# The resolution property holds away from codimension three

Siddharth Mathur  https://sites.google.com/view/sidmathur/home Mathematisches
Institut
Heinrich-Heine-Universität
40204 Düsseldorf, Germany.
and  Stefan Schröer<EMAIL_ADDRESS>Mathematisches Institut
Heinrich-Heine-Universität
40204 Düsseldorf, Germany.

###### Abstract.

The purpose of this paper is to verify a conjecture of Gross under mild
hypothesis: all reduced, separated, and excellent schemes have the resolution
property away from a closed subset of codimension $\geq 3$. Our technique uses
formal-local descent and the existence of affine flat neighborhoods to reduce
the problem to constructing certain modules over commutative rings. Once in
the category of modules we exhibit enough locally free sheaves directly,
thereby establishing the resolution property for a specific class of algebraic
spaces. A crucial step is showing it suffices to resolve a single coherent
sheaf.

###### Contents

  1. 1 Introduction
  2. 2 Common affine open neighborhoods
  3. 3 Flat neighborhoods
  4. 4 Triples
  5. 5 Locally free resolutions
  6. 6 Strong tensor generators on algebraic spaces

##  1\. Introduction

A scheme is said to have the _resolution property_ if every quasi-coherent
sheaf of finite type is the quotient of a locally free sheaf of finite rank.
Although quasi-projective and regular schemes have enough locally free
sheaves, very little is understood in the broader category of separated
schemes or algebraic spaces. Already for surfaces this is a difficult problem
(see [29], [6], and [23]) and the case of threefolds seems intractable: it is
not even known if a proper threefold always admits a single nontrivial locally
free sheaf. Although interesting progress has been made in the toric case (see
[26] and [27]), a general solution remains elusive even for this simpler
question.

In his thesis, Gross conjectured that separated schemes enjoy the resolution
property away from a closed subset of codimension $\geq 3$ (see [5, Conj.
2.0.3]). Among other things, this is an important step for establishing the
resolution property for threefolds. We verify this conjecture under mild
hypothesis:

######  Theorem A.

Let $X$ denote a scheme which is excellent, reduced and separated.

  1. (1)

Every point $x\in X$ admits an open neighborhood $U\subset X$ that contains
all points of codimension $\leq 2$ and has the resolution property.

  2. (2)

Every coherent sheaf $\mathcal{F}$ on $X$ is the quotient of a coherent sheaf
$\mathcal{E}$ which is locally free at all points of codimension $\leq 2$.

In fact, our arguments yield stronger results, many of which cover algebraic
spaces, see Theorem, 5.4, Theorem 6.7, Corollary 6.9, and Corollary 6.10 for
more precise statements.

The existence of locally free resolutions on schemes is a well studied
question and perhaps the most striking result in this direction is the
following result due to Totaro and Gross.

######  Theorem.

([33, Thm. 1.1] and [7, Thm. 1.1]) A quasi-compact and quasi-separated
algebraic space $X$ has the resolution property if and only if $X$ can be
written as the quotient of a free action of $\mathrm{GL}_{n}$ on a quasi-
affine scheme.

As such, the resolution property generalizes the notion of quasi-projectivity:
varieties embedded in $\mathbb{P}^{n}$ always arise as quotients of quasi-
affine schemes by a free action of $\operatorname{GL}_{1}$. Moreover, a
defining property of ample line bundles is that they (positively) generate all
the coherent sheaves on a space. Since this property is so commonly used, it
is no surprise that the resolution property has important and fundamental
consequences across different areas of algebraic geometry: Hilbert’s
fourteenth problem over a general base (see [30, Thm. 2] and [32, Thm 3.8]),
the moduli of polarized varieties (see [20, Ques. 43] and [20, Conj. 42]), the
theory of group schemes ([17]), the Brauer group ([23]), and algebraic
K-theory ([33, Sec. 2]). This justifies the importance of the following
question of Totaro. It was originally stated for algebraic stacks but is still
tantalizingly open in the absence of stabilizers:

Totaro’s question (Totaro, [33, Ques. 1]) Let $X$ be a separated scheme (or
algebraic space), does $X$ have the resolution property?

Despite the utility of the resolution property, little progress has been made
beyond algebraic surfaces. Even the case of smooth three dimensional algebraic
spaces over a field is completely wide open and the problem seems to be beyond
the reach of current methods. Before describing our strategy, we recall how to
resolve a coherent sheaf $\mathcal{F}$ on a surface $X$:

  1. (1)

First, we realize $\mathcal{F}$ as a quotient of a coherent sheaf which is
locally free in codimension $\leq 1$ (see [6, Prop. 1.8]). The crucial idea
here is that, using Chow’s lemma, one can find a quasiprojective open subset
containing all height one points. Now we may assume $\mathcal{F}$ is locally
free away from finitely many points $p_{1},...,p_{n}$ and so it suffices to
extend local resolutions at the $p_{i}$ to all of $X$.

  2. (2)

This extension problem is solved by constructing locally free sheaves with
certain positivity properties (see [6, Thm. 4.5]). The idea is to use Chow’s
lemma to produce a projective birational cover and one then descends a
positive bundle by modifying a sum of ample line bundles along a
$1$-dimensional exceptional locus. To conclude, one annihilates the
obstruction to extending these local resolutions by twisting with the
aforementioned locally free sheaves.

The first step automatically fails in our setting because the codimension $2$
points need not admit a quasi-projective open neighborhood. Thus, we have to
find a tractable replacement for such opens. Our solution is to algebraize
formal-local neighborhoods of any fixed finite collection of codimension $2$
points, this process yields affine _flat neighborhoods_ (see Definition 3.1)
and when glued along an affine _open_ subset of $X$, we obtain open subspaces
$U_{i}\subset X$ which contains this finite collection of codimension $2$
points (see Proposition 3.6). Moreover recent results on flat descent allow us
to reduce the problem on the various $U_{i}$ to the following result involving
only modules over commutative rings (see Theorem 4.2).

######  Theorem B.

Suppose we have ring maps $R_{i}\rightarrow R_{12}$ which are flat for $i=1,2$
with the property that the multiplication map
$R_{12}\otimes_{R_{2}}R_{12}\rightarrow R_{12}$ is bijective. Then every
object
$(M_{1},M_{2},\varphi)\in\mathrm{Mod}(R_{1})\times_{\mathrm{Mod}(R_{12})}\mathrm{Mod}(R_{2})$
with finitely generated $M_{1}$ and $M_{2}$ is the quotient of the triple
$(R_{1}^{\oplus n},R_{2}^{\oplus n},A)$ for a suitable $n\geq 0$ and
$A\in\operatorname{GL}_{n}(R_{12})$.

The second step fails in our context for two reasons: first, we work without
properness assumptions so cohomological vanishing arguments need not apply.
The second issue, even if we only consider proper $k$-schemes, is that we do
not know how to descend a desirable locally free sheaf along a higher
dimensional projective birational cover. In particular, recall that it is not
even known how to descend a single non-trivial locally free sheaf for the case
of proper threefolds. We evade these issues entirely by delicately
constructing the open subsets $U_{i}$, then pushing forward local surjections
$\mathcal{V}_{i}\to\mathcal{F}|_{U_{i}}$ and taking their direct sum. To
ensure that the push-forwards retain the essential information of
$\mathcal{F}$, we also choose the opens $U_{i}$ so they contain most of the
non-Cohen-Maucalay locus of $\mathcal{F}$, then depth considerations finish
the argument.

Structure of the paper: In the first section we establish a necessary
refinement of the fact that any finite collection of codimension $\leq 1$
points admits a common affine open neighborhood. Since this is not adequate
for codimension $2$ points, we show how to improve this result further using
affine flat neighborhoods in Section 2. Then we show that these new
neighborhoods have the resolution property using an algebraic argument in
Section 3. In Section 4, we prove that on a very general class of algebraic
spaces, any coherent sheaf can be written as the quotient of a locally free
sheaf on an open set containing all codimension $\leq 2$ points. In the final
section, we explain why it is often enough to resolve a _single_ coherent
sheaf by a locally free sheaf. This is known for schemes but is nontrivial for
algebraic spaces. This yields the main theorem and we conclude by showing one
may remove the reducedness hypothesis when working over a field of positive
characteristic.

Acknowledgments We would like to thank Jack Hall, David Rydh, and Jarod Alper
for many useful discussions. This research in the framework of the research
training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and
Topology, which is funded by the Deutsche Forschungsgemeinschaft. The first
author was also supported by the Swedish Research Council under grant no.
2016-06596 while he was in residence at Institut Mittag-Leffler in Djursholm,
Sweden during the Fall of 2021.

Notation and conventions We briefly recall the definition of an _algebraic
space_ $X$ and some associated notions: points, local rings, and the
topological space associated to $X$. A functor
$X:\textbf{Sch}_{S}^{\text{op}}\to\textbf{Set}$ is called an _algebraic space_
if $X$ is a sheaf on the big fppf site $S_{\text{fppf}}$, the diagonal
$\Delta_{S}:X\to X\times_{S}X$ is representable by schemes, and there is a
surjective étale morphism $U\to X$ where $U$ is a scheme.

A _point_ of $X$ is an equivalence class of morphisms from fields
$x:\operatorname{Spec}K\to X$ where two morphisms are considered equivalent if
there is a third dominating both. The points form a set $|X|$. Moreover, there
exists a unique topology on the sets of points of algebraic spaces so that it
coincides with the underlying topology of $X$ if $X$ is a scheme, morphisms
$X\to Y$ induce continuous morphisms $|X|\to|Y|$, and a étale morphism from a
scheme $U\to X$ induces a continuous open map, see Section [31, Tag 03BT]. If
we suppose that $X$ is quasi-separated, then $|X|$ naturally corresponds to
the set of monomorphisms $x:\operatorname{Spec}k\to X$ and the topology is
well-behaved (see [31, Tags 03IG, 06NI]). For instance, if $X$ is quasi-
separated every irreducible closed subset of $|X|$ has a unique generic point
(see [31, Tag 06NJ]). _All algebraic spaces considered in this paper will be
quasi-separated._

Following [31, Tag 04KG], every point $x\in|X|$ can be represented by a choice
of geometric point $\bar{x}:\operatorname{Spec}\overline{k}\to X$ and taking
the colimit over all étale neighborhoods of $\bar{x}$ yields a strictly
henselian local ring $\mathcal{O}_{X,\bar{x}}$ and a morphism
$\iota_{\bar{x}}:\operatorname{Spec}(\mathcal{O}_{X,\bar{x}})\to X$ which we
call the _strict henselization_ of $x$ at $X$. Note that the isomorphism type
of $\operatorname{Spec}(\mathcal{O}_{X,\bar{x}})$ over $X$ does not depend on
the choice of geometric point $\bar{x}$. Lastly, if $\mathcal{F}$ is a quasi-
coherent sheaf on an algebraic space $X$, and $x\in|X|$ is a point, then we
define the associated stalk to be
$\iota_{\bar{x}}^{*}(\mathcal{F})=\mathcal{F}_{\bar{x}}$ for some geometric
point $\bar{x}$ lying in $x$. We will often write $x\in X$ to refer to a point
$x\in|X|$.

For an integer $n\geq 0$, a quasi-coherent sheaf $\mathcal{F}$ on an algebraic
space $X$ satisfies Serre’s condition $S_{n}$ if for every $x\in X$, we have

$\text{depth}_{\mathcal{O}_{X,\bar{x}}}(\mathcal{F}_{\bar{x}})\geq\text{min}(n,\text{dim}(\text{Supp}(\mathcal{F}_{\bar{x}})))$

and an algebraic space is $S_{n}$ if $\mathcal{O}_{X}$ is. Lastly, when we
write locally free we always mean locally free of finite rank.

##  2\. Common affine open neighborhoods

The goal of this section is to show that certain finite collections of points
admit common affine open neighborhoods. In [31, Tag 0ANN] it is shown that
this holds true for points with codimension $\leq 1$ but we require a stronger
version that allows us to add in a further finite collections of points which
themselves already admit an affine open neighborhood. This was achieved in the
presence of noetherian ground rings in the proof of [6, Thm. 1.5] using a
version of Chow’s lemma. However, since we wish to remove this restriction, we
require a new argument.

The following lemma from elementary topology will be used throughout the
paper.

######  Lemma 2.1.

Let $X$ be a topological space, $U_{1}\subset X$ an open set,
$D=X\smallsetminus U_{1}$ the complementary closed set, and $D^{\prime}\subset
D$ open. Then $U_{1}\cup D^{\prime}\subset X$ is an open set in $X$.

###### Proof.

Write $D^{\prime}=D\cap V$ for some some open set $V\subset X$. We have to
check that $X\smallsetminus(U_{1}\cup D^{\prime})$ is closed. The latter
equals

$(X\smallsetminus U_{1})\cap\left((X\smallsetminus D)\cup(X\smallsetminus
V)\right)=(D\cap(X\smallsetminus D))\cup(D\cap(X\smallsetminus V)),$

which is the intersection of the closed sets $D$ and $X\smallsetminus V$. ∎

Recall that if $X$ is a noetherian scheme then there exists a normalization
morphism $X^{\prime}\to X$ by [31, Tag 035N] which is integral and that
$X^{\prime}$ can be viewed as the normalization of the reduction
$X_{\text{red}}$ by [31, Tag 035O].

######  Theorem 2.2.

Let $X$ be an scheme that is separated and noetherian, and assume that the
normalization map $X^{\prime}\rightarrow X$ remains noetherian. Suppose
$a_{1},\ldots,a_{r}\in X$ are points that admit a common affine open
neighborhood, and $\zeta_{1},\ldots,\zeta_{s}\in X$ be points of codimension
at most one. Then $a_{1},\ldots,a_{r},\zeta_{1},\ldots,\zeta_{s}\in X$ admit a
common affine open neighborhood.

Before giving the proof, we explain some of the necessary background. The
following definition is due to Ferrand (see [4, Thm. 5.4]).

######  Definition 2.3.

A scheme $X$ is said to have the _AF-property_ if every finite set of points
$a_{1},\ldots,a_{r}\in X$ admits a common affine open neighborhood.

######  Example 2.4.

This holds if $X=\text{Proj}(A)$ is the homogeneous spectrum of a graded ring,
but fails in general, as observed by Nagata [25]. The simplest separated
examples seem to be the normal surfaces $X$ constructed in [28], which are
birational to the product of an elliptic curve and the projective line and
have $\operatorname{Sing}(X)=\\{a_{1},a_{2}\\}$. Note that a normal scheme $X$
that is separated and of finite type over a ground field $k$ has the AF-
property if and only if it admits an ample sheaf (see [2, Cor. 2]). On the
other hand, if $X$ has the AF-property and $f:X\rightarrow Y$ is a finite
birational morphism, then $Y$ also has the AF-property (see [4, Thm. 5.4] or
[7, Thm. 3.3] for a generalization), but often fails to carry an ample sheaf
(see e.g. [16, Cor. 1]).

Some preliminary observations are in order: if $U_{\lambda}\subset X$,
$\lambda\in L$ is a filtered inverse system of affine open sets in a scheme
$X$, then the inverse limit $U_{\infty}=\varprojlim U_{\lambda}$ exists as a
scheme, (see [8, Sec. 8.2]). It is affine, and the coordinate ring of
$U_{\infty}$ is the direct limit of the coordinate rings for $U_{\lambda}$.
Moreover, the formation commutes with passing to the underlying topological
space. The canonical map $\iota:U_{\infty}\rightarrow X$ is injective, its
set-theoretical image is the intersection $\bigcap U_{\lambda}$, and the
Zariski topology on the domain coincides with the subspace topology on the
image. Moreover, for all image points $\iota(x)\in X$ the canonical map
$\mathcal{O}_{X,\iota(x)}\rightarrow\mathcal{O}_{U_{\infty},x}$ is bijective.
By abuse of notation, we also write the monomorphism $\iota$ as an inclusion
$U_{\infty}\subset X$, and call it an _affine pro-open set_.

Now let $a_{1},\ldots,a_{r}\in X$ be points admitting a common affine open
neighborhood. Let $U_{\lambda}$, $\lambda\in L$ be the ordered set of all such
neighborhoods. We call $U_{\infty}\subset X$ the _minimal affine pro-
neighborhood_ of the $a_{1},\ldots,a_{r}\in X$. It comprises all points $x\in
X$ that specializes to one of the $a_{i}$. One may compute it explicitly as
follows: Fix an index $\lambda\in L$ and write $U=U_{\lambda}$. Let
$R=\Gamma(U,\mathcal{O}_{X})$ be the coordinate ring, and
$\mathfrak{p}_{i}\subset R$ be the prime ideals corresponding to the $a_{i}\in
U$. With the multiplicative system
$S=R\smallsetminus(\mathfrak{p}_{1}\cup\ldots\cup\mathfrak{p}_{r})$ we get a
canonical identification $U_{\infty}=\operatorname{Spec}(S^{-1}R)$. If $X$ is
integral one furthermore has

$S^{-1}R=R_{\mathfrak{p}_{1}}\cap\ldots\cap
R_{\mathfrak{p}_{r}}=\mathcal{O}_{X,a_{1}}\cap\ldots\cap\mathcal{O}_{X,a_{r}},$

where the intersection takes place in the function field
$F=\mathcal{O}_{X,\eta}$ of our scheme ([3, Ch. 2, Sec. 3, No. 5 Prop. 17]).

###### Proof.

In light of [10, Cor. 4.5.9], it suffices to treat the case that $X$ is
reduced. We proceed by induction on $s\geq 0$. The case $s=0$ is trivial.
Suppose now that $s\geq 1$, and that the assertion is true for $s-1$. Setting
$a_{r+j}=\zeta_{j}$ for $1\leq j\leq s-1$ and replacing $r$ by $r+s-1$, we are
reduced to the case $s=1$. To simplify notation put $\zeta=\zeta_{1}$.

We first treat the case that the scheme $X$ is integral and normal. The
assertion is immediate if $\zeta\in X$ is the generic point. Suppose now that
the local ring $\mathcal{O}_{X,\zeta}$ has dimension one. Choose affine open
sets $U,V\subset X$ that contain the $a_{1},\ldots,a_{r}$ and $\zeta$,
respectively. We are done if $\zeta$ belongs to $U$, so assume that
$\zeta\not\in U$. Without restriction, we may assume $X=U\cup V$. Write
$U=\operatorname{Spec}(R)$, let $\mathfrak{p}_{i}\subset R$ be the prime
ideals corresponding to the $a_{i}\in U$, and consider the minimal affine pro-
neighborhood $U_{\infty}=\operatorname{Spec}(S^{-1}R)$ as above. This lies in
$U$, and thus does not contain $\zeta$.

Let $F=\text{Frac}(R)=\mathcal{O}_{X,\eta}$ be the function field of the
integral scheme $X$. Each point $x\in X$ yields a subring
$\mathcal{O}_{X,x}\subset F$. Recall that $x\in\overline{\\{y\\}}$ if and only
if $\mathcal{O}_{X,x}\subset\mathcal{O}_{X,y}$, according to [10, Cor. 8.5.7].
We now check that similar facts hold for semi-local rings instead of local
rings. Seeking a contradiction, assume $S^{-1}R\subset\mathcal{O}_{X,\zeta}$.
The fiber product
$\operatorname{Spec}(S^{-1}R)\times_{X}\operatorname{Spec}(\mathcal{O}_{X,\zeta})$
is affine, and its coordinate ring $C$ is generated by $S^{-1}R$ and
$\mathcal{O}_{X,\zeta}$, because $X$ is separated. Moreover, the projections
turn $\operatorname{Spec}(C)$ into pro-affine sets inside the factors
$\operatorname{Spec}(S^{-1}R)$ and
$\operatorname{Spec}(\mathcal{O}_{X,\zeta})$, and we may regard
$\operatorname{Spec}(C)\subset X$ as a pro-affine set. It comprises the points
$x\in X$ that specialize to one of the $a_{i}$, and also to $\zeta$. In
particular $\zeta$ does not belong to $\operatorname{Spec}(C)$. Using that $C$
is contained in $F$, and is generated inside $F$ by the subrings $S^{-1}R$ and
$\mathcal{O}_{X,\zeta}$, we infer $C=\mathcal{O}_{X,\zeta}$ and conclude
$\zeta\in\operatorname{Spec}(C)$, contradiction. Summing up,
$S^{-1}R\not\subset\mathcal{O}_{X,\zeta}$. Choose an element $f\in S^{-1}R$
that does not belong to $\mathcal{O}_{X,\zeta}$. After shrinking the affine
open set $U$, we may assume that $f\in R$, and regard it as a local section
$f\in\Gamma(U,\mathcal{O}_{X})$.

Recall that we are assuming that $X$ is normal. By the Serre Criterion, the
local ring $\mathcal{O}_{X,\zeta}$ is a discrete valuation ring. Let
$\operatorname{val}:F^{\times}\rightarrow\mathbf{Z}$ be the corresponding
valuation. Then $\operatorname{val}(f)<0$, thus $f^{-1}\in F^{\times}$ belongs
to the maximal ideal $\mathfrak{m}_{\zeta}$. After shrinking the affine open
set $V$, we may assume $f^{-1}\in\Gamma(V,\mathcal{O}_{X})$. All irreducible
components of $D=X\smallsetminus U$ have codimension one ([9, Cor. 21.12.7]).
Choose an open set $D_{0}\subset D$ that contains $\zeta\in D$ but no other
generic point of $D$. Then $U\cup D_{0}\subset X$ is an open set, according to
Lemma 2.1. Shrinking $V$ further, we may assume that
$D=\overline{\\{\zeta\\}}$. In turn, the effective Cartier-divisor
$V(f^{-1})\subset V$ coincides with the complement $D=X\smallsetminus U$ as
closed sets.

By construction we have $f,f^{-1}\in\Gamma(U\cap V,\mathcal{O}_{X})$. In turn,
this gives a cocycle $f^{-1}\in\Gamma(U\cap V,\mathcal{O}_{X}^{\times})$ and
thus an invertible sheaf $\mathcal{L}$ on $X$. It has
$\mathcal{L}|_{U}=\mathcal{O}_{U}$ and $\mathcal{L}|_{V}=\mathcal{O}_{V}$, and
the global sections are pairs $s=(s_{U},s_{V})$ satisfying $f^{-1}s_{U}=s_{V}$
on the overlap $U\cap V$. In particular, we have the global sections

$s_{0}=(1,f^{-1})\quad\text{and}\quad s_{1}=(f,1).$

Consequently, $\mathcal{L}$ is globally generated, and thus defines a morphism
$h:X\rightarrow P(X,\mathcal{L})$, taking values in the homogeneous spectrum
$P=P(X,\mathcal{L})$ of the graded ring $R(X,\mathcal{L})=\bigoplus_{t\geq
0}H^{0}(X,\mathcal{L}^{\otimes t})$. The quasi-coherent sheaf
$\mathcal{O}_{P}(1)$ is invertible on the union of all $D_{+}(s)\subset P$,
$s\in\Gamma(X,\mathcal{L})$. The morphism $h:X\rightarrow P$ factors over this
union, and we have $\mathcal{L}=h^{*}(\mathcal{O}_{P}(1))$. Note that the
points $h(a_{1}),...,h(a_{r}),h(\zeta)$ all land in an open subscheme
$Y\subset P$ over which $\mathcal{O}_{P}(1)$ is an ample invertible sheaf.
Thus, there is an affine open subscheme $W\subset P$ which contains these
points and where $\mathcal{O}_{P}(1)|_{W}$ is a trivial line bundle. So by
replacing $P$ with $W$ and $X$ with $h^{-1}(W)$ we may suppose that
$\mathcal{L}$ is a trivial line bundle. We will show that it is ample after
perhaps further shrinking $X$.

By construction, the non-zero locus $X_{s_{0}}=U$ is affine. Choose a local
section $\varphi\in\Gamma(U,\mathcal{L})$ that vanishes on the closed set
$U\smallsetminus V$, but not at the generic point $\eta\in U$. For some $n\geq
0$, the local section $\varphi\cdot s_{0}^{\otimes n}|U$ extends to some
global section $s_{2}\in\Gamma(X,\mathcal{L}^{\otimes 1+n})$. The
corresponding effective Cartier divisor $H\subset X$ is of the form
$H=mD+H^{\prime}$, for some multiplicity $m\geq 0$ and $D\not\subset
H^{\prime}$. Likewise, our section $s_{0}\in\Gamma(X,\mathcal{L})$ yields an
effective Cartier divisor $H_{0}=dD$ for some $d\geq 1$. Thus
$s=s_{2}^{\otimes d}\otimes
s_{0}^{\otimes-m}\in\Gamma(X,\mathcal{L}^{dn+d-m})$ vanishes on
$X\smallsetminus V$ but not at $\zeta$. Thus the non-zero locus $X_{s}$
contains $\zeta$, lies in $V$, and there takes the form
$V^{\prime}=D(s|_{V})$, whence is affine. It follows that
$\mathcal{L}\cong\mathcal{O}_{X}$ is ample. Therefore $X$ is quasi-affine by
[31, Tag 01QE] and so it must have the AF-property.

This settles the case that $X$ is integral and normal. It remains to treat the
general case where $X$ is reduced. Let $\nu:X^{\prime}\rightarrow X$ be the
normalization map, which is integral with set-theoretically finite fibers by
[18, Prop. 4.8.2], in particular closed and affine. The fiber
$\nu^{-1}(\zeta)$ consists of finitely many codimension-one points, and
$\nu^{-1}(\\{a_{1},\ldots,a_{r}\\})$ is a finite set contained in some affine
open set. We saw above that there is an affine open set $U^{\prime}\subset
X^{\prime}$ containing $\nu^{-1}(\zeta)$ and the $\nu^{-1}(a_{i})$. Consider
the complementary closed set $Z^{\prime}=X^{\prime}\smallsetminus U^{\prime}$.
The image $\nu(Z^{\prime})\subset X$ is closed and disjoint from
$a_{1},\ldots,a_{r}$ and $\zeta$. Thus

$\nu^{-1}(X\smallsetminus\nu(Z^{\prime}))=X^{\prime}\smallsetminus\nu^{-1}(\nu(Z^{\prime}))\subset
X^{\prime}\smallsetminus Z^{\prime}=U^{\prime}$

is quasiaffine. Replacing $X$ by the complement of $\nu(Z^{\prime})$, we thus
may assume that $Z^{\prime}$ is empty and $X^{\prime}$ is quasiaffine. In
particular, $X^{\prime}$ has the AF-property and by [6, Thm. 3.3] the scheme
$X$ has the AF-property. In particular, the points
$a_{1},\ldots,a_{r},\zeta\in X$ admit a common affine open neighborhood. ∎

The preceding result on schemes can be improved using the fact that
codimension $\leq 1$ points on algebraic spaces are schematic.

######  Corollary 2.5.

Let $X$ denote a separated noetherian algebraic space whose normalization
$X^{\prime}$ remains noetherian. Suppose
$\zeta_{1},...,\zeta_{n},a_{1},...,a_{r}\in X$ are as in the statement of
Theorem 2.2. Then there is an open affine subscheme $U\subset X$ which
contains all of $a_{1},..,a_{r},\zeta_{1},...,\zeta_{n}$.

###### Proof.

By [31, Tag 0ADD], there is an open subscheme $U\subset X$ containing all
points $x\in|X|$ whose strictly local rings have
$\text{dim}\mathcal{O}^{\text{sh}}_{X,x}\leq 1$. Thus, all the points in the
statement belong to the largest open subscheme $W\subset X$. Thus, we may
suppose that $X$ is a scheme and to conclude we apply Theorem 2.2. ∎

######  Remark 2.6.

The previous results are sharp in the sense that there exists a smooth proper
$k$-scheme $X$ and two points $x,\zeta\in X$ where $x$ is closed and $\zeta$
is codimension $2$ for which there is no quasiprojective open subset $U\subset
X$ containing both $x$ and $\zeta$. As an example consider Hironaka’s well-
known construction of a smooth non-projective threefold which has two rational
curves $C,D\subset X$ whose union is algebraically equivalent to $0$. In this
situation, let $x\in C$ and $\zeta$ be the generic point of $D$. If $U\subset
X$ was a quasi-projective open subset containing both points, then there would
be an affine open subset $V\subset U$ containing both points. However, the
divisor $H=X\smallsetminus V$ would intersect $C\cup D$ nontrivially,
contradicting the fact that this reducible curve is algebraically trivial.
Later we will show that we may find a desirable open neighborhood of two such
points $U\subset X$ if we relax the quasi-projectivity requirement (see
Proposition 3.6).

##  3\. Flat neighborhoods

In general, algebraic spaces have very few affine open subschemes. However,
given any finite collection of codimension $\leq 2$ points
$\zeta_{1},...,\zeta_{n}$, we will construct an open neighborhood around the
$\zeta_{i}$ which satisfies the resolution property. We begin with some
definitions.

######  Definition 3.1.

Let $C\subset X$ be a closed subspace of an algebraic space, a _flat
neighborhood_ of $C$ is a flat morphism $U\to X$ with the following property:
for any test scheme $T$ and any morphism $T\to X$ with
$T\times_{X}(X\smallsetminus C)=\varnothing$, the first projection
$T\times_{X}U\to T$ is an isomorphism.

######  Remark 3.2.

Suppose $C\subset X$ is a closed subspace defined by the ideal sheaf
$\mathcal{I}_{C}$ and $C_{i}$ the closed subspace defined by
$\mathcal{I}_{C}^{i+1}$ for $i\geq 1$, then for any flat neighborhood $U\to X$
of $C$, the first projection $C_{i}\times_{X}U\to C_{i}$ is an isomorphism. In
fact, if $X$ is locally noetherian the converse holds: if $U\to X$ is a flat
morphism and for every $i\geq 1$ the map $C_{i}\times_{X}U\to C_{i}$ is an
isomorphism, then $U\to X$ is a flat neighborhood of $C$ (see [14, Lem 3.2,
Lem. 3.3]).

We will study fpqc coverings of the form $(U_{i}\to X)_{i=1,2}$ where
$U_{1}\to X$ is a quasi-compact open immersion and $U_{2}\to X$ is a flat
neighborhood of the closed subspace $C=X\smallsetminus U_{1}$ endowed with the
reduced structure. Using the notation of [14], the resulting cartesian squares
are called _flat Mayer –Vietoris_ squares:

${U_{1}\times_{X}U_{2}}$${U_{2}}$${U_{1}}$${X}$

Now we explain how to construct affine flat neighborhoods. Let $X$ be a
noetherian algebraic space with affine diagonal, and $Z\subset X$ be an affine
closed subspace, corresponding to some coherent sheaf of ideals
$\mathcal{I}\subset\mathcal{O}_{X}$. Then the powers $\mathcal{I}^{i+1}$
define the infinitesimal neighborhoods $Z^{(i)}\subset X$, which form a direct
system of closed subspaces. One may regard the ind-scheme as a formal
neighborhood of $Z\subset X$.

According to [31, Tag 05YU] the infinitesimal neighborhoods are affine as
well, and the direct system $Z^{(i)}$ corresponds to an inverse system of
noetherian rings $A_{i}$ with surjective transition maps. Form the inverse
limit $A=\varprojlim A_{i}$ and write $U=\operatorname{Spec}(A)$ for its
spectrum. The induced projections $A\to A_{i}$ are surjective, so the
canonical morphisms $Z^{(i)}\to U$ are closed embeddings.

######  Proposition 3.3.

Let $X$ denote a noetherian algebraic space and $Z\subset X$ an affine closed
subscheme. There is a unique morphism from an affine scheme $f:U\to X$ making
the diagrams

${U}$${Z^{(i)}}$${X}$$\scriptstyle{f}$

commutative. Moreover, the morphism $f$ is a flat neighborhood of $Z$, and the
set-theoretic image $f(U)\subset X$ consists of all points $\zeta\in X$ whose
closure $\overline{\\{\zeta\\}}$ intersects $Z$.

###### Proof.

The existence of a map $f:U\to X$ forming a flat neighborhood of $Z$ is proven
in [23, Thm. 32] and uniqueness follows from [13, Cor. 1.5].

Note that $Z$ lies in the image of $f$ and so if the closure of a point
$\zeta\in X$ intersects $Z$, $\zeta$ is a generization of a point in $f(U)$.
Since the image of a flat morphism is stable under generization $f(U)$ must
contain all such $\zeta$. On the other hand, if the closure of a point
$\zeta\in X$ misses $Z$ then there is an open neighborhood $W\subset X$ of $Z$
that is disjoint from $\overline{\\{\zeta\\}}$. However, the argument showing
the existence of $f:U\to X$ shows that it factors through $W\subset X$ since
each $Z^{(i)}$ does as well. This shows $f(U)$ consists of those points
$\zeta\in X$ whose closure intersects $Z$. ∎

Now suppose that $\mathcal{I}^{\prime}\subset\mathcal{I}$ is another sheaf of
ideal with the same radical. This gives another inverse limit
$A^{\prime}=\varprojlim A^{\prime}_{i}$, together with a flat neighborhood
$U^{\prime}=\operatorname{Spec}(A^{\prime})\to X$ of
$Z^{\prime}=V(\mathcal{I}^{\prime})\subset X$. Moreover, we have a
homomorphism $A^{\prime}\to A$ of rings and a map $g:U\to U^{\prime}$ which is
a morphism of $X$-schemes by the uniqueness in Proposition 3.3.

######  Proposition 3.4.

The natural morphism $g:U\to U^{\prime}$ is an isomorphism.

###### Proof.

There are positive integers $n$ and $m$ with
$\mathcal{I}^{n}\subset\mathcal{I}^{\prime}\subset\mathcal{I}$ and
$\mathcal{I}^{\prime m}\subset\mathcal{I}^{n}\subset\mathcal{I}^{\prime}$.
Thus, if $U_{n}$ and $U^{\prime}_{m}$ denote the flat neighborhoods of
$Z^{(n)}$ and $Z^{\prime(m)}$, we obtain maps

$U\to U^{\prime}\to U_{n}$ $U^{\prime}\to U_{n}\to U_{m}^{\prime}$

and if we show both compositions are isomorphisms, it will follow that $U\to
U^{\prime}$ is an isomorphism. Thus, it suffices to assume that
$\mathcal{I}^{\prime}=\mathcal{I}^{n}$ by symmetry. Now, the result follows
from the fact that the natural map
$\varprojlim_{n}(B/I^{n})\to\varprojlim_{m}(B/I^{nm})$ is an isomorphism for
any commutative ring $B$ and any ideal $I\subset B$.∎

It follows that the flat neighborhood $f:U=\operatorname{Spec}A\to X$ only
depends on the closed set $|Z|\subset|X|$, or equivalently on the
complementary open subspace $X\smallsetminus Z$, and not on the chosen scheme
structure of $Z$. To conclude, we show that there are plenty of open
neighborhoods $U\subset X$ which appear in a flat Mayer–Vietoris square with
affine schemes.

######  Corollary 3.5.

Let $X$ be a noetherian algebraic space with affine diagonal, if
$V=\operatorname{Spec}A\subset X$ is an affine open subset such that the
reduced closed subset $C=(X\smallsetminus
V)_{\mathrm{red}}=\operatorname{Spec}B\subset X$ is an affine scheme, then $X$
appears in a flat Mayer–Vietoris square with affine schemes:

${C\times_{X}V}$${U=\operatorname{Spec}B^{\prime}}$${V=\operatorname{Spec}A}$${X}$$\scriptstyle{f}$$\scriptstyle{i}$

###### Proof.

Proposition 3.3 applied to the closed immersion $C\to X$ yields an affine flat
neighborhood $U\to X$ of $C$. Thus, by taking the fiber product of $U$ and $V$
over $X$ yields a flat Mayer–Vietoris square as desired. ∎

We conclude this section by using flat neighborhoods to improve Corollary 2.5.

######  Proposition 3.6.

Let $X$ denote a separated noetherian algebraic space and suppose that
$z_{1},...,z_{n}\in X$ all have codimension $\leq 2$ and that
$x_{1},...,x_{r}$ all admit a common affine open neighborhood. Then there is
an open neighborhood $U\subset X$ of the $z_{1},...,z_{n},x_{1},...,x_{r}$
that appears in a Mayer–Vietoris square with affine schemes.

###### Proof.

We will assume that the set $\\{z_{1},...,z_{n}\\}$ contains all the generic
points of $X$. Suppose $z_{1},...,z_{m}\in\\{z_{1},...,z_{n}\\}$ denotes the
points with codimension $\leq 1$. Then, by Corollary 2.5 there is an affine
open subscheme $\operatorname{Spec}A\subset X$ containing the
$z_{1},...,z_{m},x_{1},..,x_{r}$. If $D$ denotes
$X\smallsetminus\operatorname{Spec}A$ with the reduced structure, the points
$z_{m+1},...,z_{n}$ which are not already in $\operatorname{Spec}A$ now have
strictly local rings $\mathcal{O}^{\text{sh}}_{D,z_{i}}$ of dimension $\leq 1$
because $\operatorname{Spec}A\subset X$ is dense. By Proposition 2.5 again,
there is a affine open $\operatorname{Spec}B\subset D$ containing the
$z_{m+1},...,z_{n}$ which are not already in $\operatorname{Spec}A$. Consider
the open set $W=\operatorname{Spec}B\cup\operatorname{Spec}A\subset X$ and
note that it satisfies $|W|\cap|D|=|\operatorname{Spec}B|$. Then the open
neighborhood $W$ appears in a flat Mayer–Vietoris square with affine schemes.
∎

######  Remark 3.7.

The reason the above argument works is that codimension $\leq 1$ points admit
affine neighborhoods on algebraic spaces. So even if a codimension $2$ point
of $X$ isn’t scheme-like, it becomes so after viewing the point on the
complement of the dense affine open neighborhood $\operatorname{Spec}A\subset
X$. The same argument above would work for non-schematic points $x\in X$ if we
could find a desirable dense open $\operatorname{Spec}A\subset X$ so that
$x\in X\smallsetminus A$ admits an affine open neighborhood.

We do not know if this holds for algebraic spaces in general, however there is
one interesting case where we can say something: separated and finite type
algebraic spaces over $\overline{\mathbb{F}}_{p}$ of dimension three. Indeed,
if $X$ is such an algebraic space and $\operatorname{Spec}A\subset X$ is any
dense open subset, then the reduced complement $Y$ is a scheme because the
normalization $\tilde{Y}\to Y$ yields a quasi-projective scheme $\tilde{Y}$ by
[1, Cor. 2.11]) and by [6, Thm. 3.3] $Y$ is an AF-scheme.

In general all we can say is that $X$ admits a stratification of open
immersions:

$\varnothing=U_{0}\subset U_{1}\subset\dots\subset U_{n}=X$

where each $U_{i+1}\smallsetminus U_{i}$ is an affine scheme. This follows by
repeatedly applying Corollary 2.5.

This motivates the following question

######  Question 3.8.

Let $x$ denote point on a separated and noetherian algebraic space $X$ which
is not scheme-like. Does there exist a dense affine open set $U\subset X$ such
that $x\in X\smallsetminus U$ admits an affine open neighborhood? More
precisely, what is the least integer $p$ so that there is a stratification as
in Remark 3.7 where $x\in U_{p}$?

A positive answer would help us to extend Theorem 6.7 to points which are not
scheme-like and Corollary 6.9 to algebraic spaces which are not schemes.

##  4\. Triples

In this section we study a very concrete and simple situation which lies
entirely in the realm of commutative algebra. Suppose that we have a diagram
of rings

(4.0.1) $\begin{CD}R_{12}@<{}<{}<R_{2}\\\ @A{}A{}A\\\ R_{1}.\end{CD}$

Write $\operatorname{Mod}(R_{i})$ for the abelian categories of
$R_{i}$-modules, and consider the resulting 2-fiber product

$\mathcal{A}=\operatorname{Mod}(R_{1})\times_{\operatorname{Mod}(R_{12})}\operatorname{Mod}(R_{2}).$

The objects in this category are _triples_ $(M_{1},M_{2},\varphi)$ where
$M_{1}$ is an $R_{1}$-module, $M_{2}$ is an $R_{2}$-module, and
$\varphi:M_{1}\otimes R_{12}\rightarrow M_{2}\otimes R_{12}$ is an
isomorphism. The homomorphisms
$(M_{1},M_{2},\varphi)\rightarrow(N_{1},N_{2},\psi)$ are pairs $(f_{1},f_{2})$
of homomorphisms $f_{i}:M_{i}\rightarrow N_{i}$ such that
$\psi\circ(f_{1}\otimes\operatorname{id})=(f_{2}\otimes\operatorname{id})\circ\varphi$.
Note that $(f_{1},f_{2})$ is an isomorphism in $\mathcal{A}$ if and only if
the maps $f_{1}$ and $f_{2}$ are bijective.

Also note that one may complete the diagram (4.0.1) with the ring
$R=R_{1}\times_{R_{12}}R_{2}$ to a cartesian square, and study the resulting
functor

$\text{Mod}(R)\longrightarrow\mathcal{A},\quad
M\longmapsto(M\otimes_{R}R_{1},M\otimes_{R}R_{2},\text{\rm can}).$

This was done, under suitable assumptions, by Milnor [24, Sec. 2] or Ferrand
[4, Sec. 5]. However, we shall see below that often one has to replace the
fiber product ring $R$ by a suitable scheme or algebraic space $X$ in order to
uncover the hidden geometry of the situation. For the time being, we make a
purely algebraic study of the diagram (4.0.1). We start with the following
observation:

######  Proposition 4.1.

The category $\mathcal{A}$ is additive. It is an abelian category provided the
homomorphisms $R_{1}\rightarrow R_{12}$ and $R_{2}\rightarrow R_{12}$ are
flat.

###### Proof.

Obviously, addition of the maps $f_{i}$ endows the Hom sets with a group
structure for which composition is bilinear. Moreover
$(0,0,\operatorname{id})$ is an initial object, and

$(M_{1},M_{2},\varphi)\times(N_{1},N_{2},\psi)=(M_{1}\oplus N_{1},M_{2}\oplus
N_{2},\varphi\oplus\psi)$

is a product. Thus $\mathcal{A}$ is an additive category (see [19, Lem.
8.2.9]). Given a morphism
$f=(f_{1},f_{2}):(M_{1},M_{2},\varphi)\rightarrow(N_{1},N_{2},\psi)$ one sets
$C_{i}=\text{Cokernel}(f_{i})$ and writes $\beta:C_{1}\otimes
R_{12}\rightarrow C_{2}\otimes R_{12}$ for the induced map. The latter is
bijective, by the Five Lemma, and one can check that $(C_{1},C_{2},\beta)$ is
a cokernel for $(f_{1},f_{2})$ in the category $\mathcal{A}$.

To construct kernels we assume that $R_{i}\rightarrow R_{12}$ are flat. Set
$K_{i}=\operatorname{Ker}(f_{i})$. Then $K_{i}\otimes_{R_{i}}R_{12}$ is the
kernel for $M_{i}\otimes_{R_{i}}R_{12}\rightarrow N_{i}\otimes_{R_{i}}R_{12}$,
and we write $\alpha:K_{1}\otimes_{R_{1}}R_{12}\rightarrow
K_{2}\otimes_{R_{2}}R_{12}$ for the induced map. Again one can check that
$(K_{1},K_{2},\alpha)$ is a kernel for $(f_{1},f_{2})$.

Lastly, we need to check that the natural map from the cokernel of
$(K_{1},K_{2},\alpha)\to(M_{1},M_{2},\varphi)$ to the kernel of
$(M_{1},M_{2},\varphi)\to(C_{1},C_{2},\beta)$ is an isomorphism. However, this
is true because it holds on each component. ∎

In our setting, objects of the form $(R_{1}^{\oplus n},R_{2}^{\oplus n},A)$
for some $n\geq 0$ and some $A\in\operatorname{GL}_{n}(R_{12})$ take over the
role of locally free sheaves of constant rank which are trivial over $R_{1}$
and $R_{2}$. In light of this, the following is relevant for the resolution
property.

######  Theorem 4.2.

Suppose that $R_{i}\rightarrow R_{12}$ are flat for $i=1,2$ and that the
multiplication map $R_{12}\otimes_{R_{2}}R_{12}\rightarrow R_{12}$ is
bijective. Then every object $(M_{1},M_{2},\varphi)$ with finitely generated
$M_{1}$ and $M_{2}$ is the quotient of some $(R_{1}^{\oplus n},R_{2}^{\oplus
n},A)$ for a suitable $n\geq 0$ and $A\in\operatorname{GL}_{n}(R_{12})$.

###### Proof.

We first check that $(M_{1},M_{2},\varphi)$ is the quotient of some
$(R_{1}^{\oplus d},N_{2},\psi)$ with some integer $d\geq 0$ and some finitely
generated $N_{2}$. Since $M_{1}$ is finitely generated, there is indeed a
surjection $R_{1}^{\oplus d}\rightarrow M_{1}$. Write $h:R_{12}^{\oplus
d}\rightarrow M_{2}\otimes_{R_{2}}R_{12}$ for the composition of the induced
map $R_{12}^{\oplus d}\rightarrow M_{1}\otimes_{R_{1}}R_{12}$ with $\varphi$.
The cartesian diagram

$\begin{CD}R_{12}^{\oplus d}@>{h}>{}>M_{2}\otimes_{R_{1}}R_{12}\\\
@A{}A{}A@A{}A{\text{\rm can}}A\\\ F@>{}>{}>M_{2}\end{CD}$

defines some $R_{2}$-module $F$. The lower horizontal arrow is surjective,
because this holds for the upper horizontal arrow. Tensoring this diagram with
$R_{12}$ yields another diagram

$\begin{CD}R_{12}^{\oplus
d}\otimes_{R_{2}}R_{12}@>{}>{}>M_{2}\otimes_{R_{1}}R_{12}\otimes_{R_{2}}R_{12}\\\
@A{}A{}A@A{}A{\text{\rm can}}A\\\
F\otimes_{R_{2}}R_{12}@>{}>{}>M_{2}\otimes_{R_{2}}R_{12},\end{CD}$

which remains cartesian because $R_{12}$ is flat over $R_{1}$. The vertical
arrow on the right is bijective, because
$R_{12}\otimes_{R_{2}}R_{12}\rightarrow R_{12}$ is bijective. Hence the
vertical arrow to the left is bijective as well. In a similar way we get an
identification

$R_{12}^{\oplus d}\otimes_{R_{2}}R_{12}=R_{12}^{\oplus
d}\otimes_{R_{12}}R_{12}\otimes_{R_{2}}R_{12}=R_{12}^{\oplus d}.$

Let $F_{\lambda}\subset F$, $\lambda\in L$ be the family of finitely generated
submodules. For some sufficiently large index $\mu$, the composite map
$F_{\mu}\rightarrow M_{2}$ is surjective and the inclusion
$F_{\mu}\otimes_{R_{2}}R_{12}\subset R_{12}^{\oplus d}$ is bijective. This can
be seen as an isomorphism $\psi:R_{12}^{\oplus d}\rightarrow
F_{\mu}\otimes_{R_{2}}R_{12}$. Setting $N_{2}=F_{\lambda}$ we obtain the
desired object $(R_{1}^{\oplus d},N_{2},\psi)$.

To see that $(M_{1},M_{2},\varphi)$ is the quotient of some $(R_{1}^{\oplus
n},R_{2}^{\oplus n},A)$ it thus suffices to treat the case
$M_{1}=R_{1}^{\oplus d}$. Choose a surjection $f_{2}:R_{2}^{\oplus
n}\rightarrow M_{2}$ for some $n\geq 0$. Since $M_{2}\otimes_{R_{2}}R_{12}$ is
free of rank $d$, the kernel $K$ for the induced surjection $R_{12}^{\oplus
n}\rightarrow M_{2}\otimes_{R_{2}}R_{12}$ is locally free of rank $n-d$, and
in particular $n\geq d$. Hence $K\oplus R_{12}^{\oplus m}$ is free, for some
integer $m\geq 0$. Replacing $f_{2}$ by the surjection
$(f_{2},0):R_{2}^{\oplus n}\oplus R_{2}^{\oplus m}\rightarrow M_{2}$, we thus
may assume that $K$ is free. Choose an isomorphism $\alpha:R_{12}^{\oplus
n-d}\rightarrow K$. The surjection $f_{2}\otimes\operatorname{id}\colon
R_{12}^{\oplus n}\rightarrow M_{2}\otimes_{R_{2}}R_{12}$ admits a section
$\sigma$, because the range is free. This yields a commutative diagram

$\begin{CD}R_{12}^{\oplus d}\oplus R_{12}^{\oplus
n-d}@>{(\sigma\circ\varphi,\alpha)}>{}>R_{12}^{\oplus n}\\\
@V{(\operatorname{id},0)}V{}V@V{}V{f_{2}\otimes\operatorname{id}}V\\\
R_{12}^{\oplus d}@>{}>{\varphi}>M_{2}\otimes_{R_{2}}R_{12}\end{CD}$

with bijective horizontal arrows. Let $A\in\operatorname{GL}_{n}(R_{12})$ be
the matrix corresponding to the upper horizontal arrow, and let
$f_{1}:R_{1}^{\oplus n}\rightarrow R_{1}^{\oplus d}=M_{1}$ be the projection
onto the first $d$ factors. It follows that $(f_{1},f_{2})$ defines the
desired surjection from $(R_{1}^{\oplus n},R_{2}^{\oplus n},A)$ onto
$(M_{1},M_{2},\varphi)$. ∎

The condition $R_{12}\otimes_{R_{1}}R_{12}=R_{12}$ in Theorem 4.2 may seem
strange at first glance but, as the following proposition shows, it is quite
natural.

######  Proposition 4.3.

Let $\varphi:R\rightarrow A$ be a homomorphism of rings. Then the following
are equivalent:

  1. (1)

There are elements $f_{1},\ldots,f_{r}\in R$ such that the induced maps on
localizations $R_{f_{i}}\rightarrow A_{f_{i}}$ are bijective for $1\leq i\leq
r$, and that $\sum_{i=1}^{r}Af_{i}=A$.

  2. (2)

The induced morphism
$u:\operatorname{Spec}(A)\rightarrow\operatorname{Spec}(R)$ is an open
immersion.

  3. (3)

The $R$-algebra $A$ is flat, of finite presentation, and the multiplication
map $A\otimes_{R}A\rightarrow A$ is bijective.

###### Proof.

The statements $(1)\Rightarrow(2)\Rightarrow(3)$ are straightforward and we
omit them. It remains to show that if $u$ is a fppf morphism such that
$A\otimes_{R}A\to A$ is a bijection, then $u$ is an open immersion. Since the
multiplication map is bijective, the obvious section of the map
$\operatorname{Spec}(A\otimes_{R}A)\to\operatorname{Spec}(A)$ is an
isomorphism. In other words, $u$ is an open immersion fppf-locally and the
result now follows from [31, Tag 02L3]. ∎

######  Corollary 4.4.

Let $X$ denote a quasi-compact and quasi-separated algebraic space which
appears in a flat Mayer–Vietoris square of the following form:

${\operatorname{Spec}D}$${\hat{Z}=\operatorname{Spec}\hat{C}}$${U=\operatorname{Spec}A}$${X}$$\scriptstyle{f}$$\scriptstyle{i}$

where $i$ is an open immersion and $f$ is a flat neighborhood of
$X\smallsetminus U$. Then $X$ satisfies the resolution property.

###### Proof.

By [14, Thm. B (1)], the natural functor

$\textbf{Qcoh}(X)\to\textbf{Qcoh}(\operatorname{Spec}A)\times_{\textbf{Qcoh}(\operatorname{Spec}D)}\textbf{Qcoh}(\operatorname{Spec}\hat{C})$

is an equivalence of categories. Moreover, the ring maps $D\to\hat{C}$ and
$D\to A$ are both flat, the top horizontal arrow satisfies the equivalent
properties of Proposition 4.3 and so we may apply Theorem 4.2. In particular,
this shows that every quasi-coherent sheaf of finite type in $X$ is the
quotient of a locally free sheaf on $X$ which is trivial when restricted to
$U$ and $\hat{Z}$. ∎

##  5\. Locally free resolutions

In this section, we show that given a coherent sheaf $\mathcal{F}$ on an
algebraic space $X$, it can be resolved by a locally free sheaf away from a
closed subset of codimension $\geq 3$. Crucially, this closed subset depends
on the sheaf $\mathcal{F}$. We begin by recording a few technical conditions.

######  Assumption 5.1.

Let $X$ be a noetherian algebraic space satisfying the following conditions.

  1. (1)

Every integral closed subscheme $C\subset X$ of codimension $\leq 0$ contains
a dense open regular subset, and

  2. (2)

the normalization $\tilde{X}$ of $X$ is a noetherian algebraic space.

These conditions are satisfied if, for example, $X$ is a quasi-excellent
algebraic space (see [13, Remark 7.2]).

Next, we record a few lemmas showing that we may replace a given coherent
sheaf $\mathcal{F}$ with one that enjoys better properties. We say a sheaf
$\mathcal{F}$ on $X$ is _supported everywhere_ if
$\text{Supp}(\mathcal{F})=X$.

######  Lemma 5.2.

Let $X$ be a $S_{1}$ noetherian algebraic space with affine diagonal. Then any
coherent sheaf $\mathcal{H}$ is the quotient of a coherent sheaf $\mathcal{F}$
which is $S_{1}$ and is supported everywhere.

###### Proof.

Since $X$ is noetherian and has affine diagonal, there is an faithfully flat
affine morphism $f:U\to X$ where $U$ is an affine scheme. Now [7, Lem. 2.14]
implies $\mathcal{H}$ is the quotient of a quasi-coherent subsheaf
$\mathcal{F}\subset f_{*}\mathcal{O}_{U}^{(I)}$ for some index set $I$. Since
$\mathcal{H}$ is of finite type, and $f_{*}\mathcal{O}_{U}^{(I)}$ is supported
everywhere, we may choose $\mathcal{F}$ to be of finite type and supported
everywhere as well.

To see that $\mathcal{F}$ is $S_{1}$ it suffices to show that at any point
$p\in X$, there is an element of $\mathcal{O}_{X,p}^{\text{sh}}$ which is a
nonzero divisor of
$\mathcal{F}_{p}=\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X,p}^{\text{sh}}$.
Note that because $\mathcal{O}_{X,p}^{\text{sh}}$ is $S_{1}$, it admits a non-
zero divisor $r\in\mathcal{O}_{X,p}^{\text{sh}}$ and because $f$ is faithfully
flat, $r$ is not a zero divisor for
$f_{*}\mathcal{O}_{U}^{(I)}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X,p}^{\text{sh}}$
either. Since $\mathcal{F}_{p}$ is a subsheaf of
$f_{*}\mathcal{O}_{U}^{(I)}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X,p}^{\text{sh}}$
it follows that $r$ is not a zero divisor for $\mathcal{F}_{p}$. ∎

Another lemma we will require is the following.

######  Lemma 5.3.

Let $X$ be a noetherian and separated algebraic space which is reduced and
satisfies the assumptions in 5.1. If $\mathcal{F}$ is a $S_{1}$ coherent sheaf
on $X$ which is supported everywhere, then the set

$\mathbf{U}_{S_{2}}(\mathcal{F})=\\{x\in X|\mathcal{F}_{\bar{x}}\textrm{ is
}S_{2}\\}$

is open in $X$ and contains all codimension $\leq 1$ points of $X$. In
particular, the locus where $X$ is $S_{2}$ is open and contains all
codimension $\leq 1$ points.

###### Proof.

The question is étale local so we may and do assume that $X$ is a scheme. By
[12, Cor. 6.11.7] it suffices to show that every generic point of $X$ which
belongs to $\mathbf{U}_{S_{2}}(\mathcal{F})$ admits an open neighborhood lying
entirely in $\mathbf{U}_{S_{2}}(\mathcal{F})$. Indeed, this will show
$\mathbf{U}_{S_{2}}(\mathcal{F})$ is open and since $\mathcal{F}$ is $S_{1}$,
the set $\mathbf{U}_{S_{2}}(\mathcal{F})$ contains all codimension $\leq 1$
points.

Thus, by reducedness and the assumptions we may replace $X$ with a dense open
regular subset. However, since $\mathcal{F}$ is densely supported we may
shrink $X$ further and assume $\mathcal{F}$ is locally free. Thus it suffices
to show that $\mathcal{O}_{X}$ is $S_{2}$ on $X$, but since the local rings of
$X$ are regular, this follows. ∎

######  Theorem 5.4.

Let $X$ be a noetherian and reduced algebraic space which satisfies the
assumptions in 5.1 and let $\mathcal{F}$ be a coherent sheaf on $X$. If
$x_{1},...,x_{r}\in X$ is a finite set of points in $X$ which admits a common
affine open neighborhood, then there is an open subset $U\subset X$ with the
following properties:

  1. (1)

$U$ contains $x_{1},...,x_{r}$,

  2. (2)

$U$ contains all points of codimension $\leq 2$ in $X$,

  3. (3)

$U$ admits a locally free sheaf $\mathcal{V}$, and

  4. (4)

a surjection $\mathcal{V}\to\mathcal{F}|_{U}\to 0$.

###### Proof.

: By Lemma 5.2, we may assume that $\mathcal{F}$ is $S_{1}$ and densely
supported. Since $\mathcal{F}$ is $S_{1}$, the set
$\mathbf{U}_{S_{2}}(\mathcal{F})$ is open and contains all codimension $\leq
1$ points of $X$ by Lemma 5.3. This implies the set of codimension $2$ points
$p\in X$ where $\text{depth}(\mathcal{F}_{p})<2$ is finite, denote them by
$p_{1},...,p_{n}$.

Take $\operatorname{Spec}A\subset X$ to be a dense affine open subset which
contains all the singular codimension $1$ points of $X$ as well as
$x_{1},...,x_{r}$. Indeed, since $X$ satisfies the conditions in 5.1 and is
reduced, this is a finite set and we may apply Corollary 2.5 to produce such
an open subset. Let $D$ denote the reduced complement of
$\operatorname{Spec}A$ and consider the points in the intersection

$(\\{p_{1},...,p_{n}\\}\cup|\text{Sing}(X)|)\cap|D|$

which have codimension $\leq 2$ in $X$, call it $Z$. Since
$\operatorname{Spec}A$ contains all singular codimension $1$ points, $Z$ must
be a finite set $Z=\\{z_{1},...,z_{n}\\}$.

If $y_{0}\in D$ is any point which has codimension $\leq 2$ _viewed as a point
in_ $X$, then by Corollary 2.5 there is a dense affine open
$\operatorname{Spec}B_{0}\subset D$ which contains $Z\cup\\{y_{0}\\}$. Observe
that the open subset in $X$ defined as
$U_{0}=\operatorname{Spec}A\cup\operatorname{Spec}B_{0}$ satisfies the
following properties:

  1. (1)

$U_{0}$ contains $x_{1},...,x_{r},y_{0}$,

  2. (2)

$U_{0}$ contains all codimension $\leq 1$ points of $X$,

  3. (3)

$U_{0}$ contains all singular codimension $\leq 2$ points of $X$, and

  4. (4)

$U_{0}$ contains all codimension $2$ points $p$ of $X$ where
$\mathrm{depth}(\mathcal{F}_{p})<2$.

Note that $U_{0}$ satisfies the hypothesis of Theorem 4.4 by Corollary 3.5 and
so $U_{0}$ has the resolution property. In particular, there is a locally free
sheaf, $\mathcal{V}_{0}^{\prime}$, on $U_{0}$ and a surjection

$\mathcal{V}_{0}^{\prime}\to\mathcal{F}|_{U_{0}}$

If we push this morphism along the immersion $i_{0}:U_{0}\to X$ we obtain a
map

$\mathcal{V}_{0}=(i_{0})_{*}\mathcal{V}_{0}^{\prime}\to(i_{0})_{*}\mathcal{F}|_{U_{0}}$

Some observations are in order.

  1. (1)

The sheaf $\mathcal{V}_{0}=(i_{0})_{*}\mathcal{V}_{0}^{\prime}$ is coherent in
a neighborhood of every codimension $\leq 2$ point by a result of Kollár (see
[31, Tag 0BK3] or [21]). Indeed, $\mathcal{V}_{0}$ is coherent on $U_{0}$
because it is locally free there, hence near every codimension $\leq 1$ point
and every singular codimension $2$ point so it suffices to show this near
regular codimension $2$ points. Thus we may assume $X$ is the spectrum of a
regular local ring $R$ of dimension $2$ and $U_{0}$ is the punctured spectrum.
Now $\mathcal{V}_{0}^{\prime}$ is locally free so only has the generic point
as an associated point, hence the aforementioned lemma applies because
$\hat{R}$ is regular of dimension $2$ as well (see [31, Tags 07NV, 07NY]). It
follows that after possibly removing a closed codimension $\geq 3$ subset of
$X$, $\mathcal{V}_{0}$ is a coherent sheaf.

  2. (2)

The sheaf $\mathcal{V}_{0}$ is locally free of finite rank at every
codimension $\leq 2$ point. As above, we only need to check this at regular
codimension $2$ points so we may assume $X$ is the spectrum of a regular local
ring $R$ of dimension $2$ and $U_{0}$ is the complement of the closed point
$p$. Since $\mathcal{V}_{0}$ is the pushforward of a sheaf on $U_{0}$ and is
coherent, we have $H^{i}_{p}(X,\mathcal{V}_{0})=0$ for $i=0,1$, so
$\mathcal{V}_{0}$ has depth $2$ by [15, Thm. 3.8]. Thus, $\mathcal{V}_{0}$ has
projective dimension $0$ by the Auslander-Buchsbaum formula (see, for
instance, [31, Tag 090V]). In conclusion, after possibly throwing out a closed
subset of codimension $\geq 3$ from $X$, we may assume $\mathcal{V}_{0}$ is
locally free.

  3. (3)

The natural map $h:\mathcal{F}\to(i_{0})_{*}\mathcal{F}|_{U_{0}}$ is an
isomorphism at every codimension $\leq 2$ point. Indeed, it is true at every
point in $U_{0}$ and so is true at every codimension $\leq 1$ point of $X$.
Moreover, since $U_{0}$ contains all codimension $2$ points where
$\mathcal{F}$ is not $S_{2}$, it remains to be checked at codimension $2$
points $p$ where $\text{depth}(\mathcal{F}_{p})=2$. Thus we may assume $X$ is
the spectrum of a local ring of dimension $2$ where $\mathcal{F}$ is $S_{2}$
and $U_{0}$ is the punctured spectrum. Now the result follows by applying the
standard long exact sequence in local cohomology to $\mathcal{F}$ with respect
to the closed point $p\in X$ and [15, Thm. 3.8]. Namely, it shows that

$H^{0}(X,\mathcal{F})\xrightarrow{\sim}H^{0}(U_{0},\mathcal{F})=H^{0}(X,(i_{0})_{*}\mathcal{F}|_{U_{0}})$

which implies $h$ is an isomorphism. It follows that we may identify
$\mathcal{F}$ with $(i_{0})_{*}\mathcal{F}|_{U_{0}}$ after perhaps throwing
out a closed subset of codimension $\geq 3$.

To summarize, after throwing out a codimension $\geq 3$ subset of $X$, we
obtain a map of sheaves

$f_{0}:\mathcal{V}_{0}\to\mathcal{F}$

satisfying the following properties.

  1. (1)

$\mathcal{V}_{0}$ is a locally free sheaf,

  2. (2)

$f_{0}$ is surjective away from the closure of finitely many codimension $2$
points of $X$, and

  3. (3)

$f_{0}$ is surjective at $x_{1},...,x_{r},y_{0}$.

We are not finished because the cokernel of $f_{0}$ may be supported on
codimension $2$ points of $X$. However, since $U_{0}$ contains all codimension
$\leq 1$ points, the set of such codimension $2$ points is finite and we
denote them by $y_{1},...,y_{l}\in X$. Now, we may repeat the procedure above
but with $y_{i}$ instead of $y_{0}$ to obtain locally free sheaves
$\mathcal{V}_{i}$, and a surjective map

$\bigoplus_{i=0}^{l}f_{i}:\bigoplus_{i=0}^{l}\mathcal{V}_{i}\to\mathcal{F}$

after throwing out a closed subset of codimension $\geq 3$ from $X$. ∎

##  6\. Strong tensor generators on algebraic spaces

To establish the resolution property one has to show that _every_ coherent
sheaf is the quotient of a locally free sheaf. The purpose of this section is
to reduce to the case of a _single_ coherent sheaf and thereby verify the
conjecture of Gross using the results of the previous section.

Let $\mathbb{N}[s,t]$ denote the set of polynomials in $s$ and $t$ which have
non-negative integer coefficients. Note that if $p(s,t)\in\mathbb{N}[s,t]$ is
a polynomial $p(s,t)=\Sigma_{m,n}a_{m,n}s^{m}t^{n}$ and $\mathcal{F}$ is a
quasi-coherent sheaf of finite presentation then then we can form a new quasi-
coherent sheaf of finite presentation

$p(\mathcal{F},\mathcal{F}^{\vee})=\bigoplus_{m,n}(\mathcal{F}^{\otimes
m}\otimes(\mathcal{F}^{\vee})^{\otimes n})^{\oplus a_{m,n}}.$

######  Definition 6.1.

Let $X$ be an algebraic space and suppose $\mathcal{F}$ is a quasi-coherent
sheaf of finite type.

  1. (1)

We say $\mathcal{F}$ is a _strong tensor generator_ if for every quasi-
coherent sheaf of finite type $\mathcal{G}$ there is a polynomial $p(s,t)$ in
the set $\mathbb{N}[s,t]$ and a surjection
$p(\mathcal{F},\mathcal{F}^{\vee})\to\mathcal{G}$.

  2. (2)

We say $\mathcal{F}$ is a _resolving sheaf_ if for every open subset $U$ such
that $\mathcal{F}|_{U}$ is the quotient of a locally free sheaf, the open set
$U$ has the resolution property.

######  Remark 6.2.

Gross defines _strong tensor generators_ only in the context of locally free
sheaves so our definition generalizes his (see [7, Definition 5.1]). However,
he describes the weaker notion of _tensor generators_ (also confined to the
locally free case) and we refer to [7, Def. 5.1] for the definition since we
will not need it here.

######  Proposition 6.3.

Let $X$ denote a noetherian algebraic space and suppose that $\mathcal{F}$ is
a strong tensor generator, then $\mathcal{F}$ is a resolving sheaf.

###### Proof.

Let $j:U\subset X$ be an open immersion and suppose that there is a locally
free sheaf $\mathcal{V}$ on $U$ along with a surjection
$\mathcal{V}\to\mathcal{F}|_{U}$. We will show that $\mathcal{V}$ is a locally
free strong tensor generator on $U$. This will imply that the frame bundle of
$\mathcal{V}$ is quasi-affine and hence that $U$ has the resolution property,
as desired (see [7, Thm. 1.1]).

Let $\mathcal{M}_{U}$ be a coherent sheaf on $U$, then by [22, Cor. 15.5],
there exists a coherent subsheaf $\mathcal{M}^{\prime}\subset
j_{*}\mathcal{M}_{U}$ which extends $\mathcal{M}_{U}$. Now since $\mathcal{F}$
is a strong tensor generator, there is a polynomial $p(s,t)\in\mathbb{N}[s,t]$
and a surjection

$p(\mathcal{F},\mathcal{F}^{\vee})\to\mathcal{M}^{\prime}\to 0$

and upon restriction to $U$ we obtain the compostion of surjections

$p(\mathcal{V},\mathcal{V}^{\vee})\to
p(\mathcal{F}|_{U},\mathcal{F}^{\vee}|_{U})=p(\mathcal{F},\mathcal{F}^{\vee})|_{U}\to\mathcal{M}^{\prime}|_{U}=\mathcal{M}_{U}\to
0$

which shows that $\mathcal{V}$ is a strong tensor generator on $U$. ∎

Schröer and Vezzosi in [29, Prop. 2.2] (see also [7, Ex. 1.3]) proved that any
noetherian scheme admits a coherent strong tensor generator (and thus, a
resolving sheaf). Unfortunately, the argument relies on the existence of an
affine open covering. In the following, we show that normal algebraic spaces
always admits an $S_{2}$ resolving sheaf $\mathscr{F}$. In the absence of the
normality hypothesis, we find a coherent tensor generator after removing a
very small subset as long as we are in characteristic $0$. We begin with the
following refinement of [31, Tag 0BD1].

######  Lemma 6.4.

Fix a quasi-compact and quasi-separated algebraic space $Y$. Suppose there is
a finite surjective morphism $\pi:X\to Y$ morphism of algebraic spaces where
$Y$ and $X$ are integral, and $Y$ is normal. Then if $X$ is a quasi-affine
scheme, $Y$ must also be quasi-affine.

###### Proof.

Since $X$ is quasi-affine it is an AF-scheme and hence $Y$ is an AF-scheme by
[7, Thm. 1.2]. We are thus reduced to the case of schemes and we may conclude
by [31, Tags 0BD1, 0BD3]. ∎

######  Proposition 6.5.

Let $X$ be a separated noetherian algebraic space.

  1. (1)

If $X$ is normal, then there is a coherent $S_{2}$ resolving sheaf
$\mathcal{F}$ on $X$ with $\mathrm{Supp}(\mathcal{F})=X$.

  2. (2)

If $X$ lies over $\mathbb{Q}$, there exists a closed subspace $C\subset X$
with codimension $\geq 3$ such that the complement $U=X\smallsetminus C$ has a
strong tensor generator $\mathcal{F}_{U}$. If $X$ is $S_{1}$, then we may take
$\mathcal{F}_{U}$ to be $S_{1}$ with $\mathrm{Supp}(\mathcal{F}_{U})=U$.

###### Proof.

If $X$ is normal, then there is a normal scheme $Y$ equipped with an action of
a finite abstract group $G$ and an isomorphism $X=Y/G$ (see [22, Cor.
16.6.2]). In particular, there is a finite morphism $\pi:Y\to X$. We know that
$Y$ admits a strong tensor generator $\mathcal{G}$ which is $S_{2}$ everywhere
and locally free on the regular locus of $Y$ by the proof of [29, Prop. 2.2].
Indeed, there it is shown that every coherent sheaf $\mathcal{M}$ is the
quotient of a (finite) direct sum $\bigoplus_{i}\mathcal{O}(-t_{i}D_{i})$
where $D_{i}$ is a Weil divisor, $t_{i}\in\mathbb{N}$ and $\\{X\smallsetminus
D_{i}\\}$ is an affine open cover of $Y$ (see also [31, Tag 0F89]).

We claim that $\mathcal{F}=\pi_{*}\mathcal{G}$ is a $S_{2}$ resolving sheaf.
To show it is resolving, assume $\mathcal{V}$ is a locally free sheaf on an
open subset $U\subset X$ and we have a surjection
$\mathcal{V}\to\mathcal{F}|_{U}$, then we obtain another surjection on
$Y_{U}=Y\times_{X}U$:

$\pi_{U}^{*}\mathcal{V}\to\pi_{U}^{*}\mathcal{F}|_{U}\to\mathcal{G}|_{U}$

because $\pi$, and hence $\pi_{U}$, is affine. Thus, the frame bundle of
$\pi^{*}\mathcal{V}$, call it $W$, is quasi-affine because
$\pi^{*}\mathcal{V}$ is a strong tensor generator on $Y_{U}$ (see [7, Thm.
6.4]) and we have the following diagram all of whose squares are cartesian

${W=\text{Fr}(\pi^{*}\mathcal{V})}$${Y_{U}}$${Y}$${\left[W/G\right]}$${\left[Y_{U}/G\right]}$${\left[Y/G\right]}$${\text{Fr}(\mathcal{V})}$${U}$${Y/G=X}$

Indeed, the three top vertical arrows are $G$-torsor maps since the rightmost
one is and this is stable under base change. The bottom three vertical arrows
are coarse moduli space maps since the rightmost one is and this is stable
under flat base change. Thus we see that $W/G$ can be identified with the
frame bundle of $\mathcal{V}$. Next, observe that $W\to W/G$ is a finite
surjective morphism between normal algebraic spaces, since it is proper and
quasi-finite. Hence by Corollary 6.4, the fact that $W$ is quasi-affine
implies $W/G$ is. Therefore the frame bundle of $\mathcal{V}$ is quasi-affine.
In particular, $U$ has the resolution property by [33, Thm. 1.1].

It remains to see why $\mathcal{F}=\pi_{*}\mathcal{G}$ is $S_{2}$ but this
follows because $\mathcal{G}$ is densely supported and $S_{2}$ (see [12, Cor.
5.7.11 (ii)]). Lastly, because $\pi$ is dominant, $\text{Supp}(\mathcal{G})=Y$
implies $\text{Supp}(\mathcal{F})=X$.

If $X$ lies over $\operatorname{Spec}\mathbb{Q}$, first note that there is a
dense open set $U\subset X$ so that $U$ contains all codimension $\leq 2$
points and where we may write $U=\bigcup_{i=1}^{m}U_{i}$ so that each $U_{i}$
has the resolution property. To see this, fix a $p\in X$ of codimension $\leq
2$ and choose any dense affine open $\operatorname{Spec}A\subset X$ and note
that $p\in C=(X\smallsetminus\operatorname{Spec}A)_{\text{red}}$ now has
codimension $\leq 1$ as a point of $C$. Thus, it admits an affine open
neighborhood $p\in\operatorname{Spec}B\subset
X\smallsetminus\operatorname{Spec}A$ by Corollary 2.5. Thus, there is a open
subset $W_{p}\subset X$ containing $p$ which satisfies the hypothesis of
Proposition 4.4. Indeed, $W_{p}$ can be taken to be the open set
$\operatorname{Spec}B\cup\operatorname{Spec}A$. Thus, we may take
$U=\bigcup_{i=1}^{m}W_{p_{i}}$ where $p_{i}\in X$ are the finitely many points
of codimension $\leq 2$ which are not contained in $\operatorname{Spec}A$.

This means that after replacing $X$ with $U$, we may suppose that $X$ is
covered by (finitely many) dense open subsets $U_{i}$ each of which has the
resolution property. For each $1\leq i\leq m$, let $\mathcal{F}_{i}$ denote a
coherent extension (to $X$) of a (locally free) strong tensor generator for
$U_{i}$. Indeed, a locally free tensor generator exists on $U_{i}$ by [7, Thm.
1.1] and in characteristic $0$, all tensor generators are strong tensor
generators by [7, Prop. 6.5].

Now, if $\mathcal{I}_{i}$ denotes the ideal sheaf of the reduced complement
$X\smallsetminus U_{i}$ then we claim that

$\mathcal{G}=\bigoplus_{i=1}^{n}\mathcal{I}_{i}\oplus\mathcal{F}_{i}\oplus\mathcal{F}_{i}^{\vee}$

is a strong tensor generator. Indeed, if $\mathcal{H}$ is an arbitrary
coherent sheaf, we know that there is a surjection

$f_{i}:p_{i}(\mathcal{F}_{i}|_{U_{i}},\mathcal{F}_{i}^{\vee}|_{U_{i}})\to\mathcal{H}|_{U_{i}}$

for some polynomial $p_{i}(s,t)\in\mathbb{N}[s,t]$. Using [11, Prop. 6.9.17],
there is a $n_{i}$ and a map

$\mathcal{I}_{i}^{n_{i}}p_{i}(\mathcal{F}_{i},\mathcal{F}_{i}^{\vee})\to\mathcal{H}$

which extends $f_{i}$ and is therefore surjective on $U_{i}$. Doing this for
every $1\leq i\leq m$, we obtain a surjection

$\bigoplus_{i=1}^{m}f_{i}:\bigoplus_{i=1}^{m}\mathcal{I}_{i}^{n_{i}}p_{i}(\mathcal{F}_{i},\mathcal{F}_{i}^{\vee})\to\mathcal{H}.$

Thus every coherent sheaf on $X$ is a quotient of finitely many sheaves, each
a graded piece of the tensor algebra $\text{T}(\mathcal{G})$:

$\text{T}^{k}(\mathcal{G})=\text{T}^{k}(\bigoplus_{i=1}^{m}\mathcal{I}_{i}\oplus\mathcal{F}_{i}\oplus\mathcal{F}_{i}^{\vee}).$

This implies $\mathcal{G}$ is a strong tensor generator, and, in particular, a
resolving sheaf on $X$, as desired. For the last statement, apply Lemma 5.2 to
obtain a surjection $\mathcal{G}^{\prime}\to\mathcal{G}$ where
$\mathcal{G}^{\prime}$ is $S_{1}$. Since $\mathcal{G}$ is a strong tensor
generator with $\text{Supp}(\mathcal{G})=X$, the same is true for
$\mathcal{G}^{\prime}$. ∎

######  Remark 6.6.

We do not know if Proposition 6.5 holds in the absence of the characteristic
$0$ hypothesis. Indeed, for the proof to go through we need to show that non-
normal algebraic spaces which satisfy the hypothesis of Corollary 3.5 admit
strong locally free tensor generators. However, we only know that locally free
tensor generators exist on such spaces.

We now use this to verify the first part of the conjecture of Gross (see [5,
Conj. 2.0.3]) when $X$ is reduced. In fact, using the machinery developed thus
far, we are even able to establish this in certain non-schematic cases as
well.

######  Theorem 6.7.

Let $X$ be a noetherian and separated algebraic space which is reduced and
satisfies the assumptions in 5.1. Let $x_{1},...,x_{r}\in X$ denote a set of
points which admit a common affine open neighborhood and suppose one of the
following holds:

  1. (1)

$X$ is a scheme, or

  2. (2)

$X$ is a normal algebraic space, or

  3. (3)

$X$ lies over $\operatorname{Spec}\mathbb{Q}$.

Then there exists an open subset $U\subset X$ such that

  1. (1)

$U$ contains the points $x_{1},...,x_{r}$,

  2. (2)

$U$ contains every point of codimension $\leq 2$ in $X$, and

  3. (3)

$U$ has the resolution property.

###### Proof.

By Proposition 6.5, a resolving sheaf $\mathcal{F}$ exists on $X$. Then, by
Theorem 5.4, there is an open subset $U\subset X$ which contains
$x_{1},...,x_{r}$ and every point of codimension $\leq 2$, a locally free
sheaf $\mathcal{V}$ on $U$, and a surjection

$\mathcal{V}\to\mathcal{F}|_{U}\to 0,$

as desired. Since $\mathcal{F}$ is a resolving sheaf, this shows $U$ satisfies
the resolution property, as desired. ∎

Unfortunately, the previous result relies on the reducedness hypothesis.
However, over a field of positive characteristic this may be obviated using
the following result.

######  Proposition 6.8.

Suppose $X$ be a noetherian algebraic space that lies over $\mathbb{F}_{p}$.
Then $X_{\mathrm{red}}$ has the resolution property if and only if $X$ has the
resolution property.

###### Proof.

If $X$ has the resolution property, then the pushforward of any coherent sheaf
on $X_{\mathrm{red}}$ is the quotient of a locally free sheaf on $X$. By
restricting, we see that $X_{\text{red}}$ has the resolution property.

For the other direction, note that the closed immersion $i:X_{\text{red}}\to
X$ is defined by a nilpotent ideal sheaf $\mathcal{I}\subset\mathcal{O}_{X}$.
Then because there is a $n>0$ with $\mathcal{I}^{p^{n}}=0$ we obtain a
commutative diagram

${X}$${X_{\text{red}}}$${X}$$\scriptstyle{f}$$\scriptstyle{F^{n}}$

where $F^{n}$ is the $n$-fold iterated (absolute) Frobenius. Moreover, the
composition $f\circ i=X_{\text{red}}\to X\to X_{\text{red}}$ is also the
$n$-fold iterated (absolute) Frobenius on $X_{\text{red}}$.

Let $\mathcal{V}$ denote a locally free sheaf on $X_{\text{red}}$ whose frame
bundle is quasi-affine, this exists by our hypothesis and [7, Thm. 1.1]. To
show $X$ has the resolution property it suffices to show that
$f^{*}\mathcal{V}$ also has quasi-affine frame bundle. But since $i$ is a
nilpotent immersion it suffices to show that the frame bundle of
$i^{*}f^{*}(\mathcal{V})=(F_{X_{\text{red}}}^{n})^{*}\mathcal{V}$ is quasi-
affine. Indeed, if $W^{\prime}\to W$ is a nilpotent immersion of algebraic
spaces where $W^{\prime}$ is quasi-affine then by [7, Prop. 3.5] $W$ is a
scheme and [31, Tag 0B7L] implies $W$ is also quasi-affine.

Thus, we are reduced to checking that if the locally free sheaf $\mathcal{V}$
has a quasi-affine frame bundle, then
$(F_{X_{\text{red}}}^{n})^{*}\mathcal{V}$ does as well. But the induced map on
frame bundles $W^{\prime}\to W$ is finite and since $W$ is quasi-affine, it
follows that $W^{\prime}$ is quasi-affine as well. This shows that $X$ has the
resolution property. ∎

######  Corollary 6.9.

Let $k$ be a field of positive characteristic. If $X/\operatorname{Spec}k$ is
a noetherian and separated scheme satisfying the conditions in 5.1, then for
any finite collection of points $x_{1},...,x_{r}\in X$ which admit a common
affine open neighborhood, there exists an open subset $U\subset X$ such that

  1. (1)

$U$ contains the points $x_{1},...,x_{r}$,

  2. (2)

$U$ contains all codimension $\leq 2$ points of $X$, and

  3. (3)

$U$ has the resolution property.

###### Proof.

Let $i:X_{\text{red}}\to X$ denote the canonical nilpotent immersion. Then by
Theorem 6.7, there is an open subset $x_{1},...,x_{n}\in U^{\prime}\subset
X_{\text{red}}$ with $\mathrm{codim}(X_{\text{red}}\smallsetminus
U^{\prime})\geq 3$ such that $U^{\prime}$ has the resolution property.
However, $i$ is a homeomorphism so there is a open subspace $U\subset X$ which
contains all codimension $\leq 2$ points of $X$ with
$U_{\text{red}}=U^{\prime}$. Now $U$ has the resolution property by
Proposition 6.8. ∎

To conclude, we verify the second part of Gross’ conjecture (see [5, Conj.
2.0.3]) under some mild hypothesis.

######  Corollary 6.10.

Let $X$ denote a noetherian scheme which satisfies the assumptions of 5.1 and
at least one of the following conditions:

  1. (1)

$X$ is reduced, or

  2. (2)

$X$ lies over a field of positive characteristic.

If $\mathcal{M}$ is a coherent sheaf on $X$, then there is a surjection
$\mathcal{F}\to\mathcal{M}$ where $\mathcal{F}$ is coherent and $\mathcal{F}$
is locally free away from a closed subset of codimension $\geq 3$.

###### Proof.

For each $x\in X$, Theorem 6.7 (or Corollary 6.9 in the non-reduced case)
implies there is an open neighborhood $U_{x}\subset X$ with the resolution
property and with $\text{codim}(X\smallsetminus U_{x})\geq 3$. Thus, it
follows that there is a surjection
$f_{x}:\mathcal{F}_{x}^{\prime}\to\mathcal{M}|_{U_{x}}\to 0$ where
$\mathcal{F}_{x}^{\prime}$ is a locally free sheaf on $U_{x}$. Let
$\mathcal{F}_{x}^{\prime\prime}$ denote a coherent extension of
$\mathcal{F}_{x}^{\prime}$ to all of $X$. Then, using Deligne’s formula [11,
Prop. 6.9.17] for the inclusion $i:U_{x}\to X$, there is an integer $n_{x}$
and a morphism

$\mathcal{I}_{x}^{n_{x}}\mathcal{F}_{x}^{\prime\prime}\to\mathcal{M}$

extending $f_{x}$, where $\mathcal{I}_{x}$ is the ideal sheaf of
$Z_{x}=X\smallsetminus U_{x}$. Thus, for each $x\in X$, there is a coherent
sheaf $\mathcal{F}_{x}=\mathcal{I}_{x}^{n_{x}}\mathcal{F}_{x}^{\prime\prime}$,
a morphism $\mathcal{F}_{x}\to\mathcal{M}$ which is surjective near $x$, with
$\mathcal{F}_{x}$ locally free away from a closed subset of codimension $\geq
3$. Since $X$ is noetherian, there are finitely many such sheaves
$\mathcal{F}_{x_{1}},...,\mathcal{F}_{x_{n}}$ so that

$\mathcal{F}=\bigoplus_{i=1}^{n}\mathcal{F}_{x_{i}}\to\mathcal{M}\to 0$

is surjective. Moreover, $\mathcal{F}$ is locally free away from a closed
subset of codimension $\geq 3$. ∎

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