that $X$ is affine and ${{\mathcal F}}$ corresponds to a free module of finite rank. In this case, representability of both ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is obvious; we have proved the implications 1) $\Rightarrow$ 2) and 1) $\Rightarrow$ 2’). Finally, the implications 2) $\Rightarrow$ 3) and 2’) $\Rightarrow$ 3’) are trivial.

 {#s22}

[**Proof of lemma \[t1\]:**]{} Let $(A , {{\mathfrak{m}}})$ be a local ring and $M$ a finitely presented $A$-module which is not free. We will find the required local homomorphism $A \to B$ as a suitable quotient of $A$:\
Let $$\label{eq:2}
  A^m \xrightarrow{\alpha} A^n \xrightarrow{\beta} M \rightarrow 0$$ be a minimal presentation of $M$, i.e. $n = \dim_k (M / {{\mathfrak{m}}}M)$ where $k := A / {{\mathfrak{m}}}$ is the residue field of $A$. Then $M$ is free if and only if $\alpha = 0$: clearly $\alpha = 0$ is sufficient for freeness of $M$ and conversely, if $M$ is free, it is necessarily so of rank $n$, hence $\beta$ is a surjective endomorphism of $A^n$ which must be an isomorphism by a standard application of Nakayama’s lemma, c.f. [@M], thm. 2.4., hence $\alpha = 0$.\
For any $J \subseteq {{\mathfrak{m}}}$, (\[eq:2\]) $\otimes_A A / J$ is a minimal presentation of the $A / J$-module $M / JM$. If we denote by $I \subseteq A$ the ideal generated by the coefficients of any matrix representation of $\alpha$ and note that the minimality of (\[eq:2\]) implies $I\subseteq {{\mathfrak{m}}}$ we find that $M / JM$ is $A / J$-free if and only if $\alpha\otimes id_{A/J}=0$ if and only if $I \subseteq J$. As $M$ is not $A$-free we have $I \neq 0$ and as $I$ is finitely generated we get ${{\mathfrak{m}}}I \subsetneq I$, again by Nakayama’s lemma. By Zorn’s lemma, using again that $I$ is finitely generated, there is an ideal $J$ with ${{\mathfrak{m}}}I \subseteq J \subsetneq I$ and which is maximal subject to these conditions (indeed, any ascending chain of such ideals admits its union as an upper bound because I is finitely generated).

We claim that $B := A / J$ is as required:\
By the maximality of $J$ the ideal ${\overline{I}}:= I / J$ is non-zero principal: ${\overline{I}}= (b) , 0 \neq b \in B$ and we neccessarily have $b^2 = 0$: if not, we would have $b \in (b^2)$, i.e. $b = xb^2$ or $b (1-xb) = 0$ for some $x \in B$. As $b \in {\overline{{{\mathfrak{m}}}}}:= {{\mathfrak{m}}}/ J$, the maximal ideal of $B ,  1-xb$ was a unit of $B$, so we would have $b = 0$.

We now show that $M\otimes_{A} B$ has the desired structure: any coefficient $\alpha_{ij}$ of a matrix representation of $\alpha \otimes {\mathrm{id}}_B$ is of the form $\alpha_{ij} = bu_{ij} , u_{ij} \in B$. As by construction ${\overline{{{\mathfrak{m}}}}}b = 0$ we see that if $\alpha_{ij} \neq 0$, then $u_{ij} \in B^*$. We get a matrix equation $(\alpha_{ij}) = b (u_{ij})$ and $(u_{ij})$ can be chosen with $u_{ij} = 0$ or $u_{ij} \in B^*$, all $i,j$. Then the usual Gau[ß]{}-algorithm can be applied to $(u_{ij})$, showing that indeed $M \otimes_A B \cong B^n \oplus (B / b)^m$ for some $m,n\geq 0$. As, by construction, $M \otimes_A B$ is not $B$-free, we finally see that $m \ge 1$.

If $A$ is noetherian we can start the construction of $B$ by first dividing out a suitable high power of ${{\mathfrak{m}}}$: Indeed, if $M / {{\mathfrak{m}}}^n M$ was free for all $n \ge 1$ we would have $I \subseteq \bigcap_{n \ge 1} {{\mathfrak{m}}}^n = (0)$. Then the ring $B$ we obtain in the above construction is noetherian local with ${\overline{{{\mathfrak{m}}}}}$ nilpotent, hence zero-dimensional, i.e. $B$ is artin local.

[**Proof of lemma \[t2\]:**]{} We can assume that the ideal sheaf ${{\mathcal I}}$ of $S_0 \subseteq S$ satisfies ${{\mathcal I}}^2 = 0$. Our assertion is local on $S, X$ and $Y$ and thus reduces to the following:\
Given a ring $k$ and an ideal $I \subseteq k$ of square zero, if $f : A \to B$ is a morphism of $k$-algebras with $B$ $k$-flat and such that $f \otimes_k {\mathrm{id}}_{k / I}$ is an isomorphism, then $f$ is an isomorphism:\
1) $f$ is surjective: any $b \in B$ can be written $$b = f (a) + \sum_j\alpha_jb_j'\; \mbox{ ,some} \; \alpha_j \in I , b'_j \in B , a \in A \; .$$ Applying this to the $b'_j$ we get (for some $\alpha_{ij} \in I , b''_{ij} \in B , a_j \in A)$: $$b = f (a) + \sum_j\alpha_j(f(a_j)+\sum_{ij}\alpha_{ij}b_{ij}'')=f(a+\sum_j\alpha_j a_j )\; .$$ 2) $f$ is injective: For $K := $ker$ (f)$ the $k$-flatness of $B$ implies $K / IK = 0$ and the same argument as in 1) shows that $K = 0$.

[**Proof of 2) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme and, by contradiction, that ${{\mathcal F}}$ is not locally free. Note that the assumption on representability is stable under base-change $Y\rightarrow X$. So, base-changing to a suitable local ring of $X$, we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ local and ${{\mathcal F}}$ corresponding to a finitely presented $A$-module $M$ which is not free. According to lemma \[t1\] we can assume $M \cong A^n \oplus (A / a)^m$ for some $0 \neq a \in A$ with $a^2 = 0$ and $m \ge 1$. Let $G \to S:= {\mathrm{Spec}\,}(A)$ be the group-scheme representing ${\underline{{\mathrm{Aut}\,}}}_M$ and put $S_0 := {\mathrm{Spec}\,}(A / a)$. The sub-functor $G' \hookrightarrow G$ of automorphisms preserving (base-changes of) the direct summand $(A / a)^m$ is represented by a closed sub-group scheme (still to be denoted $G'$) according to lemma \[t4\].

Let $P \subseteq {\mathrm{Gl}\,}_{n+m , S}$ denote the standard parabolic sub-group of automorphisms preserving the rank $m$ direct summand. $P$ is flat over $S$, as can be seen over ${\mathrm{Spec}\,}({{\mathbb{Z}}})$. There is a morphism of $S$-groups $f : P \to G'$ which on points is given by sending $\left( 
  \begin{smallmatrix}
    \alpha & 0 \\ \beta & \gamma
  \end{smallmatrix} \right)$ to $\left( 
  \begin{smallmatrix}
    \alpha & 0 \\ \pi \beta & \overline{\gamma}
  \end{smallmatrix} \right)$, where $\alpha \in {\mathrm{Aut}\,}_A (A^n) , \gamma \in {\mathrm{Aut}\,}_A (A^m) , \beta : A^n \to A^m$ and $\overline{\gamma} \in {\mathrm{Aut}\,}_{A/a} ((A / a)^m)$ denotes the reduction of $\gamma$ and $\pi : A^m \to (A / a)^m$ is the natural map. This “point-wise” description of $f$ is immediately checked to be functorial and a homomorphism and hence does indeed define a morphism of $S$-groups. Obviously, $f \times {\mathrm{id}}_{S_0}$ is an isomorphism, hence so is $f$ by lemma \[t2\]. This is however a contradiction, because $f (S) : P (S) \to G' (S)$ is not injective, as $f (S) ({\mathrm{id}}_{A^n} \oplus (1-a) {\mathrm{id}}_{A^m}) = 1$ and $a \neq 0 , m \ge 1$.

 {#s23}

[**Proof of lemma \[t3\]:**]{} Given a scheme $X$, a quasi-coherent ${{\mathcal O}}_X$-module ${{\mathcal F}}$ of finite presentation and some $\varphi \in {\mathrm{End}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$ we have to show that there is an open sub-scheme $X_0 \subseteq X$ such for all $f : Y \to X , f^* (\varphi) \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) \subseteq {\mathrm{End}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}})$ if and only if $f$ factors through $X_0$. Consider ${{\cal G}}:= {\mathrm{coker}\,}(\varphi)$ and the exact sequence of ${{\mathcal O}}_X$-modules $$\label{eq:3}
      {{\mathcal F}}\xrightarrow{\varphi} {{\mathcal F}}\xrightarrow{} {{\cal G}}\xrightarrow{} 0 \; .$$ We claim that $f^* (\varphi)$ is an automorphism if and only if $f^* ({{\cal G}}) = 0$: as $f^* ((\ref{eq:3}))$ is exact, necessity is obvious. If, conversely, $f^* ({{\cal G}}) = 0$ then for any $y \in Y$ $f^* (\varphi)_y$ is a surjective endomorphism of the finitely generated ${{\mathcal O}}_{Y,y}$-module ${{\mathcal F}}_y$, hence is an isomorphism, hence so is $f^* (\varphi)$.

So the sought for $X_0 \subseteq X$ is the complement of the support of ${{\cal G}}$ which is open, because ${{\cal G}}$ is finitely presented.

[**Proof of lemma \[t4\]:**]{} Given a scheme $X$ and a short exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free and some $\alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$, we have to show the representability by a closed sub-scheme of $X$ of the following functor on $X$-schemes: $$F (Y \xrightarrow{f} X) := \left\{ 
  \begin{array}{ccl}
* & , & f^* (\alpha) (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \\
\emptyset &, & \mbox{otherwise} \; .
  \end{array} \right.$$ Clearly, $F$ is a Zariski sheaf, so the problem is local on $X$, i.e. we can assume that $X = {\mathrm{Spec}\,}(A)$ is affine, ${{\mathcal F}}''$ corresponds to some $A^n$, ${{\mathcal F}}'$ corresponds to some $A$-module $M$ for which there is a presentation $A^a \to A^b \to M \to 0$ and ${{\mathcal F}}$ corresponds to some $A$-module $N$. The exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ then becomes an exact sequence $0 \to M \xrightarrow{\iota} N \xrightarrow{\pi} A^n \to 0$ of $A$-modules and we are given some $\alpha \in {\mathrm{Aut}\,}_A (N)$. Consider $\nu := \pi \alpha \iota$. As all the above sequences are exact after [*any*]{} base-change, we have $F (Y \xrightarrow{f} X) \neq \emptyset \iff f^* (\nu) = 0$.

We have a diagram (defining $\psi$): $$\xymatrix{
A^a \ar[r] & A^b \ar[r] \ar[dr]^{\psi} & M \ar[r] \ar[d]^{\nu} & 0 \\
           &                           & A^n                   &
}$$ which is exact after any base-change, hence $f^* (\nu) = 0 \iff f^* (\psi) = 0$, for any $f : Y \to X$. So the closed sub-scheme of $X$ we are looking for is the one defined by the ideal of $A$ generated by the coefficients of any matrix representation of $\psi$.

[**Proof of 2’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. Then so is ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ by lemma \[t3\], hence ${{\mathcal F}}$ is locally free by what has been shown in \[s22\].

 {#s24}

\
[**Proof of 3) $\Rightarrow$ 1) and 3’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of 1.1 we assume that $X$ is locally noetherian and that either 3) or 3’) holds as well as, by contradiction, that ${{\mathcal F}}$ is not locally free. By lemma \[t3\] we know in either case that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. Using the last assertion of lemma \[t1\] we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ artin local. Then ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme according to [@K], p. 25, 7) contradicting what we proved in \[s22\].

[**Acknowledgements.**]{} I would like to thank H. Frommer and M. Volkov for interesting discussions and G. Weckermann for excellent type-setting.

[9999]{} S. Bosch, W. Lütkebohmert, M. Raynauld, Néron Models, Ergebnisse der Mathematik, 3. Folge, Band 21, Springer, Heidelberg 1990. D. Knutson, Algebraic Spaces, Springer LNM 203. H. Matsumura, Commutative ring theory, Cambridge studies in advanced mathematics 8, 1997. N. Nitsure, Representability of ${\mathrm{Gl}\,}_E$, arXiv:math.AG/0204047.

[Mathematisches Institut der WWU Münster\
Einsteinstr. 62\
48149 Münster\
Germany\
e-mail: naumannn@uni-muenster.de]{}
