# Tensor and direct extension of definable subcategories

Mike Prest
Department of Mathematics, University of Manchester, UK
<EMAIL_ADDRESS>
###### Abstract

Definable subcategories may be extended along a ring homomorphism directly, by
using their defining conditions in the new module category, or by tensoring up
with the new ring. We investigate what is preserved and reflected by these
processes. On the way, we introduce the notion of being atomic with respect to
a bimodule - a refinement of the Mittag-Leffler condition. Particular
attention is given to the case where the ring homomorphism is an elementary
embedding.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Pp formulas, pp-pairs and pp-types
    2. 2.2 Pure-injectives and the Ziegler spectrum
    3. 2.3 Definable subcategories of module categories
    4. 2.4 Elementary duality
    5. 2.5 Ultrapowers
  3. 3 Direct extension of definable subcategories
  4. 4 Tensor extension of definable subcategories
    1. 4.1 Tensor product and pp formulas
    2. 4.2 Mittag-Leffler and atomic modules
    3. 4.3 Mittag-Leffler and atomic modules with respect to a bimodule
    4. 4.4 Tensoring by a bimodule
  5. 5 Extension along elementary embeddings
    1. 5.1 Elementary extensions of rings and algebras
    2. 5.2 Tensor extension and restriction of definable subcategories

##  1 Introduction

A subcategory of a module category ${\rm Mod}\mbox{-}R$ is said to be
definable if it is closed in ${\rm Mod}\mbox{-}R$ under direct products,
directed colimits and pure submodules, equivalently if it is an additive
axiomatisable subcategory of ${\rm Mod}\mbox{-}R$. If $f:R\to S$ is a
homomorphism of rings, then there are two obvious ways of inducing a definable
subcategory of ${\rm Mod}\mbox{-}S$ from a definable subcategory ${\cal D}$ of
${\rm Mod}\mbox{-}R$: direct extension, which takes an axiomatisation of
${\cal D}$, transfers it along $f$ and then applies this in ${\rm
Mod}\mbox{-}S$; tensor extension, which is the smallest definable subcategory
of ${\rm Mod}\mbox{-}S$ which contains all modules of the form
$M\otimes_{R}S_{S}$ with $M\in{\cal D}$.

We consider in Sections 3 and 4 the effects of these extension processes,
paying particular attention to tensor extension, indeed considering more
generally the effect that tensoring with a bimodule ${}_{R}B_{S}$ has on
definable subcategories of ${\rm Mod}\mbox{-}R$.

This leads us to consider a strengthening of the relative Mittag-Leffler
condition. Recall that a right $R$-module $M_{R}$ is Mittag-Leffler if, for
every set $\\{\,_{R}L_{i}\\}_{i}$ of left $R$-modules, the natural map
$M\otimes_{R}\prod_{i}L_{i}\to\prod_{i}\,M\otimes_{R}L_{i}$ is monic. There is
a relative version of the definition under which the modules $L_{i}$ are
restricted to those belonging to some definable subcategory of $R\mbox{-}{\rm
Mod}$ (the category of left $R$-modules), see Section 4.2. If the $L_{i}$ all
are $(R,S)$-bimodules for some ring $S$, then that natural map, even if monic,
need not, in the relativised case, be pure as an embedding of $S$-modules, see
4.21. We characterise in 4.17 those modules $M$ where tensoring with $M$ does
give a pure embedding of $S$-modules.

We consider various conditions under which we can say more about extending
definable subcategories along a morphism of rings. In particular, in Section 5
we consider the case where $f:R\to S$ is an elementary embedding of rings.

Subcategories are assumed to be full and closed under isomorphism. Rings $R$
can be taken to be rings with one object but everything works with $R$ a
(skeletally) small preadditive category. In that more general case, elements
of modules and variables in formulas are sorted which, in some arguments,
would require more cumbersome notation. Therefore we present the results for
the usual, 1-sorted ring, case, but the arguments work just as well over rings
with many objects (see, for instance, [12]). The model theory that we use is
summarised in the next section and always refers to the usual language for
$R$-modules. One can see various other references for details about this, but
we think that enough is said here to make the paper understandable without
needing to refer elsewhere.

##  2 Background

This section is rather dense with information and notation but more leisurely
introductions can be found in many other papers. We use [11] as a reference
which gathers together many of the results that we will need.

###  2.1 Pp formulas, pp-pairs and pp-types

A pp formula for right $R$-modules is an existentially quantified
(homogeneous) system of $R$-linear equations, that is, a formula of the form

$\exists\overline{y}\,(\overline{x}A+\overline{y}B=0),$

where $A$ and $B$ are suitable-sized matrices with entries from $R$ and tuples
$\overline{x}$, $\overline{y}$ of variables are read as row vectors (and $0$
is a column vector). We unpack this compact notation to write the formula more
explicitly as

$\exists\overline{y}\,\bigwedge_{j=1}^{m}\,\sum_{i=1}^{n}x_{i}r_{ij}+\sum_{k=1}^{t}y_{k}s_{kj}=0.$

Here the $r_{ij}$ and $s_{kj}$ \- the entries of $A$ and $B$ \- are
elements111precisely, function symbols standing for multiplication by those
elements; for details of how the formal language is set up, see elsewhere of
$R$ and $\bigwedge$ is to $\wedge$ (“and”) as $\sum$ is to $+$; so this is a
system of $m$ $R$-linear equations prefaced by a quantifier which has the
effect of projecting (solution sets) along the $\overline{y}$-axis. The
variables $\overline{x}=(x_{1},\dots,x_{n})$ are free to be substituted with
values from some module. If we denote this formula by $\phi$, then we may
write $\phi(\overline{x})$ or $\phi(x_{1},\dots,x_{n})$ to display its free =
unquantified variables.

Given such a pp formula $\phi$ we have, in each module $M$, its solution set:

$\phi(M)=\\{\overline{a}\in M^{n}:\exists\overline{b}\in M^{t}\text{ such that
}\overline{a}A+\overline{b}B=0\\}.$

We say that $\phi(M)$ is a subgroup of $M^{n}$ pp-definable in $M$ or, more
briefly, a pp-definable subgroup of $M$. The notation
$M\models\phi(\overline{a})$, which is more usual in model theory, is
equivalent to $\overline{a}\in\phi(M)$.

If $M$ is a submodule of $N$, $M\leq N$, then $M$ is a pure submodule if, for
every tuple $\overline{a}$ from $M$, if $N\models\phi(\overline{a})$, then
$M\models\phi(\overline{a})$. More generally, a pure monomorphism is a
monomorphism whose image is pure in its codomain.

If $\psi$ and $\phi$ are formulas with the same free variables
$x_{1},\dots,x_{n}$, then we write $\psi\leq\phi$ if, for every module $M$,
$\psi(M)\leq\phi(M)$, that is, $\psi(M)$ is a subgroup of $\phi(M)$ (pp-
definable subgroups are modules over the centre of the ring $R$ but are not,
in general, $R$-submodules). Formulas which are equivalent in this
preordering, for instance $\phi$ and $\phi\wedge(\overline{x}=\overline{x})$,
have the same solution set in every module and, in practice are identified, so
we regard this as an ordering on (equivalence classes of) pp formulas. The
resulting poset, denoted ${\rm pp}^{n}_{R}$, is a lattice, with meet given by
conjunction, $\phi\wedge\psi$, and join given by $\phi+\psi$, where the latter
is the pp formula
$\exists\overline{x}_{1}\overline{x}_{2}\,(\phi(\overline{x}_{1})\,\wedge\,\psi(\overline{x}_{2})\,\wedge\,\overline{x}=\overline{x}_{1}+\overline{x}_{2})$.
These define $\phi(M)\cap\psi(M)$, respectively $\phi(M)+\psi(M)$ in any
module $M$. One may note that the conjunction and sum of two pp formulas are
not strictly of the form given in the definition of pp formula above but may
be rewritten as equivalent formulas in that form, so we do refer to them as pp
formulas.

A pp-pair is a pair $\psi\leq\phi$ of pp-formulas, that is, where one implies
the other. We notate such a pair as $\phi/\psi$ because often we are
interested in the factor groups $\phi(M)/\psi(M)$. We say that a pp-pair
$\phi/\psi$ is closed on a module $M$ if $\phi(M)=\psi(M)$; otherwise it is
open on $M$.

The pp-type of an element $a$ in a module $M$ is the set of all pp formulas
that $a$ satisfies in $M$; more generally for $n$-tuples:222The free variables
$x_{1},\dots x_{n}$ are just placeholders, so we keep them in the background
in our notation whenever we can.

${\rm
pp}^{M}(\overline{a})=\\{\phi(\overline{x}):M\models\phi(\overline{a})\\}.$

We say that $\overline{a}$ is a realisation of that pp-type in $M$. Every set
$p$ of pp formulas which is a filter, that is, upwards-closed (if
$\phi\leq\psi$ and $\phi\in p$ then $\psi\in p$) and closed under
intersection/conjunction ($\phi,\psi\in p$ implies $\phi\wedge\psi\in p$),
occurs in this way, so we refer to such a set as a pp-type.

A pp-type $p$ is finitely generated if there is a pp formula $\phi\in p$ such
that $p=\\{\psi:\phi\leq\psi\\}$; we write $p=\langle\phi\rangle$ and say that
$p$ is generated by $\phi$. If $A$ is finitely presented and $\overline{a}$ is
from $A$, then ${\rm pp}^{A}(\overline{a})$ is finitely generated, [11,
1.2.6].

Morphisms preserve pp formulas: if $f:M\to N$ and
$M\models\phi(\overline{a})$, then $N\models\phi(f\overline{a})$. Thus
morphisms are non-decreasing on pp-types: ${\rm
pp}^{M}(\overline{a})\subseteq{\rm pp}^{N}(f\overline{a})$ if $f$ is as above.
Note that $f$ is a pure monomorphism iff ${\rm pp}^{M}(\overline{a})={\rm
pp}^{N}(f\overline{a})$ for every $\overline{a}$ from $M$.

###  2.2 Pure-injectives and the Ziegler spectrum

A module $N$ is pure-injective if it is injective over pure monomorphisms,
equivalently if every embedding of $N$ as a pure submodule is a split
embedding.

The (right) Ziegler spectrum, [22], ${\rm Zg}_{R}$ of $R$ is the topological
space on the set, ${\rm pinj}_{R}$, of isomorphism classes of indecomposable
pure-injective (right) $R$-modules and which has, for a basis of open sets,
those of the form

$(\phi/\psi)=\\{N\in{\rm Zg}_{R}:\phi(N)/\psi(N)\neq 0\\}.$

If ${\cal T}$ is any topological space then the lattice ${\cal O}({\cal T})$
of open subsets of ${\cal T}$ is a complete Heyting algebra (a complete
lattice where meet distributes over arbitrary join), also referred to as a
frame.

###  2.3 Definable subcategories of module categories

We say that ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$ if
it satisfies the following equivalent conditions, where a subcategory is
additive if it is closed under finite direct sums and under direct summands.

######  Theorem 2.1.

(see, e.g., [11, 3.4.7] or [12, 9.1]) The following conditions on a
subcategory ${\cal D}$ of ${\rm Mod}\mbox{-}R$ are equivalent:

(i) ${\cal D}$ is additive and axiomatisable by some set of sentences in the
language for $R$-modules;

(ii) there is a set $\Phi$ of pp-pairs such that ${\cal D}=\\{M\in{\rm
Mod}\mbox{-}R:\phi(M)=\psi(M)\,\,\forall\phi/\psi\in\Phi\\}$;

(iii) ${\cal D}$ is closed in ${\rm Mod}\mbox{-}R$ under direct products,
directed colimits and pure submodules.

We say that ${\cal D}$ is defined by $\Phi$ in the situation of (ii) above.

A definable subcategory of ${\rm Mod}\mbox{-}R$ is also closed under images of
pure epimorphisms and under pure-injective hulls (e.g. [11, 3.4.8]).

If $M$ is a module then we denote by $\langle M\rangle$ the definable
subcategory generated by $M$ \- the smallest definable subcategory (of the
ambient module category) containing $M$. Therefore $\langle M\rangle$ consists
of the class of modules $N$ such that every pp-pair closed on $M$ is closed on
$N$:

$\langle M\rangle=\\{N\in{\rm
Mod}\mbox{-}R:\phi(M)=\psi(M)\implies\phi(N)=\psi(N)\,\,\forall\phi\geq\psi\text{
pp }\\}.$

Similar notation is used for the definable subcategory generated by a class of
modules. Every definable subcategory is generated by some, by no means unique,
$M$. We say that two $R$-modules $M,N$ are definably equivalent if they
generate the same definable subcategory of ${\rm Mod}\mbox{-}R$: $\langle
M\rangle=\langle N\rangle$, equivalently if every pp-pair open on $M$ is open
on $N$ and vice versa.

The set of definable subcategories of a module category ${\rm Mod}\mbox{-}R$
is closed under arbitrary intersection.

######  Lemma 2.2.

(e.g. proof of [11, 5.1.1]) Suppose that ${\cal D}_{i}$, $i\in I$, are
definable subcategories of ${\rm Mod}\mbox{-}R$, with ${\cal D}_{i}$ being
defined by $\Phi_{i}$. Then their intersection $\bigcap_{i}\,{\cal D}_{i}$ is
definable, being defined by $\bigcup_{i}\,\Phi_{i}$.

######  Lemma 2.3.

Suppose that ${\cal D}_{i}$, $i\in I$, are definable subcategories of ${\rm
Mod}\mbox{-}R$ and let $\Phi_{i}$ be the set of all pp-pairs closed on ${\cal
D}_{i}$. Then the definable join ${\cal D}=\langle\bigcup_{i}\,{\cal
D}_{i}\rangle$ is defined by $\bigcap_{i}\,\Phi_{i}$.

Proof. Certainly the category defined by $\Phi=\bigcap_{i}\,\Phi_{i}$ contains
each ${\cal D}_{i}$ and hence contains ${\cal D}$ (since that is the smallest
definable subcategory which contains each ${\cal D}_{i}$). For the converse,
if ${\cal D}$ were strictly contained in the subcategory defined by $\Phi$,
then there would be some pp-pair $\phi/\psi$ not in $\Phi$ but closed on
${\cal D}$. There is some $i$ such that $\phi/\psi\notin\Phi_{i}$ so, by
definition of $\Phi_{i}$, there is some $D\in{\cal D}_{i}$ such that
$\phi/\psi$ is open on $D$. But $D\in{\cal D}$ and that is a contradiction.
Therefore $\Phi$ defines precisely ${\cal D}$. $\Box$

Therefore the set of definable subcategories of ${\rm Mod}\mbox{-}R$, ordered
by inclusion, is a complete lattice which we denote by ${\rm DefSub}_{R}$.

######  Theorem 2.4.

(e.g. [11, 5.1.1, 5.1.5]) The lattice ${\rm DefSub}_{R}$ of definable
subcategories of ${\rm Mod}\mbox{-}R$ is naturally isomorphic to the lattice
of closed subsets of the Ziegler spectrum of $R$ and so is anti-isomorphic to
the frame ${\cal O}({\rm Zg}_{R})$ of open subsets of ${\rm Zg}_{R}$.

The first correspondence is ${\cal D}\mapsto{\cal D}\cap{\rm pinj}_{R}$ and,
in the other direction, it takes a set of indecomposable pure-injective
modules to the definable subcategory that they together generate. We write
${\rm Zg}({\cal D})$ for the set ${\cal D}\cap{\rm pinj}_{R}$ given the
relative topology induced from ${\rm Zg}_{R}$.

The join of finitely many definable subcategories is described algebraically
as follows.

######  Lemma 2.5.

[11, 3.4.9] Suppose that ${\cal D}_{1},\dots,{\cal D}_{n}$, are definable
subcategories of ${\rm Mod}\mbox{-}R$. Then their join ${\cal D}=\langle{\cal
D}_{1}\cup\dots\cup{\cal D}_{n}\rangle$ consists of the pure submodules of
modules of the form $M_{1}\oplus\dots\oplus M_{n}$ where $M_{i}\in{\cal
D}_{i}$.

If ${\cal D}$ is a definable category, then a pure-injective $N\in{\cal D}$ is
an elementary cogenerator for ${\cal D}$ if, for each $M\in{\cal D}$, there is
a pure embedding $M\to N^{I}$ for some index set $I$. Every definable category
has an elementary cogenerator (see [11, 5.3.52]). We will use the
characterisation, see [14, 3.8], that a pure-injective module $N$ is an
elementary cogenerator (for the definable subcategory that it generates) iff
every ultrapower of $N$ purely embeds in some direct power of $N$ (for the
definition of ultrapower, see Section 2.5 below).

######  Corollary 2.6.

If $N_{i}$ is an elementary cogenerator for the definable subcategory ${\cal
D}_{i}$, $i=1,\dots,n$, then $N_{1}\oplus\dots\oplus N_{n}$ is an elementary
cogenerator for $\langle{\cal D}_{1}\cup\dots\cup{\cal D}_{n}\rangle$.

The description of 2.5 does not work for infinite joins, see [14, 3.12].

If ${\cal X}$ is any subclass or subcategory of ${\rm Mod}\mbox{-}R$, then we
write $\psi\leq_{\cal X}\phi$ if $\psi(M)\leq\phi(M)$ for every $M\in{\cal
X}$, and we write $\psi=_{\cal X}\phi$ if $\psi(M)=\phi(M)$ for every
$M\in{\cal X}$. If ${\cal X}={\rm Mod}\mbox{-}R$ then we drop the subscript,
identifying pp formulas that agree on all $R$-modules.

######  Lemma 2.7.

If ${\cal X}$ is any subclass or subcategory of ${\rm Mod}\mbox{-}R$ and
${\cal D}=\langle{\cal X}\rangle$ is the definable subcategory that ${\cal X}$
generates, then $\leq_{\cal D}$ and $\leq_{\cal X}$ coincide, as do $=_{\cal
D}$ and $=_{\cal X}$.

That follows from 2.1.

The relation $=_{\cal D}$ of ${\cal D}$-equivalence between pp formulas in a
given set of, say $n$, free variables is a congruence on the lattice ${\rm
pp}^{n}_{R}$ of pp formulas in those $n$ free variables. Therefore, there is
an induced surjective homomorphism of modular lattices ${\rm
pp}^{n}_{R}\to{\rm pp}^{n}({\cal D})$, where the latter is defined to be the
ordered set of pp formulas, in $n$ free variables, modulo equivalence $=_{\cal
D}$ on ${\cal D}$, equally modulo equality when evaluated on $M$ if $\langle
M\rangle={\cal D}$.

######  Lemma 2.8.

If $\overline{a}$ is from $M\in{\cal D}$, then ${\rm pp}^{M}(\overline{a})$ is
closed under $=_{\cal D}$ and upwards closed with respect to $\leq_{\cal D}$.

If $\overline{a}$ is from $M$ and ${\cal X}$ is any class or category of
modules, then we say that ${\rm pp}^{M}(\overline{a})$ is ${\cal X}$-generated
by $\phi$, and write ${\rm pp}^{M}(\overline{a})=_{\cal X}\langle\phi\rangle$,
if ${\rm pp}^{M}(\overline{a})=\\{\psi:\phi\leq_{\cal X}\psi\\}$, in which
case we say that ${\rm pp}^{M}(\overline{a})$ is ${\cal X}$-finitely
generated. By 2.7 we can suppose that ${\cal X}$ is a definable subcategory.
We don’t assume in this definition that $M\in\langle{\cal X}\rangle$.

If ${\cal X}$ is a class of right $R$-modules then a right $R$-module $M$ is
${\cal X}$-atomic iff, for every finite tuple $\overline{a}$ from $M$, the pp-
type, ${\rm pp}^{M}(\overline{a})$, of $\overline{a}$ in $M$ is ${\cal
X}$-finitely generated. Again, by 2.7, we may assume that ${\cal X}$ is a
definable subcategory.

###  2.4 Elementary duality

The (elementary) dual ([7]) of a pp formula $\phi$

$\exists\overline{y}\,(\overline{x}A+\overline{y}B=0)$

for right modules is the pp formula $D\phi$ for left modules which is

$\exists\overline{z}\,(\overline{x}=A\overline{z}\,\wedge\,B\overline{z}=0).$

For example, the dual of the annihilator formula $xr=0$ for right $R$-modules
is (to equivalence) the divisibility formula $r|x$, meaning $\exists
y\,(x=ry)$, for left $R$-modules, and vice versa.

######  Theorem 2.9.

(see [11, §1.3.1]) For each $n$, elementary duality $D(-)$ of pp formulas
defines an anti-isomorphism ${\rm pp}^{n}_{R}\simeq{\rm pp}^{n}_{R^{\rm op}}$.

In particular $D(\psi\wedge\phi)=D\psi+D\phi$,
$D(\psi+\phi)=D\psi\,\wedge\,D\phi$ and $D^{2}\phi=\phi$ (where $D$ is also
used to denote elementary duality from left to right formulas).

If ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$, determined
by closure of some set $\Phi$ of pp-pairs, then the (elementary) dual
definable category ${\cal D}^{\rm d}$ ([3]) is the subcategory of
$R\mbox{-}{\rm Mod}$ defined by the set of dual pairs - the collection of
$D\psi/D\phi$ such that $\phi/\psi\in\Phi$. This is well-defined: the result
${\cal D}^{\rm d}$ is the same for any choice of $\Phi$ defining ${\cal D}$.
We have, e.g. [11, 3.4.18], $({\cal D}^{\rm d})^{\rm d}={\cal D}$.

If $(-)^{\ast}$ is a duality from ${\rm Mod}\mbox{-}R$ to $R\mbox{-}{\rm Mod}$
such as $M^{\ast}={\rm Hom}_{\mathbb{Z}}(M,{\mathbb{Q}}/{\mathbb{Z}})$, or
$M^{\ast}={\rm Hom}_{K}(M,K)$ if $R$ is an algebra over a field $K$, then
$M\in{\cal D}$ iff $M^{\ast}\in{\cal D}^{\rm d}$, e.g. see [11, 3.4.17].

######  Proposition 2.10.

([3, 2.6]) Let ${\cal D}$ be a definable subcategory of ${\rm Mod}\mbox{-}R$
and ${\cal D}^{\rm d}$ be its dual. If $\phi$ and $\psi$ are pp formulas in
the same free variables, then

$\psi\leq_{\cal D}\phi\text{ iff }D\phi\leq_{{\cal D}^{\rm d}}D\psi.$

###  2.5 Ultrapowers

If $I$ is a set then an ultrafilter on $I$ is a collection ${\cal U}$ of
subsets of $I$ which is a filter in the lattice of subsets of $I$ and which is
such that, for every $J\subseteq I$, either $J\in{\cal U}$ or $I\setminus
J\in{\cal U}$.

Given a structure $X$ \- a ring, a module over a ring, … (precisely, a
structure for some first order finitary language) - we can define the
corresponding ultrapower $X^{I}/{\cal U}$ of $X$ to be the structure whose
underlying set is $X^{I}/\sim_{\cal U}$ where $\sim_{\cal U}$ is the
equivalence relation on $X^{I}$ given by $(a_{i})_{i}\sim_{\cal U}(b_{i})_{i}$
iff $\\{i\in I:a_{i}=b_{i}\\}\in{\cal U}$. We denote the equivalence class of
$(a_{i})_{i}$ by $(a_{i})_{i}/{\cal U}$. The structure on $X^{I}/{\cal U}$ is
defined pointwise; let’s be more precise supposing that $X$ is a ring. First
though, we remark that the idea is that sets in ${\cal U}$ are “large” so
$\sim_{\cal U}$ identifies elements of the product which are equal on a
“large” set of coordinates.

If $X$ is a ring, then we define addition on the ultrapower $X^{I}/{\cal U}$
by $(a_{i})_{i}/{\cal U}+(b_{i})_{i}/{\cal U}=(a_{i}+b_{i})_{i}/{\cal U}$ and
similarly for multiplication. The identity and zero are $(1_{i})_{i}/{\cal U}$
and $(0_{i})_{i}/{\cal U}$ respectively. It follows easily from the definition
of ultrafilter that the subset $Z\subseteq X^{I}$ of elements which are
$\sim_{\cal U}$-equivalent to $0$ is a two-sided ideal of the ring $X^{I}$ and
that, as a set and as a ring, $X^{I}/Z$ is exactly the factor ring
$X^{I}/{\cal U}$.

Similarly for modules: if $M$ is a right $R$-module, then $M^{I}/{\cal U}$ is
a right $R$-module with $(a_{i})_{i}/{\cal U}\cdot r$ defined to be
$(a_{i}r)_{i}/{\cal U}$. In fact, defining $(a_{i})_{i}/{\cal
U}\cdot(r_{i})_{i}/{\cal U}=(a_{i}r_{i})_{i}/{\cal U}$ makes $M^{I}/{\cal U}$
a right $R^{I}/{\cal U}$-module with the $R$-action on $M^{I}/{\cal U}$
arising via the diagonal embedding (see 5.1) $R\to R^{I}/{\cal U}$.

##  3 Direct extension of definable subcategories

Suppose that $f:R\to S$ is a morphism of rings. If $\phi$ is a pp formula for
$R$-modules, then we denote by $f_{\ast}\phi$ the pp formula for $S$-modules
obtained by replacing each ring element $r$ appearing in $\phi$, by $fr$. Note
that $f_{\ast}(\phi\wedge\psi)=f_{\ast}\phi\wedge f_{\ast}\psi$ and
$f_{\ast}(\phi+\psi)=f_{\ast}\phi+f_{\ast}\psi$, so this is a lattice
homomorphism from ${\rm pp}^{n}_{R}$ to ${\rm pp}^{n}_{S}$.

######  Lemma 3.1.

[11, 3.2.7] If $M\in{\rm Mod}\mbox{-}S$, then
$f_{\ast}\phi(M_{S})=\phi(M_{R})$.

That is, the meaning of $f_{\ast}\phi$ in an $S$-module $M$ is the same as the
meaning of $\phi$ in $M$ regarded as an $R$-module via $f$. We have the
following corollaries.

######  Corollary 3.2.

If $\phi/\psi$ is a pp-pair for (right) $R$-modules, then
$f_{\ast}\phi/f_{\ast}\psi$ is a pp-pair for (right) $S$-modules.

######  Corollary 3.3.

If $\phi$ and $\psi$ are equivalent pp formulas for (right) $R$-modules, then
$f_{\ast}\phi$ and $f_{\ast}\psi$ are equivalent for (right) $S$-modules.

######  Corollary 3.4.

If $\phi$ and $\psi$ are pp formulas for $R$-modules and $M\in{\rm
Mod}\mbox{-}S$, then the pp-pair $f_{\ast}\phi/f_{\ast}\psi$ is closed on
$M_{S}$ iff the pair $\phi/\psi$ is closed on $M_{R}$.

So, in practice, particularly if $R$ is a subring of $S$ via $f$, we often
just identify $\phi$ and $f_{\ast}\phi$.

Let ${\cal D}$ be a definable subcategory of ${\rm Mod}\mbox{-}R$. Define the
direct extension of ${\cal D}$ to ${\rm Mod}\mbox{-}S$, to be

${\cal D}^{S}=\\{M_{S}:M_{R}\in{\cal D}\\}.$

(We could alternatively notate this as ${\cal D}^{f}$.) Since restriction
commutes with direct products and directed colimits, and since an $S$-pure
embedding is, in particular, $R$-pure, by 2.1 ${\cal D}^{S}$ is a definable
subcategory of ${\rm Mod}\mbox{-}S$. Indeed it is defined by closure of “the
same” pp-pairs that define ${\cal D}$.

######  Lemma 3.5.

If ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$ and if $\Phi$
is a set of pp-pairs for $R$-modules defining (by their closure) ${\cal D}$,
then $f_{\ast}\Phi=\\{f_{\ast}\phi/f_{\ast}\psi:\phi/\psi\in\Phi\\}$ defines
the direct extension ${\cal D}^{S}$ of ${\cal D}$ to a definable subcategory
of ${\rm Mod}\mbox{-}S$.

Proof. If $M\in{\cal D}^{S}$ and $\phi/\psi$ is closed on ${\cal D}$ then,
since $M_{R}\in{\cal D}$, we have that $\phi/\psi$ is closed on $M_{R}$. So,
by 3.4, $f_{\ast}\phi/f_{\ast}\psi$ is closed on $M_{S}$.

For the converse, if $M_{S}$ is such that, for every pp-pair $\phi/\psi$ in
$\Phi$ we have $f_{\ast}\phi/f_{\ast}\psi$ closed on $M_{S}$ then, by 3.4,
each $\phi/\psi\in\Phi$ is closed on $M_{R}$. It follows that $M_{R}\in{\cal
D}$, so $M\in{\cal D}^{S}$, as required. $\Box$

######  Example 3.6.

If ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$ then the
class of $R$-modules which are restrictions of $S$-modules $M\in{\cal D}^{S}$
may be strictly smaller than ${\cal D}$. For instance, take the ring morphism
$R\to S$ to be ${\mathbb{Q}}\to{\mathbb{R}}$ and ${\cal D}$ to be all of ${\rm
Mod}\mbox{-}{\mathbb{Q}}$, so ${\cal D}^{S}={\rm Mod}\mbox{-}{\mathbb{R}}$.

Examples of the form $\pi_{1}:R_{1}\times R_{2}\to R_{1}$ show how this can
happen for even simpler reasons.

We now take account of this.

Suppose that $f:R\to S$ is a ring homomorphism and that ${\cal C}$ is a
definable subcategory of ${\rm Mod}\mbox{-}S$. Set $f^{\ast}{\cal
C}=\\{M_{R}:M\in{\cal C}\\}$. Let $\Phi({\cal C})$ be the set of all pp-pairs
for $S$-modules which are closed on ${\cal C}$. Define the (definable)
restriction, ${\cal C}|_{R}$, of ${\cal C}$ to ${\rm Mod}\mbox{-}R$ along $f$
to be the subcategory of ${\rm Mod}\mbox{-}R$ defined by the subset
$\Phi({\cal C})_{R}$ of $\Phi({\cal C})$ consisting of all those pp-pairs
$\phi/\psi$ for $R$-modules such that $f_{\ast}\phi/f_{\ast}\psi\in\Phi({\cal
C})$ (that is, we use all the pp-pairs in $\Phi({\cal C})$ which can be
applied to $R$-modules).

In the special case ${\cal C}={\rm Mod}\mbox{-}S$, we denote ${\cal C}|_{R}$
by ${\rm DefTr}^{S}_{R}$, or ${\rm DefTr}(f)$, the definable trace of ${\rm
Mod}\mbox{-}S$ in ${\rm Mod}\mbox{-}R$.

######  Lemma 3.7.

Suppose that $f:R\to S$ is a ring homomorphism and that ${\cal C}$ is a
definable subcategory of ${\rm Mod}\mbox{-}S$. Then $f^{\ast}{\cal
C}=\\{M_{R}:M\in{\cal C}\\}\subseteq{\cal C}|_{R}$, and the definable
subcategory of ${\rm Mod}\mbox{-}R$ generated by $f^{\ast}{\cal C}$ is ${\cal
C}|_{R}$; that is, $\langle f^{\ast}{\cal C}\rangle={\cal C}|_{R}$.

Proof. The first statement is immediate by 3.4. If the containment $\langle
f^{\ast}{\cal C}\rangle\subseteq{\cal C}|_{R}$ were proper, then there would
be a pp-pair $\phi/\psi$ for $R$-modules closed on $f^{\ast}{\cal C}$ but not
on ${\cal C}|_{R}$. The former implies, by 3.4 again, that
$f_{\ast}\phi/f_{\ast}\psi$ is closed on ${\cal C}$, hence that
$f_{\ast}\phi/f_{\ast}\psi\in\Phi({\cal C})$. Therefore, by definition,
$\phi/\psi\in\Phi({\cal C})_{R}$ and so that pair is closed on ${\cal
C}|_{R}$, contradiction, as required. $\Box$

We give an algebraic description of ${\cal C}|_{R}$ in 3.9, using the
following result.

######  Lemma 3.8.

Suppose that $R\to S$ is a ring homomorphism and ${\cal C}$ is a definable
subcategory of ${\rm Mod}\mbox{-}S$. If $N$ is an elementary cogenerator for
${\cal C}$, then $N_{R}$ is an elementary cogenerator for ${\cal C}|_{R}$.

Proof. First we show that $N_{R}$ is an elementary cogenerator for the
definable category $\langle N_{R}\rangle$ that it generates.

Since $N$ is pure-injective as an $S$-module it is pure-injective as an
$R$-module (e.g. [11, 4.3.7]). So, by [14, 3.8], it is enough to show that
every ultrapower of $N_{R}$ purely embeds in a power of $N_{R}$. Consider an
ultrapower $(N_{R})^{I}/{\cal U}$ where ${\cal U}$ is an ultrafilter on a set
$I$. We can form the corresponding ultrapower $N^{I}/{\cal U}$ of the
$S$-module $N$. Since $N$ is an elementary cogenerator, there is an $S$-pure
embedding $N^{I}/{\cal U}\to N^{\kappa}$ for some $\kappa$. Note that
$(N^{I}/{\cal U})_{R}$ is exactly $(N_{R})^{I}/{\cal U}$ (form the ultrapower,
then apply the module restriction functor $f^{\ast}$ or apply $f^{\ast}$, then
form the ultrapower; these are the same since the restriction functor has both
a left and right adjoint so commutes with powers and directed colimits, hence
(e.g. [11, 3.3.1], with ultrapowers). Therefore, on restriction, we have an
$R$-pure embedding $(N_{R})^{I}/{\cal U}\to(N_{R})^{\kappa}$, as required.

Just for interest, we note a different, more model-theoretic, proof which uses
a different characterisation of elementary cogenerators - as pure-injectives
which realise every neg-isolated type (see [11, 5.3.50]). So we have to show
that if $p$ is a pp-type for $\langle N_{R}\rangle$ which is neg-isolated, say
by the pp formula $\psi$, then $p$ is realised in $N$. So we consider
$f_{\ast}p$ \- a set of pp formulas for $S$-modules, which we can extend to a
set $q$ of pp formulas which is maximal, relative to $\langle N\rangle={\cal
C}$, not containing $f_{\ast}\psi$. Since $N$ is an elementary cogenerator for
${\cal C}$, there is $\overline{a}$ from $N$ such that ${\rm
pp}^{N}(\overline{a})=q$. Then ${\rm pp}^{N_{R}}(\overline{a})=p$, as
required.

So $N_{R}$ is an elementary cogenerator for $\langle N_{R}\rangle$. Since each
$M\in{\cal C}$ purely embeds in some power $N^{\kappa}$, each $M_{R}$ purely
embeds in some power $(N_{R})^{\kappa}$ and hence $\langle
N_{R}\rangle=\langle f^{\ast}{\cal C}\rangle={\cal C}|_{R}$. $\Box$

######  Corollary 3.9.

Suppose that $R\to S$ is a ring homomorphism and that ${\cal C}$ is a
definable subcategory of ${\rm Mod}\mbox{-}S$. Then

${\cal C}|_{R}=\\{M^{\prime}\in{\rm Mod}\mbox{-}R:M^{\prime}\text{ is a pure
submodule of }M_{R}\text{ for some }M\in{\cal C}\\}.$

That is, to obtain ${\cal C}|_{R}$ from ${\cal C}$, we just have to close
$f^{\ast}{\cal C}$ under ($R$-)pure submodules.

Given a definable subcategory ${\cal D}$ of ${\rm Mod}\mbox{-}R$, set

${\cal D}^{S}_{R}=({\cal D}^{S})|_{R}.$

Directly from the definitions we have the following.

######  Lemma 3.10.

Suppose that $R\to S$ is a ring homomorphism and that ${\cal D}$ is a
definable subcategory of ${\rm Mod}\mbox{-}R$; then

${\cal D}^{S}_{R}={\cal D}\,\cap\,{\rm DefTr}^{S}_{R}$

and ${\cal D}^{S}=({\cal D}^{S}_{R})^{S}$.

A definable subcategory of ${\rm Mod}\mbox{-}S$ of the form ${\cal D}^{S}$ for
some definable ${\cal D}\subseteq{\rm Mod}\mbox{-}R$ will be said to be
directly induced (from (${\rm Mod}\mbox{-}$)$R$ along $f$). By 3.10 every
definable subcategory of ${\rm Mod}\mbox{-}S$ directly induced along $f$ has
the form ${\cal D}^{S}$ for some definable subcategory of ${\rm Mod}\mbox{-}R$
contained in ${\rm DefTr}^{S}_{R}$.

######  Proposition 3.11.

Suppose that $R\to S$ is a ring homomorphism. The map $(-)^{S}:{\cal
D}\mapsto{\cal D}^{S}$ is a morphism from the lattice ${\rm DefSub}_{R}$ of
definable subcategories of ${\rm Mod}\mbox{-}R$ to ${\rm DefSub}_{S}$.

The lattice of definable subcategories of ${\rm Mod}\mbox{-}S$ directly
induced from $R$ is a sublattice of ${\rm DefSub}_{S}$. It is a complete
lattice and its inclusion into ${\rm DefSub}_{S}$ preserves infinite meets.

The map, $(-)^{S}:{\cal D}\mapsto{\cal D}^{S}$, restricted to the definable
subcategories of ${\rm DefTr}^{S}_{R}$ and corestricted to the directly
induced (from $R$) definable subcategories of ${\rm Mod}\mbox{-}S$ is a
lattice isomorphism.

Proof. If $({\cal D}_{\lambda})_{\lambda}$ are definable subcategories of
${\rm Mod}\mbox{-}R$, then their intersection ${\cal D}$ is definable (2.2)
and hence is the meet of these in the lattice of definable subcategories of
${\rm Mod}\mbox{-}R$. We have $M\in\bigcap_{\lambda}\,({\cal
D}_{\lambda})^{S}$ iff $M_{R}\in{\cal D}_{\lambda}$ for each $\lambda$, iff
$M_{R}\in{\cal D}$ iff $M\in{\cal D}^{S}=\big{(}\bigcap_{\lambda}\,{\cal
D}_{\lambda})^{S}$, so $(-)^{S}$ commutes with arbitrary intersections.

Take two directly induced definable subcategories of ${\rm Mod}\mbox{-}S$, by
3.10 without loss of generality of the form $({\cal D}_{1})^{S}$ and $({\cal
D}_{2})^{S}$ for some definable subcategories ${\cal D}_{1}$, ${\cal D}_{2}$
contained in ${\rm DefTr}^{S}_{R}$. Let $N_{i}$ be an elementary cogenerator
for $({\cal D}_{i})^{S}$. Since, by 3.10, we have $({\cal
D}_{i})^{S}|_{R}={\cal D}_{i}$, it follows by 3.8 that $(N_{i})_{R}$ is an
elementary cogenerator for ${\cal D}_{i}$ so, by 2.6, also
$(N_{1})_{R}\oplus(N_{2})_{R}=(N_{1}\oplus N_{2})_{R}$ is an elementary
cogenerator for the join ${\cal D}_{1}\vee{\cal D}_{2}$. Since $N_{1}\oplus
N_{2}$ is an elementary cogenerator for $({\cal D}_{1})^{S}\,\vee\,({\cal
D}_{2})^{S}$, we see that $(-)^{S}$ preserves finite joins.

If ${\cal D}_{1}\subseteq{\cal D}_{2}\subseteq{\rm DefTr}^{S}_{R}$ have the
same image under $(-)^{S}$ and if $M_{2}\in{\cal D}_{2}$, then, since ${\cal
D}_{2}\subseteq{\rm DefTr}^{S}_{R}$, there is, by 3.9, $M_{S}\in({\cal
D}_{2})^{S}$ with $M_{2}$ pure in $M_{R}$. By assumption $M\in({\cal
D}_{1})^{S}$ so $M_{2}$, being pure in $M_{R}$, is in ${\cal D}_{1}$, showing
that $(-)^{S}$ is injective on definable subcategories of ${\rm
DefTr}^{S}_{R}$. $\Box$

The inclusion does not preserve infinite joins, as is shown by the next
example.

######  Example 3.12.

Take $R=k[x]$, $S=k[x,y,z:y(y-x)=0=y(yz-1)]$, where $k$ is a field, let
$f:R\to S$ be the natural inclusion. Let $M_{n}$ be the $R$-module
$k[x]/(x^{n})$.

Suppose that $M$ is an $S$-module whose restriction to $R$ is isomorphic to
$M_{n}$, say $M$ is generated by $a$ which satisfies $ax^{n}=0$. We show that
$ay=0$. From the relations on $S$ we have
$ay=ay^{2}z=ay.yz=ay^{3}z^{2}=\dots=ay^{n+1}z^{n}$. Also $ay^{2}=ayx$, so
$ay^{n+1}=axy^{n}=\dots=ax^{n}y=0$. So $ay=0$.

Let ${\cal D}_{n}$ be the definable subcategory of ${\rm Mod}\mbox{-}R$
generated by $k[x]/(x^{n})$ and suppose that $M\in({\cal D}_{n})^{S}$, that is
$M_{R}\in{\cal D}_{n}$. Since $x^{n}$ acts as $0$ on $k[x]/(x^{n})$, the same
is true of every module in ${\cal D}_{n}$, in particular on $M_{R}$. So the
computation above shows that $My=0$.

Let ${\cal D}$ be the join of the definable subcategories ${\cal D}_{n}$. Then
[22, 5.1] ${\cal D}$ contains the module $k(x)_{k[x]}$, where $k(x)$ is the
ring of rational functions. Note that $x$ acts invertibly on this module. We
may make $k(x)$ into an $S$-module by defining the action of $y$ to agree with
that of $x$ (and $z$ to act as $x^{-1}$). Thus we obtain a module in ${\cal
D}^{S}$ on which the action of $y$ is nonzero.

On the other hand, since the pp-pair $v=v/vy=0$ is closed on each $({\cal
D}_{n})^{S}$ it is also closed on the join of these definable subcategories in
${\rm DefSub}_{S}$ which, therefore, does not coincide with ${\cal D}^{S}$.

By 3.11, if ${\cal C}$ is a definable subcategory of ${\rm Mod}\mbox{-}S$,
then there is a smallest definable subcategory of ${\rm Mod}\mbox{-}S$
directly induced from $R$ and containing ${\cal C}$. We refer to this as the
$R$-closure of ${\cal C}$ (with respect to the given ring morphism $R\to S$).

######  Lemma 3.13.

If ${\cal C}$ is a definable subcategory of ${\rm Mod}\mbox{-}S$, then the
definable subcategory, $({\cal C}|_{R})^{S}$ of ${\rm Mod}\mbox{-}S$ directly
induced from the restriction ${\cal C}|_{R}$ is the $R$-closure of ${\cal C}$.
The $R$-closure of ${\cal C}$ is defined by the set of those pp-pairs which
are closed on ${\cal C}$ and which are of the form $f_{\ast}\phi/f_{\ast}\psi$
for some $\phi,\psi$ in the language of $R$-modules. The definable subcategory
of ${\rm Mod}\mbox{-}R$ defined by the set of corresponding pp-pairs
$\phi/\psi$ is ${\cal C}|_{R}$.

Proof. The first statement is clear from the definitions, the second and third
statements follow directly from the definitions, 3.4 and 3.5. $\Box$

It follows from 2.4 and 3.11 that direct extension is an embedding at the
level of topology, that is, a frame embedding, from the Ziegler spectrum of
$R$ to that of $S$.

######  Corollary 3.14.

If $f:R\to S$ is a morphism of rings, then direct extension of definable
subcategories gives a morphism of frames ${\cal O}({\rm Zg}_{R})\to{\cal
O}({\rm Zg}_{S})$ which, restricted to the subspace ${\cal O}({\rm Zg}({\rm
DefTr}^{S}_{R}))$, is an embedding.

Proof. Each open subset of ${\rm Zg}_{R}$ has the form ${\rm
Zg}_{R}\setminus{\rm Zg}({\cal D})$ for some definable subcategory ${\cal D}$
of ${\rm Mod}\mbox{-}R$. So the map ${\rm Zg}_{R}\setminus{\rm Zg}({\cal
D})\mapsto{\rm Zg}_{S}\setminus{\rm Zg}({\cal D}^{S})$ between open subsets of
${\rm Zg}_{R}$ and ${\rm Zg}_{S}$ preserves, by 3.11, finite intersections and
arbitrary unions of open subsets, that is, is a morphism of frames. Also by
3.11 that map is injective when restricted to a map on the open subsets of
${\rm Zg}({\rm DefTr}^{S}_{R})$ where this subset is given the relative
topology. $\Box$

#### Extension along ring epimorphisms

We recall that if $f:R\to S$ is an epimorphism of rings, then the situation is
very straightforward. Indeed, the action of each element of $S$ is pp-
definable in terms of the actions of elements of $R$, so the model theory of
$S$-modules is contained in that of $R$-modules, see [11, 6.1.9]. For
instance, every (pp) formula for $S$-modules is equivalent, on every
$S$-module, to a (pp) formula for $R$-modules. Indeed, restriction along $f$
is a full and faithful embedding of ${\rm Mod}\mbox{-}S$ into ${\rm
Mod}\mbox{-}R$ and, see [11, 5.5.4], the image of that embedding is a
definable subcategory of ${\rm Mod}\mbox{-}R$ and so ${\cal D}^{S}={\cal
D}\cap{\rm Mod}\mbox{-}S$ for every definable subcategory ${\cal D}$ of ${\rm
Mod}\mbox{-}R$.

##  4 Tensor extension of definable subcategories

Suppose that $f:R\to S$ is a ring homomorphism. Given a definable subcategory
${\cal D}$ of ${\rm Mod}\mbox{-}R$, we may consider its tensor extension to
${\rm Mod}\mbox{-}S$:

${\cal D}\otimes_{R}S=\langle\\{M\otimes_{R}S:M\in{\cal D}\\}\rangle$

\- the definable subcategory of ${\rm Mod}\mbox{-}S$ generated by what is
obtained by applying $-\otimes_{R}S$ to ${\cal D}$.

######  Example 4.1.

The class $\\{M\otimes_{R}S:M\in{\cal D}\\}$ appearing above might not be a
definable subcategory of ${\rm Mod}\mbox{-}S$. Take $R\to S$ to be, for
instance, the diagonal map $K\to K\times K$ where $K$ is a field, and take
${\cal D}$ to be ${\rm Mod}\mbox{-}K$. Then the modules in
$\\{M\otimes_{R}S:M\in{\cal D}\\}$ all will be even-dimensional so will not
include, for instance, the $K\times K$-module $(K,0)$, which clearly is a
direct summand of $(K,K)=K\otimes_{K}(K\times K)$.

######  Example 4.2.

Closing the class $\\{M\otimes_{R}S:M\in{\cal D}\\}$ under pure submodules is,
in general, not enough to give a definable subcategory: take $R\to S$ to be
$K\to K[X]$ and ${\cal D}$ to be ${\rm Mod}\mbox{-}K$. Then
$\\{M\otimes_{R}S:M\in{\cal D}\\}$ is the category of free $K[X]$-modules
which is closed under neither direct products nor directed colimits (by
coherence of $K[X]$, its definable closure is the category of flat
$K[X]$-modules, see e.g. [11, 3.4.24]).

This example also shows that if $N$ is an elementary cogenerator for ${\cal
D}$ then $N\otimes_{R}S$ need not be an elementary cogenerator for ${\cal
D}\otimes_{R}S$. For, in this example, $K\otimes_{K}K[X]$ is not even pure-
injective and, even if we replace the ring $K[X]$ by, say, its completion
$\overline{K[X]_{(X)}}$ at the ideal generated by $X$, that module, though
pure-injective, is not an elementary cogenerator for the flat
$\overline{K[X]_{(X)}}$-modules (the ring of quotients of
$\overline{K[X]_{(X)}}$ doesn’t embed as a module in any power of
$\overline{K[X]_{(X)}}$).

Tensoring up with $S$ preserves neither infinite join nor finite meet of
definable categories.

######  Example 4.3.

For join, take $f:R\to S$ to be the inclusion ${\mathbb{Z}}\to{\mathbb{Q}}$
and let ${\cal D}_{n}$ be the definable subcategory generated by
${\mathbb{Z}}_{p^{n}}={\mathbb{Z}}/p^{n}{\mathbb{Z}}$. So ${\cal D}_{n}$
consists of the direct sums of copies of ${\mathbb{Z}}_{p^{n}}$. Then ${\cal
D}_{n}\otimes_{\mathbb{Z}}{\mathbb{Q}}=0$, so $\bigvee_{n}\,({\cal
D}_{i}\otimes_{\mathbb{Z}}{\mathbb{Q}})=0$. On the other hand, the torsionfree
module ${\mathbb{Z}}_{(p)}$ belongs to $\bigvee_{n}{\cal D}_{i}$ and so
$(\bigvee_{n}{\cal D}_{i})\otimes_{\mathbb{Z}}{\mathbb{Q}}\neq 0$.

For meet, take $R\to S$ to be the surjection of rings
${\mathbb{Z}}\to{\mathbb{Z}}_{p}$. Let ${\cal
D}_{1}=\langle{\mathbb{Z}}\rangle={\rm Flat}\mbox{-}{\mathbb{Z}}$, ${\cal
D}_{2}=\langle{\mathbb{Z}}_{p}\rangle$. Then ${\cal D}_{1}\cap{\cal D}_{2}=0$
but ${\cal D}_{1}\otimes{\mathbb{Z}}_{p}={\cal
D}_{2}\otimes{\mathbb{Z}}_{p}={\rm Mod}\mbox{-}{\mathbb{Z}}_{p}$.

Depending on the type of morphism $f$ and the properties of the (bi)module
${}_{R}S_{S}$, there may be little relation between ${\cal D}^{S}$ and ${\cal
D}\otimes_{R}S$ or between, say, ${\cal D}$ and $({\cal D}\otimes_{R}S)|_{R}$.
For instance, in Example 4.2, ${\cal D}^{K[X]}={\rm Mod}\mbox{-}K[X]$ whereas
${\cal D}\otimes_{K}K[X]={\rm Flat}\mbox{-}K[X]$. Or take $f:R\to S$ to be
${\mathbb{Z}}\to{\mathbb{Z}}_{2}$ and ${\cal D}$ to be the subcategory of
torsionfree abelian groups; then ${\cal D}\otimes{\mathbb{Z}}_{2}$ is ${\rm
Mod}\mbox{-}{\mathbb{Z}}_{2}$, whereas ${\cal D}\cap{\rm
Mod}\mbox{-}{\mathbb{Z}}_{2}$ is $0$.

Later we will consider some special cases where there is more connection,
particularly that where $f:R\to S$ is an elementary extension of rings. We
will also, in Section 4.4, generalise the functor $-\otimes_{R}S_{S}$ and
consider the effect of a functor $-\otimes_{R}B_{S}$ where $B$ is an
$(R,S)$-bimodule. Before that, we review some results on the interaction of pp
formulas and tensor product and also consider a generalisation of the Mittag-
Leffler condition.

###  4.1 Tensor product and pp formulas

First, we state Herzog’s characterisation in terms of pp formulas of when a
tensor product is $0$. Recall from Section 2.3 that if $\phi$ is a pp formula
(in $n$ free variables) for right modules then its elementary dual $D\phi$ is
a pp formula (in $n$ free variables) for left modules.

######  Theorem 4.4.

(Herzog’s Criterion [3, 3.2]) Suppose that $\overline{a}$ is an $n$-tuple from
the right $R$-module $M$ and that $\overline{b}$ is an $n$-tuple from the left
$R$-module $L$. Then $\sum_{i=1}^{n}\,a_{i}\otimes b_{i}$,
$\overline{a}\otimes\overline{b}$ for short, is $0$ in $M\otimes_{R}L$ iff
there is a pp formula $\phi$ (in $n$ free variables) for right modules such
that $M_{R}\models\phi(\overline{a})$ and ${}_{R}L\models
D\phi(\overline{b})$.

######  Corollary 4.5.

If ${\rm pp}^{M}(\overline{a})$ is generated by $\phi$, then
$\overline{a}\otimes\overline{b}=0$ in $M\otimes_{R}L$ iff ${}_{R}L\models
D\phi(\overline{b})$.

######  Corollary 4.6.

Suppose that $\overline{a}$ is an $n$-tuple from the right $R$-module $M$ and
let ${\cal D}$ be a definable subcategory of ${\rm Mod}\mbox{-}R$. Suppose
that the left $R$-module $L$ belongs to the dual definable subcategory, ${\cal
D}^{\rm d}$ and let $\overline{b}$ be an $n$-tuple from $L$. If ${\rm
pp}^{M}(\overline{a})$ is ${\cal D}$-generated by $\phi$, then
$\overline{a}\otimes\overline{b}=0$ in $M\otimes_{R}L$ iff ${}_{R}L\models
D\phi(\overline{b})$.

Proof. One direction is by 4.4. For the other, if
$\overline{a}\otimes\overline{b}=0$ then, by 4.4, we have
$M\models\psi(\overline{a})$ and $L\models D\psi(\overline{b})$ for some pp
$\psi$. By assumption $\phi\leq_{\cal D}\psi$ so, by 2.10, $D\phi\geq_{{\cal
D}^{\rm d}}D\psi$ and so, since $L\in{\cal D}^{\rm d}$, we have $L\models
D\phi(\overline{b})$, as claimed. $\Box$

Note that, in the above result, we don’t have to assume that $M\in{\cal D}$.

There is a more general theorem, recalled below (4.8), which is applicable
when tensoring with a bimodule ${}_{R}B_{S}$. Herzog’s criterion is the case
in 4.8 where the pp formula $\sigma$ for right $S$-modules is $y=0$.

First we need a generalisation of the dual $D\phi$ of a pp formula. Given
rings $R$ and $S$, a pp formula $\phi(\overline{x})$ for right $R$-modules and
a pp formula $\sigma(\overline{y})$ for right $S$-modules, we produce, 4.7
below, a pp formula $(\sigma:\phi)$ for $(R,S)$-bimodules which expresses the
property, of a tuple of appropriate length, that “tensored with a tuple
satisfying $\phi$ the result will satisfy $\sigma$”. The dual $D\phi$ is the
special case $(y=0\,;\,\phi)$ (with $S={\mathbb{Z}}$ or, indeed, $S$
arbitrary).

We give the statement, then explain what we mean by a partitioned pp formula.
We include a proof since, in the original reference, in order to avoid
notational complexity, the proof was given only in the case where $\sigma$ has
one free variable. Here we take the opportunity to give the proof of the
general case.

######  Theorem 4.7.

([10, 2.1(1,2)]) Suppose that $\sigma$ is a pp (partitioned) formula for right
$S$-modules and that $\phi$ is a pp formula for right $R$-modules. Then there
is a (matching partitioned) pp formula $(\sigma:\phi)$ for $(R,S)$-bimodules,
equivalently for right $R^{\rm op}\otimes S$-modules, such that, for every
right $R$-module $M_{R}$ and $(R,S)$-bimodule ${}_{R}B_{S}$, for every
$\overline{a}$ from $M$ and matching $\overline{b}$ from $B$:

(1) if $M_{R}\models\phi(\overline{a})$ and
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$, then
$(M\otimes_{R}B)_{S}\models\sigma(\overline{a}\otimes\overline{b})$;

(2) if ${\rm pp}^{M}(\overline{a})$ is generated by $\phi$, then
$M\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$ iff
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$.

If the formula $\sigma$ has just one free variable, then the references to
partition can be ignored. Otherwise, say if $\sigma$ has $k$ free variables,
we have to specify, as well as the formula $\phi$, the way in which the free
variables of $\phi$ are partitioned into $k$ blocks. For instance, if $k=2$
and $\phi$ has $3$ free variables, then we have to specify how we tensor two
$3$-tuples to get a 2-tuple: whether $(a_{1}\otimes b_{1},a_{2}\otimes
b_{2}+a_{3}\otimes b_{3}$) or $(a_{1}\otimes b_{1}+a_{2}\otimes
b_{2},a_{3}\otimes b_{3})$. If $\zeta$ is a partition of $n$ into $k$ parts
then by a $\zeta$-partitioned formula we mean one with $n$ free variables and
whose free variables $x_{1},\dots,x_{n}$ are partitioned according to $\zeta$.
A matching formula is another $\zeta$-partitioned formula. We conveniently
regard such a formula as having free variables presented as a $k$-tuple
$(\overline{x}_{1},\dots,\overline{x_{k}})$ of tuples $\overline{x}_{t}$ of
variables. Usually we don’t have to specify more than this (such as the length
of each part) and we may just refer to a formula as above as being a
$k$-partitioned formula. Of course, we assume that formulas and tuples of
elements match as needed, even if that is not specified.

Proof. The proof in [10] is given just for the case where the partition has a
single part, that is, where $\sigma$ has just one free variable. The general
case proceeds similarly and we give that here for completeness.

Suppose that $\sigma=\sigma(v_{1},\dots,v_{k})$ has $k$ free variables and
take $\sigma$ explicitly to be the formula

$\exists\overline{w}\,\bigwedge_{j}\sum_{l}\,v_{l}s_{lj}+\sum_{i}w_{i}t_{ij}=0.$

Next consider the partitioned formula
$\phi=\phi(x_{1},\dots,x_{n})=\phi(\overline{x}_{1},\dots,\overline{x}_{k})$
and choose a free realisation
$(C_{\phi},\overline{a}=(\overline{a}_{1}\,\dots\,\overline{a}_{k}))$ of
$\phi$, meaning that $C_{\phi}$ is finitely presented and ${\rm
pp}^{C_{\phi}}(\overline{a})=\langle\phi\rangle$ (such exists, [11, 1.2.14]).
Take a finite generating tuple $\overline{c}$ for the $R$-module $C_{\phi}$
and let $\theta$ be a conjunction333finite, since $C_{\phi}$ is finitely
presented of $R$-linear equations which together generate all the relations on
$\overline{c}$ in $C_{\phi}$; suppose that $\overline{c}$ is an $m$-tuple.
Also choose a matrix $H$ such that $\overline{a}=\overline{c}H$ and note that
we can write $H$ as a partitioned matrix $(H_{1}\,\dots\,H_{k})$ so that
$\overline{a}_{l}=\overline{c}H_{l}$.

We set the pp formula (for $(R,S)$-bimodules) $(\sigma:\phi)(\overline{y})$,
with free variables $\overline{y}=(\overline{y}_{1}\,\dots\,\overline{y}_{k})$
an $n$-tuple partitioned as is $\overline{x}$, to be

$\exists\overline{u}\,\bigwedge_{j}\,D\theta(\sum_{l}H_{l}\overline{y}_{l}\cdot
s_{lj}+\sum_{i}\overline{u}_{i}\cdot t_{ij})$

where $\overline{u}$ is a tuple (of length equal to that of $\overline{w}$) of
$m$-tuples.

Proof of (2)($\Leftarrow$) and (1): With notation as in the statement of (2),
write $\overline{b}$ from ${}_{R}B_{S}$ as
$(\overline{b}_{1}\,\dots\,\overline{b}_{k})$ to match the partitioned
$n$-tuple $\overline{a}$.

Suppose first that ${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$, so there
is $\overline{d}$, a tuple of $m$-tuples $\overline{d}_{i}$ from $B$, such
that

$\bigwedge_{j}\,D\theta(\sum_{l}H_{l}\overline{b}_{l}\cdot
s_{lj}+\sum_{i}\overline{d}_{i}\cdot t_{ij}).$

Also, since $M\models\phi(\overline{a})$, then there is $\overline{c}$ from
$M$ such that $\overline{a}=\overline{c}H$ and such that
$M\models\theta(\overline{c})$. Then, for each $j$, we have, since
$M\models\theta(\overline{c})$,

$\sum_{l}\overline{c}\otimes H_{l}\overline{b}_{l}\cdot
s_{lj}+\sum_{i}\overline{c}\otimes\overline{d}_{i}\cdot t_{ij}=0,$

(regard tuples occurring on the right of tensors as column vectors)

from which we obtain

$\sum_{l}\overline{c}H_{l}\otimes\overline{b}_{l}\cdot
s_{lj}+\sum_{i}\overline{c}\otimes\overline{d}_{i}\cdot t_{ij}=0,$

that is

$\sum_{l}\overline{a}_{l}\otimes\overline{b}_{l}\cdot
s_{lj}+\sum_{i}\overline{c}\otimes\overline{d}_{i}\cdot t_{ij}=0.$

Thus $M\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$, as
required.

For the converse, 2)($\Rightarrow$), suppose that
$C_{\phi}\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$, that
is
$C_{\phi}\otimes_{R}B_{S}\models\sigma(\overline{a}_{1}\otimes\overline{b}_{1},\dots,\overline{a}_{k}\otimes\overline{b}_{k})$.
So there are $\overline{m}_{i}$ from $C_{\phi}$ and $\overline{d}_{i}$ from
$B$ such that

$\bigwedge_{j}\,\sum_{l}\overline{a}_{l}\otimes\overline{b}_{l}.s_{lj}+\sum_{i}\overline{m}_{i}\otimes\overline{d}_{i}.t_{ij}=0.$

Since $C_{\phi}$ is generated by $\overline{c}$, we have, for each $i$,
$\overline{m}_{i}=\overline{c}K_{i}$ for some matrix $K_{i}$ with entries in
$R$. Therefore
$\overline{m}_{i}\otimes\overline{d}_{i}t_{ij}=\overline{c}K_{i}\otimes\overline{d}_{i}t_{ij}=\overline{c}\otimes
K_{i}\overline{d}_{i}t_{ij}$. Writing $\overline{e}_{i}=K_{i}\overline{d}_{i}$
and recalling that $\overline{a}_{l}=\overline{c}H_{l}$, this gives us

$\bigwedge_{j}\,\sum_{l}\overline{c}_{l}\otimes
H_{l}\overline{b}_{l}.s_{lj}+\sum_{i}\overline{c}\otimes\overline{e}_{i}.t_{ij}=0,$

that is,

$\bigwedge_{j}\,\overline{c}\otimes(\sum_{l}H_{l}\overline{b}_{l}.s_{lj}+\sum_{i}\overline{e}_{i}.t_{ij})=0.$

By Herzog’s criterion, 4.4, and since ${\rm
pp}^{C_{\phi}}(\overline{c})=\langle\theta\rangle$, it follows that, for each
$j$, we have ${}_{R}B_{S}\models
D\theta(\sum_{l}H_{l}\overline{b}_{l}.s_{lj}+\sum_{i}\overline{e}_{i}.t_{ij})$
and hence that

${}_{R}B_{S}\models\bigwedge_{j}\,D\theta(\sum_{l}H_{l}\overline{b}_{l}.s_{lj}+\sum_{i}\overline{e}_{i}.t_{ij}).$

That is

${}_{R}B_{S}\models\exists\overline{u}\,\bigwedge_{j}\,D\theta(\sum_{l}H_{l}\overline{b}_{l}.s_{lj}+\sum_{i}\overline{u}_{i}.t_{ij}),$

as required.

The proof for the general case, with $M$ in place of $C_{\phi}$, is at the end
of the proof of 4.8 below. $\Box$

The next result is the generalisation of 4.4.

######  Theorem 4.8.

([10, 2.1(3)]) Suppose that $M_{R}$ is a right $R$-module, ${}_{R}B_{S}$ is an
$(R,S)$-bimodule, $\sigma$ is a pp formula for right $S$-modules,
$\overline{a}$ is a partitioned tuple from $M$ and $\overline{b}$ is a
matching partitioned tuple from $B$. Then
$(M\otimes_{R}B)_{S}\models\sigma(\overline{a}\otimes\overline{b})$ iff there
is a (matching partitioned) pp formula $\phi$ for right $R$-modules such that
$M_{R}\models\phi(\overline{a})$ and
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$.

Proof. The proof is just as in [10] and we repeat it here. We already have one
direction by 4.7, so suppose that
$M\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$. With
notation as in the above proof, that means we have $\overline{m}_{i}$ from $M$
and $\overline{d}_{i}$ from $B$ such that

$\bigwedge_{j}\,\sum_{l}\overline{a}_{l}\otimes\overline{b}_{l}.s_{lj}+\sum_{i}\overline{m}_{i}\otimes\overline{d}_{i}.t_{ij}=0.$

Then, by 4.4, for each $j$, there is a pp formula $\phi_{j}$ for $R$-modules
such that

$M\models\phi_{j}(\overline{a},\overline{m}_{1},\dots,\overline{m}_{s}),$

where $\overline{a}=(\overline{a}_{1}\,\dots\,\overline{a}_{k})$ and $i$ runs
from 1 to $s$, and

${}_{R}B_{S}\models
D\phi_{j}(\overline{b},\overline{d}_{1},\dots,\overline{d}_{s}).$

Let $\phi(\overline{x})$ be the pp formula
$\exists\overline{u}_{1},\dots,\overline{u}_{s}\,\bigwedge_{j}\phi_{j}(\overline{x},\overline{u}_{1},\dots,\overline{u}_{s})$,
so $M\models\phi(\overline{a})$. We claim that
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$. By 4.7(2), it is sufficient
to show that if $(C_{\phi},\overline{a}^{\prime})$, with
$\overline{a}^{\prime}=(\overline{a}^{\prime}_{1}\,\dots\,\overline{a}^{\prime}_{k})$,
is a free realisation of $\phi$, then $C_{\phi}\otimes
B\models\sigma(\overline{a}^{\prime}\otimes\overline{b})$. Since
$C_{\phi}\models\phi(\overline{a}^{\prime})$, there are
$\overline{c}_{1},\dots,\overline{c}_{s}$ such that
$C_{\phi}\models\phi_{j}(\overline{a}^{\prime},\overline{c}_{1},\dots,\overline{c}_{s})$
for each $j$. Therefore

$C_{\phi}\otimes
B\models\sum_{l}\overline{a}^{\prime}_{l}\otimes\overline{b}_{l}.s_{lj}+\sum_{i}\overline{c}_{i}\otimes\overline{d}_{i}.t_{ij}=0$

for each $j$ and hence
$C_{\phi}\otimes_{R}B_{S}\models\sigma(\overline{a}^{\prime}\otimes\overline{b})$,
as required.

To finish the proof of 4.7, if $M\otimes
B\models\sigma(\overline{a}\otimes\overline{b})$, so
$M\models\psi(\overline{a})$ and $B\models(\sigma:\psi)(\overline{b})$ for
some $\psi$ then, since, by assumption $\phi\leq\psi$, we have
$(\sigma:\psi)\leq(\sigma:\phi)$ and so $B\models(\sigma:\phi)(\overline{b})$,
as required. $\Box$

Following easily from the definitions, we have some elementary properties of
$(-:-)$.

######  Lemma 4.9.

([10, §2.2 (1), (2)])

(1) $(\sigma:\phi+\psi)=(\sigma:\phi)\wedge(\sigma:\psi)$, in particular, if
$\phi\leq\psi$, then $(\sigma:\psi)\leq(\sigma:\phi)$.

(2) $(\sigma:\phi\wedge\psi)\geq(\sigma:\phi)+(\sigma:\psi)$.

######  Remark 4.10.

We will often apply this with ${}_{R}B_{S}=\,_{R}S_{S}$ where $R\to S$ is a
morphism of rings. Note that, if $\sigma$ is a pp formula for right
$R$-modules, then $(\sigma:\phi)$ will be a formula for $(R,R)$-bimodules:
this can be seen from the proof of 4.7 but also follows from the statement
since, if we read the result with ${}_{R}S_{R}$ in place of ${}_{R}S_{S}$,
then nothing changes about the properties of $(\sigma:\phi)$, hence that
formula is unchanged (up to equivalence).

######  Corollary 4.11.

Suppose that $f:R\to S$ is any morphism of rings, that $M$ is an $R$-module
and that $\overline{a}=(a_{1},\dots,a_{n})$ is a tuple from $M$. Consider
$\overline{a}\otimes\overline{1}=(a_{1}\otimes 1_{S},\dots,a_{n}\otimes
1_{S})\in M\otimes_{R}S_{S}$.

Then

${\rm pp}^{M\otimes S}(\overline{a}\otimes\overline{1})=\\{\sigma:\text{there
is }\phi\in{\rm pp}^{M}(\overline{a})\text{ with }(\sigma:\phi)\in{\rm
pp}^{{}_{R}S_{S}}(\overline{1})\\}$

and ${\rm pp}^{M\otimes S}(\overline{a}\otimes\overline{1})$ is generated by
$f_{\ast}{\rm pp}^{M}(\overline{a})$.

In particular, if ${\rm pp}^{M}(\overline{a})$ is finitely generated (by
$\phi$) then so is ${\rm pp}^{M\otimes S}(\overline{a}\otimes\overline{1})$
(by $f_{\ast}\phi$).

Proof. The first part is immediate from 4.8.

Since $M_{R}\to M\otimes_{R}S_{R}$ given by $a\mapsto a\otimes 1_{S}$ is an
$R$-linear map ($ar\otimes 1=a\otimes fr$) we have $f_{\ast}\phi\in{\rm
pp}^{M\otimes S}(\overline{a}\otimes\overline{1})$ for every $\phi\in{\rm
pp}^{M}(\overline{a})$. For the converse, if $\sigma\in{\rm pp}^{M\otimes
S}(\overline{a}\otimes\overline{1})$, then for some $\phi\in{\rm
pp}^{M}(\overline{a})$ we have $(\sigma:\phi)\in{\rm
pp}^{{}_{R}S_{S}}(\overline{1})$; we claim that $f_{\ast}\phi\leq\sigma$. So
suppose that $N$ is an $S$-module and that $N\models
f_{\ast}\phi(\overline{c})$. Then (3.1) $N_{R}\models\phi(\overline{c})$ and
so, since ${}_{R}S_{S}\models(\sigma:\phi)(\overline{1})$, we have
$N\otimes_{R}S_{S}\models\sigma(\overline{c}\otimes\overline{1})$. We have the
$S$-linear map $N\otimes_{R}S_{S}\to N\otimes_{S}S_{S}\simeq N$, so it follows
that $N\models\sigma(\overline{c})$. Therefore $f_{\ast}\phi\leq\sigma$, as
claimed. The statement about finite generation is then immediate. $\Box$

###  4.2 Mittag-Leffler and atomic modules

Recall that a module $M$ is Mittag-Leffler [15] if the equivalent conditions
of the following theorem are fulfilled.

######  Theorem 4.12.

([17, 2.2], see [18]) Suppose that $M$ is a right $R$-module. Then the
following conditions are equivalent.

(i) For every set $\\{L_{i}\\}_{i\in I}$ of left $R$-modules, the canonical
map

$M\otimes_{R}\,(\prod_{i\in I}\,L_{i})\to\prod_{i\in I}\,(M\otimes_{R}L_{i})$

is monic.

(ii) The pp-type ${\rm pp}^{M}(\overline{a})$ of any finite tuple in $M$ is
finitely generated.

For example, finitely presented modules, more generally pure-projective
modules (direct summands of direct sums of finitely presented modules), are
Mittag-Leffler.

There are relative versions of these definitions and results, [17], see [19,
§3], as follows.

If ${\cal K}$ is a class of left $R$-modules then a right $R$-module is ${\cal
K}$-Mittag-Leffler if, for every set $\\{L_{i}\\}_{i\in I}$ of modules in
${\cal K}$, the canonical map $M\otimes_{R}\,(\prod_{i\in
I}\,L_{i})\to\prod_{i\in I}\,(M\otimes_{R}L_{i})$ is monic.

######  Lemma 4.13.

(see [19, 3.6]) If ${\cal K}$ is a class of left $R$-modules and $\langle{\cal
K}\rangle$ is the definable subcategory of $R\mbox{-}{\rm Mod}$ that it
generates, then a right module $M$ is ${\cal K}$-Mittag-Leffler iff it is
$\langle{\cal K}\rangle$-Mittag-Leffler.

Then, generalising 4.12 we have the following, where, given a definable
subcategory ${\cal D}$ of ${\rm Mod}\mbox{-}R$, we say that $M_{R}$, a module
not necessarily in ${\cal D}$, is ${\cal D}$-atomic if the pp-type ${\rm
pp}^{M}(\overline{a})$ of any finite tuple in $M$ is ${\cal D}$-finitely
generated (definition just after 2.8).

######  Theorem 4.14.

[17, 2.2] (see [19, 3.1, 3.6]) Suppose that ${\cal K}$ is a class of left
$R$-modules and $M$ is a right module. The following conditions on $M$ are
equivalent:

(i) $M$ is ${\cal K}$-Mittag-Leffler;

(ii) $M$ is ${\cal D}$-atomic where ${\cal D}=\langle{\cal K}\rangle^{\rm d}$
is the definable subcategory of ${\rm Mod}\mbox{-}R$ dual to the definable
subcategory $\langle{\cal K}\rangle$ of $R\mbox{-}{\rm Mod}$ generated by
${\cal K}$.

If $\langle{\cal K}\rangle=\langle L\rangle$, then another equivalent is:

(iii) for every finite tuple $\overline{a}$ from $M$, there is $\phi\in{\rm
pp}^{M}(\overline{a})$ such that the tensor-annihilator $\\{\overline{b}\text{
from }L:\overline{a}\otimes\overline{b}=0\\}$ of $\overline{a}$ in ${}_{R}L$
is exactly $D\phi(L)$.

Proof. The equivalence of (i) and (ii) is in the references but the
equivalence of (ii) and (iii) is not explicitly there. (ii)$\Rightarrow$(iii)
is by 4.6. For the other direction, we have, by 4.4, that the tensor-
annihilator of $\overline{a}$ in $L$ is $\bigcup_{\psi\in{\rm
pp}^{M}(\overline{a})}\,D\psi(L)$. Therefore, for $\psi\in{\rm
pp}^{M}(\overline{a})$, we have $D\psi(L)\leq D\phi(L)$, that is,
$D\psi\leq_{\langle L\rangle}D\phi$, equally $D\psi\leq_{\langle{\cal
K}\rangle}D\phi$. Therefore, by 2.10, $\psi\geq_{\cal D}\phi$, as required.
$\Box$

###  4.3 Mittag-Leffler and atomic modules with respect to a bimodule

######  Definition 4.15.

Suppose that $M_{R}$ is a right $R$-module and that ${}_{R}B_{S}$ is an
$(R,S)$-bimodule. Say that $M_{R}$ is atomic with respect to the bimodule
${}_{R}B_{S}$ if, for every $k$-partitioned finite tuple $\overline{a}$ from
$M$, and for every pp formula $\sigma(\overline{u})$, with $k$ free variables,
in the language of right $S$-modules, there is $\phi\in{\rm
pp}^{M}(\overline{a})$ such that $(\sigma:\phi)$ is maximal (and hence, by
4.9(2), the unique maximum) among pp-definable subgroups of ${}_{R}B_{S}$ of
the form $(\sigma:\psi)$ with $\psi\in{\rm pp}^{M}(\overline{a})$.

This definition might seem somewhat technical, though 4.17 shows its
naturality, so we explain how it generalises the usual atomic/Mittag-Leffler
conditions (both the original and the relative conditions as in Section 4.2).

Consider the case where $\sigma$ is the pp formula $u=0$ and consider the
above atomicity condition on $M_{R}$ and ${}_{R}B_{S}$ but required only for
this formula. Note first that the bimodule formula $(\sigma:\phi)$ then
becomes (see Section 4.1) $D\phi$ \- the elementary dual formula (usually
regarded as a formula for left $R$-modules but equally a pp formula for
$(R,{\mathbb{Z}})$-bimodules). Then, bearing in mind Herzog’s criterion, 4.4,
the above becomes the requirement that, for every finite tuple $\overline{a}$
from $M$, there is $\phi\in{\rm pp}^{M}(\overline{a})$ such that the tensor-
annihilator $\\{\overline{b}\in B:\overline{a}\otimes\overline{b}=0\\}$ of
$\overline{a}$ in ${}_{R}B$ is exactly $D\phi(_{R}B)$. By 4.14 that is exactly
the condition that $M$ is Mittag-Leffler with respect to (the definable
subcategory of $R\mbox{-}{\rm Mod}$ generated by) the left module ${}_{R}B$
(equally, in our terminology, with respect to the bimodule
${}_{R}B_{\mathbb{Z}}$). Equivalently, by 4.14, $M$ is atomic with respect to
the dual of that definable subcategory.

Therefore, by 4.14, the condition of $M$ being atomic with respect to
${}_{R}B_{S}$, restricted just to the pp formula $\sigma$ being $u=0$, is
equivalent to the condition that, for all
$L_{\lambda}\in\langle_{R}B\rangle=\langle_{R}B_{\mathbb{Z}}\rangle$, the
canonical morphism
$M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\to\prod_{\lambda}\,M\otimes_{R}L_{\lambda}$
is monic. We will show, 4.17, that the full atomicity condition is equivalent
to the condition that this morphism be a pure embedding of right $S$-modules.
Before showing that, we note the following (which extends 4.13).

######  Lemma 4.16.

If $R$ and $S$ are $K$-algebras for some commutative ring $K$ and if $M_{R}$
is atomic with respect to ${}_{R}B_{S}$, where $K$ acts centrally, then
$M_{R}$ is atomic with respect to every bimodule ${}_{R}L_{S}$ in the
definable subcategory of $R\otimes_{K}S^{\rm op}\mbox{-}{\rm Mod}$ generated
by $B$.

More precisely, for each pp formula $\sigma$ for right $S$-modules, and for
each partitioned tuple $\overline{a}$ from $M$, if
$(\sigma:\phi)(B)\geq(\sigma:\psi)(B)$ for every $\psi\in{\rm
pp}^{M}(\overline{a})$, then the same is true for each
$L\in\langle_{R}B_{S}\rangle$.

Proof. Given a pp formula $\sigma$ for right $S$-modules and a tuple
$\overline{a}$ from $M$, let $\phi\in{\rm pp}^{M}(\overline{a})$ be chosen for
$\sigma$ as in the definition (of atomic with respect to ${}_{R}B_{S}$). That
is, for every $\psi\in{\rm pp}^{M}(\overline{a})$, we have
$(\sigma:\psi)(_{R}B_{S})\leq(\sigma:\phi)(_{R}B_{S})$ and therefore, by 2.7,
this inclusion also holds for every $L\in\langle_{R}B_{S}\rangle$, so $M$ is
indeed atomic with respect to $L$. $\Box$

######  Theorem 4.17.

Suppose that $M_{R}$ is a right $R$-module and that ${}_{R}B_{S}$ is an
$(R,S)$-bimodule. Then $M_{R}$ is atomic with respect to ${}_{R}B_{S}$ iff,
for all $L_{\lambda}\in\langle_{R}B_{S}\rangle$, the canonical morphism
$M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\to\prod_{\lambda}\,M\otimes_{R}L_{\lambda}$
is a pure embedding of right $S$-modules.

Proof. The proof is just as for the case where $\sigma$ is $u=0$.

($\Rightarrow$) Just to simplify notation, we recall that, to check purity, it
is enough to consider only pp formulas $\sigma$ with one free variable, see
[11, 2.1.6]. We have already noted that the hypothesis implies that
$\iota:M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\to\prod_{\lambda}\,M\otimes_{R}L_{\lambda}$
is monic, so let $\overline{a}$ be from $M$ and
$\overline{l}=(\overline{l}_{\lambda})_{\lambda}$ be from
$\prod_{\lambda}\,L_{\lambda}$ and suppose that
$\prod_{\lambda}\,M\otimes_{R}L_{\lambda}\models\sigma(\iota(\overline{a}\otimes\overline{l}))$,
that is
$\prod_{\lambda}\,M\otimes_{R}L_{\lambda}\models\sigma((\overline{a}\otimes\overline{l}_{\lambda})_{\lambda})$
and so
$M\otimes_{R}L_{\lambda}\models\sigma(\overline{a}\otimes\overline{l}_{\lambda})$
for each $\lambda$. By 4.8 there is, for each $\lambda$, some
$\psi_{\lambda}\in{\rm pp}^{M}(\overline{a})$ such that
$L_{\lambda}\models(\sigma:\psi_{\lambda})(\overline{l}_{\lambda})$. We need a
uniform choice of $\psi_{\lambda}$. We have that because $M$ is atomic with
respect to $B$, and hence there is $\phi\in{\rm pp}^{M}(\overline{a})$ such
that $(\sigma:\phi)(B)\geq(\sigma:\psi)(B)$ for each $\psi\in{\rm
pp}^{M}(\overline{a})$. By 4.16, we have
$(\sigma:\phi)(L)\geq(\sigma:\psi_{\lambda})(L_{\lambda})$ and so
$L_{\lambda}\models(\sigma:\phi)(\overline{l}_{\lambda})$ for each $\lambda$.
Therefore $\prod_{\lambda}\,L_{\lambda}\models(\sigma:\phi)(\overline{l})$ and
hence, by 4.7,
$M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\models\sigma(\overline{a}\otimes\overline{l})$,
as required.

($\Leftarrow$) The argument reverses: given a pp formula $\sigma$ for right
$S$-modules and a partitioned tuple $\overline{a}$ from $M$, let $I$ be the
set of pairs $(\psi,\overline{l}^{\prime})$ where $\psi\in{\rm
pp}^{M}(\overline{a})$ and $\overline{l}^{\prime}\in(\sigma:\psi)(_{R}B_{S})$.
Form the product $B^{I}$ and define $\overline{l}\in B^{I}$ to be the element
whose coordinate at $(\psi,\overline{l}^{\prime})$ is $\overline{l}^{\prime}$.
Then, by 4.7, at each coordinate $(\psi,\overline{l}^{\prime})$ we have
$M\otimes_{R}B_{(\psi,\overline{l}^{\prime})}\models\sigma(\overline{a}\otimes\overline{l}^{\prime})$.
Therefore $M\otimes_{R}B^{I}\models\sigma(\overline{a}\otimes\overline{l})$
and hence, by 4.8, there is $\phi\in{\rm pp}^{M}(\overline{a})$ such that
$\overline{l}\in(\sigma:\phi)(B^{I})$. Therefore, looking at each coordinate,
we see $\overline{l}^{\prime}\in(\sigma:\phi)(B)$. So we have shown
$(\sigma:\phi)(B)\geq(\sigma:\psi)(B)$ for every $\psi\in{\rm
pp}^{M}(\overline{a})$, and $M$ is atomic with respect to $B$, as claimed.
$\Box$

######  Corollary 4.18.

If the bimodule ${}_{R}B_{S}$ has the ascending chain condition on pp-
definable subgroups, then every right $R$-module $M$ is atomic with respect to
$B$ and hence the canonical morphism $M\otimes_{R}B^{I}\to(M\otimes_{R}B)^{I}$
is a pure embedding of $S$-modules for each $I$, as is
$M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\to\prod_{\lambda}\,M\otimes_{R}L_{\lambda}$
if each $L_{\lambda}\in\langle_{R}B_{S}\rangle$.

Proof. It was already noted (parenthetically) in the definition at the start
of this section that, by 4.9(2), the set of $(\sigma:\phi)$ with $\phi$ in
some pp-type, is upwards-directed (since pp-types are closed under $\wedge$,
hence downwards-directed). So this is immediate from the definition. $\Box$

######  Corollary 4.19.

Given an $(R,S)$-bimodule $B$, the class of right $R$-modules which are atomic
with respect to $B$ is closed under arbitrary direct sums and pure submodules.

That easily follows from the definition but is also immediate from 4.17.

######  Corollary 4.20.

If $M_{R}$ is a Mittag-Leffler module and, for each $\lambda$, $L_{\lambda}$
is an $(R,S)$-bimodule, then the canonical map
$M\otimes_{R}\prod_{\lambda}\,L_{\lambda}\to\prod_{\lambda}\,M\otimes_{R}L_{\lambda}$
is a pure embedding of right $S$-modules.

Proof. That follows from 4.17 since any Mittag-Leffler module is atomic with
respect to any bimodule because each pp-type realised in a Mittag-Leffler
module is finitely generated, 4.12. But we note that this is easy to prove
directly since, in order to prove $S$-purity of the map it is enough to show
that tensoring it with any finitely presented left $S$-module $F$ gives an
embedding. The tensored map is

$\big{(}M\otimes_{R}\prod_{\lambda}\,L_{\lambda})\otimes_{S}F\to\big{(}\prod_{\lambda}\,M\otimes_{R}L_{\lambda})\otimes_{S}F.$

Since ${}_{S}F$ is finitely presented, $-\otimes_{S}F$ commutes with direct
products [5] so, on the left, we have

$\big{(}M\otimes_{R}\prod_{\lambda}\,L_{\lambda})\otimes_{S}F\simeq
M\otimes_{R}\big{(}\prod_{\lambda}\,L_{\lambda})\otimes_{S}F\simeq
M\otimes_{R}\prod_{\lambda}\,(L_{\lambda}\otimes_{S}F)$

and, on the right, we have

$\big{(}\prod_{\lambda}\,M\otimes_{R}L_{\lambda})\otimes_{S}F\simeq\prod_{\lambda}\,M\otimes_{R}L_{\lambda}\otimes_{S}F.$

The map between these two modules is monic by applying the Mittag-Leffler
property of $M$ with the left $R$-modules ${}_{R}L_{\lambda}\otimes_{S}F$, as
required. $\Box$

######  Remark 4.21.

In view of 4.20, we should show that $M_{R}$ being atomic with respect to a
bimodule ${}_{R}L_{S}$ is genuinely stronger than requiring that it be atomic
with respect to the left module ${}_{R}L$. Looking at the proof of 4.20, we
see that we must show that, given $M_{R}$ and ${}_{R}L_{S}$, if $M_{R}$ is
atomic with respect to ${}_{R}L$, it need not be atomic with respect to every
module ${}_{R}L\otimes_{S}F$ where ${}_{S}F$ is a finitely presented left
$S$-module.

To get examples, consider ${}_{R}L_{R}=\,_{R}R_{R}$. By [1, 2.7], $M_{R}$ is
atomic with respect to ${}_{R}R$ iff it is atomic with respect to all flat
left $R$-modules iff for every finitely generated submodule $A$ of $M$, the
inclusion of $A$ into $M$ factors through a finitely presented module. On the
other hand, if $M_{R}$ is atomic with respect to every module of the form
${}_{R}R\otimes_{R}F$ with ${}_{R}F$ finitely presented, then $M_{R}$ is, by
4.13, atomic with respect to $R\mbox{-}{\rm Mod}$, that is $M_{R}$ is Mittag-
Leffler.

So, if $R$ is right noetherian, then every module $M_{R}$ is atomic with
respect to ${}_{R}R$ but, unless $R$ is right pure semisimple, not every such
module is atomic with respect to ${}_{R}R_{R}$.

The conclusion of the next result is similar to that of 4.17 but the
hypothesis is rather different.

######  Proposition 4.22.

Suppose that ${}_{R}B$ is a pure-projective left $R$-module with endomorphism
ring $S$. Then for any set $(M_{\lambda})_{\lambda}$ of right $R$-modules, the
canonical map
$f:(\prod_{\lambda}\,M_{\lambda})\otimes_{R}B_{S}\to\prod_{\lambda}\,(M_{\lambda}\otimes_{R}B_{S})$
is a pure embedding of right $S$-modules.

Proof. The map is an embedding because ${}_{R}B$ is Mittag-Leffler by 4.12. To
check $S$-purity it is sufficient to show that, for every finitely presented
left $S$-module $L$, the map
$f\otimes_{S}L:(\prod_{\lambda}\,M_{\lambda})\otimes_{R}B\otimes_{S}L\to\big{(}\prod_{\lambda}\,(M_{\lambda}\otimes_{R}B_{S})\big{)}\otimes_{S}L$
is monic. As seen in the proof of 4.20, since ${}_{S}L$ is finitely presented
$\big{(}\prod_{\lambda}\,(M_{\lambda}\otimes_{R}B_{S})\big{)}\otimes_{S}L\simeq\prod_{\lambda}\,(M_{\lambda}\otimes_{R}B\otimes_{S}L)$
so, to show that $f\otimes_{S}L$ is monic, it is sufficient to show that
${}_{R}B\otimes_{S}L$ is Mittag-Leffler.

Suppose first that $B$ is a direct sum of finitely presented modules. Take a
finite presentation ${}_{S}S^{m}\to\,_{S}S^{n}\to L\to 0$ of $L$, so we have
an exact sequence
${}_{R}B\otimes_{S}S^{m}\to\,_{R}B\otimes_{S}S^{n}\to\,_{R}B\otimes_{S}L\to
0$. Take any finite tuple $\overline{a}$ from $B\otimes_{S}L$. Since $B$ is a
direct sum of finitely presented modules, there is a finitely presented direct
summand $F$ of $B$ such that $\overline{a}$ is contained in $F\otimes_{S}L$
which, since $\otimes$ commutes with direct sums, is a direct summand of
$F\otimes_{S}B$; therefore the pp-type of $\overline{a}$ in
${}_{R}B\otimes_{S}L$ is equal to the pp-type of $\overline{a}$ in
${}_{R}F\otimes_{S}L$. But, from the exact sequence
${}_{R}F\otimes_{S}S^{m}\to\,_{R}F\otimes_{S}S^{n}\to\,_{R}F\otimes_{S}L\to
0$, that is ${}_{R}F^{m}\to\,_{R}F^{n}\to\,_{R}F\otimes_{S}L\to 0$, we see
that $F\otimes_{S}L$ is a finitely presented left $R$-module, and hence the
pp-type of $\overline{a}$ in ${}_{R}F\otimes_{S}L$ is finitely generated.
Therefore, by 4.12, ${}_{R}B\otimes_{S}L$ is a Mittag-Leffler left $R$-module.

For the general case, ${}_{R}B$ is a direct summand of a direct sum of
finitely presented left $R$-modules so, referring to the proof above,
${}_{R}B\otimes_{S}L$ is a direct summand of a Mittag-Leffler left $R$-module
and so is itself Mittag-Leffler, as required. $\Box$

###  4.4 Tensoring by a bimodule

The functor $M_{R}\mapsto M\otimes_{R}B_{S}$ links definable subcategories of
${\rm Mod}\mbox{-}R$, ${\rm Mod}\mbox{-}S$ and $R\mbox{-}{\rm Mod}\mbox{-}S$,
the last being the category of $(R,S)$-bimodules.

For a definable subcategory ${\cal D}$ of ${\rm Mod}\mbox{-}R$, we write
${\cal D}\otimes_{R}B_{S}$ for the definable subcategory of ${\rm
Mod}\mbox{-}S$ generated by the collection of $M\otimes_{R}B_{S}$ with
$M\in{\cal D}$. We have seen already at the beginning of Section 4 that this
operation is not very well-behaved in general but, in the case that ${}_{R}B$
is pure-projective, we obtain a simple description of ${\cal
D}\otimes_{R}B_{S}$. First a lemma.

######  Lemma 4.23.

If $M^{\prime}\to M$ is a pure embedding of right $R$-modules and if
${}_{R}B_{S}$ is a bimodule, then $M^{\prime}\otimes_{R}B_{S}\to
M\otimes_{R}B_{S}$ is a pure embedding of right $S$-modules.

If $M$ is a right $R$-module and $B^{\prime}\to B$ is a pure embedding of
$(R,S)$-bimodules, then $M\otimes_{R}B^{\prime}_{S}\to M\otimes_{R}B_{S}$ is a
pure embedding of right $S$-modules.

Proof. For the first statement, we have the obvious algebraic proof, as
follows. An embedding of right modules is pure iff tensoring with any left
module gives an embedding. So it must be shown that
$(M^{\prime}\otimes_{R}B_{S})\otimes_{S}L\to(M\otimes_{R}B_{S})\otimes_{S}L$,
that is $M^{\prime}\otimes_{R}(B_{S}\otimes_{S}L)\to
M\otimes_{R}(B_{S}\otimes_{S}L)$, is monic. But that is so since
$M^{\prime}\to M$ is pure.

We have an equally short proof using pp formulas. Suppose that
$M\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$ where
$\sigma$ is a pp formula for $S$-modules, $\overline{a}$ is from $M^{\prime}$
and $\overline{b}$ is from $B$. By 4.8 we have $M\models\phi(\overline{a})$
for some pp formula $\phi$ for $R$-modules and
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$. Since $M^{\prime}$ is pure in
$M$, we have $M^{\prime}\models\phi(\overline{a})$ and so, by 4.7, we have
$M^{\prime}\otimes B\models\sigma(\overline{a}\otimes\overline{b})$, as
required.

For the second statement, suppose that
$M\otimes_{R}B_{S}\models\sigma(\overline{a}\otimes\overline{b})$ where
$\sigma$ is a pp formula for $S$-modules, $\overline{a}$ is from $M$ and
$\overline{b}$ is from $B^{\prime}$. By 4.8 we have
$M\models\phi(\overline{a})$ for some pp formula $\phi$ for $R$-modules and
${}_{R}B_{S}\models(\sigma:\phi)(\overline{b})$. Since $B^{\prime}$ is pure in
$B$, we have ${}_{R}B^{\prime}_{S}\models(\sigma:\phi)(\overline{b})$ and so,
by 4.7, we have $M\otimes
B^{\prime}\models\sigma(\overline{a}\otimes\overline{b})$, as required. $\Box$

######  Theorem 4.24.

Suppose that the $(R,S)$-bimodule $B$ is pure-projective as a left $R$-module.
Let ${\cal D}$ be a definable subcategory of ${\rm Mod}\mbox{-}R$. Then ${\cal
D}\otimes_{R}B_{S}$ is the closure of $\\{M\otimes_{R}B_{S}:M\in{\cal D}\\}$
under $S$-pure submodules and, if $N$ is an elementary cogenerator of ${\cal
D}$, then every member of ${\cal D}\otimes_{R}B_{S}$ purely embeds in a power
of $N\otimes_{R}B_{S}$, the pure-injective hull of which is therefore an
elementary cogenerator of ${\cal D}\otimes_{R}B_{S}$.

Proof. The functor $-\otimes_{R}B_{S}$ commutes with directed colimits. Also,
by 4.22, for all right $R$-modules $M_{i}$, the canonical morphism
$(\prod_{i}\,M_{i})\otimes_{R}B\to\prod_{i}\,(M_{i}\otimes_{R}B)$ is a pure
embedding of right $S$-modules. So, since products and direct limits of pure
embeddings are pure embeddings, the only operation required to extend
$\\{M\otimes_{R}B_{S}:M\in{\cal D}\\}$ to a definable subcategory is closure
under $S$-pure submodules.

For the second statement, every $M\in{\cal D}$ is a pure submodule of some
power $N^{I}$ of $N$, so we have, by 4.23, $M\otimes_{R}B_{S}$ pure in
$N^{I}\otimes_{R}B_{S}$ which, by 4.22, is $S$-pure in
$(N\otimes_{R}B_{S})^{I}$. Therefore any $S$-pure submodule of
$M\otimes_{R}B_{S}$ will be $S$-pure in a product of copies of
$N\otimes_{R}B_{S}$, as claimed. $\Box$

If ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$ and ${\cal
E}$ is a definable subcategory of ${\rm Mod}\mbox{-}S$, then denote by $({\cal
E}:{\cal D})$ the category of $(R,S)$-bimodules which, by tensor, send ${\cal
D}$ into ${\cal E}$:

$({\cal E}:{\cal D})=\\{_{R}B_{S}:\forall\,D\in{\cal
D},\,\,D\otimes_{R}B_{S}\in{\cal E}\\}.$

It is proved in [10, 2.3] that $({\cal E}:{\rm Mod}\mbox{-}R)$ is a definable
subcategory of $R\mbox{-}{\rm Mod}\mbox{-}S$. We generalise that here as
follows.

######  Theorem 4.25.

Suppose that ${\cal D}$ is a definable subcategory of ${\rm Mod}\mbox{-}R$
which is $\varinjlim$-generated by modules which are atomic with respect to
all $(R,S)$-bimodules (for example, suppose that ${\cal D}$ is generated by
the finitely presented modules in it). Let ${\cal E}$ be a definable
subcategory of ${\rm Mod}\mbox{-}S$. Then $({\cal E}:{\cal D})$ is a definable
subcategory of $R\mbox{-}{\rm Mod}\mbox{-}S$.

Proof. In contrast to the proof in [10] we give an algebraic proof here. If
$B=\varinjlim\,B_{i}$ is a directed colimit of bimodules $B_{i}\in({\cal
E}:{\cal D})$ and if $M\in{\cal D}$, so each $M\otimes B_{i}\in{\cal E}$ then,
since $M\otimes-$ commutes with directed colimits, we have $M\otimes B\in{\cal
E}$ and hence $B\in({\cal E}:{\cal D})$. If $B^{\prime}$ is a pure subbimodule
of $B\in({\cal E}:{\cal D})$, then, by 4.23, if $M\in{\cal D}$, we have
$M\otimes B^{\prime}$ pure in $M\otimes B$, hence $M\otimes B^{\prime}\in{\cal
E}$ and so $B^{\prime}\in({\cal E}:{\cal D})$.

Suppose that $M\in{\cal D}$ is atomic for $R\mbox{-}{\rm Mod}\mbox{-}S$ and
that $B_{i}\in({\cal E}:{\cal D})$, $i\in I$. The embedding
$M\otimes\prod_{i}\,B_{i}\to\prod M\otimes B_{i}$ is $S$-pure by 4.17, so
$M\otimes\prod_{i}\,B_{i}\in{\cal E}$, and therefore $\prod_{i}B_{i}\in({\cal
E}:{\cal D})$. Now take any $M\in{\cal D}$. By assumption
$M=\varinjlim_{j}M_{j}$ with the $M_{j}$ $R\mbox{-}{\rm Mod}\mbox{-}S$-atomic
modules in ${\cal D}$. We have
$M\otimes\prod_{i}\,B_{i}=(\varinjlim\,M_{j})\otimes\prod_{i}\,B_{i}=\varinjlim\,(M_{j}\otimes\prod_{i}\,B_{i})$
and this is in ${\cal E}$ since each $M_{j}\otimes\prod_{i}\,B_{i}$ is in
${\cal E}$. Therefore $({\cal E}:{\cal D})$ is also closed under direct
products, so is definable as claimed. $\Box$

That some restriction on ${\cal D}$ is needed is shown by the following
example.

######  Example 4.26.

[10, after 2.3] Take $R=S={\mathbb{Z}}$, ${\cal
D}=\langle{\mathbb{Q}}\rangle$, ${\cal E}=0$. Then every torsion module is in
$({\cal E}:{\cal D})$, yet ${\mathbb{Q}}$, which is in the definable
subcategory generated by the torsion modules, is not.

For ${\cal D}$ a definable subcategory of ${\rm Mod}\mbox{-}R$ and ${\cal B}$
a definable subcategory of $R\mbox{-}{\rm Mod}\mbox{-}S$, we define ${\cal
D}\otimes{\cal B}$ to be the definable subcategory of ${\rm Mod}\mbox{-}S$
generated by the class of modules of the form $M\otimes_{R}B_{S}$ with
$M\in{\cal D}$ and $B\in{\cal B}$. This ‘tensor’ operation on definable
subcategories was looked at in [10, §3], though done there in terms of
theories of modules and closed subsets of the Ziegler spectrum, rather than in
terms of definable subcategories. Here we reconsider and extend those results.

In view of 4.26 it is necessary to impose some conditions on $M$ and $B$ in
order for $M\otimes B$ to be a definable generator for $\langle
M\rangle\otimes\langle B\rangle$. It was shown in [10, 3.1] that, if $M$ and
$B$ are weakly saturated, then $\langle M\otimes B\rangle=\langle
M\rangle\otimes\langle B\rangle$, where $M$ is weakly saturated if, for every
tuple $\overline{b}$ from some $M^{\prime}\in\langle M\rangle$, there is
$\overline{a}$ from $M$ with ${\rm pp}^{M}(\overline{a})={\rm
pp}^{M^{\prime}}(\overline{b})$.444Provided every pp-quotient
$\phi(M)/\psi(M)$ is infinite if nonzero, then this is equivalent to the usual
model-theoretic definition of weak saturation.

We will say that a module $M\in{\cal D}$ is a pure cogenerator for the
definable subcategory ${\cal D}$ (necessarily ${\cal D}=\langle M\rangle$) if
every $D\in{\cal D}$ purely embeds in a power $M^{I}$ of $M$. Note that an
elementary cogenerator for ${\cal D}$ is exactly a pure-injective pure
cogenerator for ${\cal D}$. These various notions of largeness in ${\cal D}$
are quite closely related.

######  Lemma 4.27.

Let ${\cal D}=\langle M\rangle$ be the definable subcategory generated by a
module $M$.

(1) If $M$ is a pure cogenerator for ${\cal D}$ then its pure-injective hull
is an elementary cogenerator for ${\cal D}$.

(2) If $M$ is a pure cogenerator then some power of $M$ is weakly saturated.

(3) If $M$ is weakly saturated and pure-injective, then $M$ is an elementary
cogenerator.

Proof. (1) is by the definitions. For (2), since each member of $\langle
M\rangle$ is purely embedded in some power of $M$, each pp-type realised in
some member of $\langle M\rangle$ is realised in some power of $M$. Since
there is just a set of pp-types, there is a power of $M$ large enough to
realise every such pp-type, so that power is weakly saturated. For (3), let
$M^{\prime}\in\langle M\rangle$. For each element $a\in M^{\prime}$, there is,
by hypothesis, some $b\in M$ such that ${\rm pp}^{M}(b)={\rm
pp}^{M^{\prime}}(a)$. By [11, 4.3.9], there is, since $M$ is pure injective, a
morphism $f_{a}:M^{\prime}\to M$ with $fa=b$. Consider $\prod_{a\in
M^{\prime}}\,f_{a}:M^{\prime}\to M^{M^{\prime}}$. By construction this is a
pure embedding and so $M$ is an elementary cogenerator. $\Box$

For a definable subcategory ${\cal D}$, say that $M\in{\cal D}$ is finitely
generic for ${\cal D}$ if, for every pp formula $\phi$, there is
$\overline{a}$ from $M$ such that ${\rm pp}^{M}(\overline{a})$ is ${\cal
D}$-generated by $\phi$, that is, ${\rm
pp}^{M}(\overline{a})=\\{\phi^{\prime}:\phi\leq_{\cal D}\phi^{\prime}\\}$.

######  Lemma 4.28.

Every definable category ${\cal D}$ contains a ${\cal D}$-atomic finitely
generic module for ${\cal D}$.

If $M$ is a pure cogenerator for ${\cal D}$, then some power of $M$ is
finitely generic for ${\cal D}$.

Proof. If $\phi$ is a pp formula then there is a ${\cal D}$-atomic module
$M_{\phi}$ in ${\cal D}$ which contains a tuple with pp-type ${\cal
D}$-generated by $\phi$ \- [6], see [13, 4.10]. Take the direct sum of these
modules as $\phi$ ranges over pp formulas. Then this is ${\cal D}$-atomic
([13, 4.4]) and, by construction, is finitely generic.

For the second statement, if $M$ is a pure cogenerator for ${\cal D}$ and if
$\phi$ is a pp formula, then $M_{\phi}$ as above purely embeds in some power
of $M$. Since there is just a set of pp formulas, a suitable large power of
$M$ embeds all such $M_{\phi}$, and hence is finitely generic for ${\cal D}$.
$\Box$

Note that a ${\cal D}$-atomic finitely generic module for ${\cal D}$ will not
normally be an elementary cogenerator, or even a pure cogenerator. For
instance, in the case ${\cal D}={\rm Mod}\mbox{-}R$, the direct sum of one
copy of each finitely presented module will be finitely generic, whereas a
pure cogenerator must have, as a direct summand, a copy of the injective hull
of each simple module.

######  Proposition 4.29.

([10, 2.4]) If ${\cal D}$ a definable subcategory of ${\rm Mod}\mbox{-}R$ and
${\cal B}$ a definable subcategory of $R\mbox{-}{\rm Mod}\mbox{-}S$, then the
following are equivalent:

(i) the pp-pair $\sigma/\tau$ is closed on ${\cal D}\otimes{\cal B}$;

(ii) for all pp formulas $\phi$ for $R$-modules, there is a pp formula
$\phi^{\prime}$ for $R$-modules such that $\phi\leq_{\cal D}\phi^{\prime}$ and
such that $(\sigma:\phi)\leq_{\cal B}(\tau:\phi^{\prime})$;

Proof. (ii)$\Rightarrow$(i) Suppose that $B\in{\cal B}$ and
$B\models\sigma(\overline{a}\otimes\overline{b})$ where $\overline{a}$ is from
$M\in{\cal D}$ and $\overline{b}$ is from $B$. Then
$M\models\phi(\overline{a})$ and $B\models(\sigma:\phi)(\overline{b})$ for
some pp formula $\phi$ for $R$-modules. By assumption there is $\phi^{\prime}$
such that $\phi\leq_{\cal D}\phi^{\prime}$, so
$M\models\phi^{\prime}(\overline{a})$, and also
$B\models(\tau:\phi^{\prime})(\overline{b})$. Therefore, by 4.7, we have
$M\otimes B\models\tau(\overline{a}\otimes\overline{b})$, as required.

(i)$\Rightarrow$(ii) Suppose that $\sigma/\tau$ is closed on ${\cal
D}\otimes{\cal B}$. Choose, by 4.28, $M$ finitely generic in ${\cal D}$ and
$B$ finitely generic in ${\cal B}$. Let $\phi$ be a pp formula for $R$-modules
and choose $\overline{a}$ whose pp-type in $M$ is ${\cal D}$-generated by
$\phi$. Also choose $\overline{b}$ from $B$ whose pp-type in $B$ is ${\cal
B}$-generated by $(\sigma:\phi)$. Certainly $M\otimes
B\models\sigma(\overline{a}\otimes\overline{b})$; by assumption $M\otimes
B\models\tau(\overline{a}\otimes\overline{b})$. Therefore there is
$\phi^{\prime}\in{\rm pp}^{M}(\overline{a})$ such that
$(\tau:\phi^{\prime})\in{\rm pp}^{B}(\overline{b})$. By choice of
$\overline{a}$, and $\overline{b}$ we have $\phi\leq_{\cal D}\phi^{\prime}$
and $(\sigma:\phi)\leq_{\cal B}(\tau:\phi^{\prime})$, as required. $\Box$

######  Theorem 4.30.

(cf. [10, 3.1]) If $M$ is finitely generic in the definable subcategory ${\cal
D}$ of ${\rm Mod}\mbox{-}R$ and $B$ is finitely generic in the definable
subcategory ${\cal B}$ of $R\mbox{-}{\rm Mod}\mbox{-}S$, then $\langle
M\otimes_{R}B_{S}\rangle={\cal D}\otimes{\cal B}$.

Proof. Certainly $M\otimes_{R}B_{S}\in{\cal D}\otimes{\cal B}$. For the
converse, suppose that $\sigma/\tau$ is open on ${\cal D}\otimes{\cal B}$. By
4.29 there is a pp formula $\phi$ for $R$-modules such that

($\ast$) for every $\phi^{\prime}$ with $\phi\leq_{\cal D}\phi^{\prime}$ we do
not have $(\sigma:\phi)\leq_{\cal B}(\tau:\phi^{\prime})$.

By assumption, there is $\overline{a}$ from $M$ with the pp-type of
$\overline{a}$ in $M$ being ${\cal D}$-generated by $\phi$. Also, by
assumption there is $\overline{b}$ from $B$ with the ${\rm
pp}^{B}(\overline{b})$ being ${\cal B}$-generated by $(\sigma:\phi)$.
Certainly then $M\otimes B\models\sigma(\overline{a}\otimes\overline{b})$. If
we had $M\otimes B\models\tau(\overline{a}\otimes\overline{b})$, then there
would be $\phi^{\prime}\in{\rm pp}^{M}(\overline{a})$ such that
$(\tau:\phi^{\prime})\in{\rm pp}^{B}(\overline{b})$. By choice of
$\overline{a}$, $\phi\leq_{\cal D}\phi^{\prime}$ and by choice of
$\overline{b}$, $(\sigma:\phi)\leq_{\cal B}(\tau:\phi^{\prime})$,
contradicting ($\ast$). Therefore $\sigma/\tau$ is open on $M\otimes B$, as
required (as commented after 2.1). $\Box$

##  5 Extension along elementary embeddings

###  5.1 Elementary extensions of rings and algebras

In this section we will suppose that $f:R\to S$ is an elementary extension of
rings. For instance, suppose that $R$ is a finite-dimensional algebra over an
algebraically closed field $K$ and let $L$ be an algebraically closed
extension field of $K$. Then the embedding $R\to S=R\otimes_{K}L$ is555Sketch
proof: choose a $K$-basis $(a_{1},\dots,a_{n})$ for $R$ over $K$ and let the
$\gamma_{ijk}\in K$ be such that $a_{i}a_{j}=\sum_{k}\,\gamma_{ijk}a_{k}$.
Then $R$ can be “coded up” within $K$, being identified with the vector space
$K^{n}$ endowed with the multiplication
$(\alpha_{1},\dots,\alpha_{n})\cdot(\beta_{1},\dots,\beta_{n})=(\sum_{ij}\,\gamma_{ij1}\alpha_{i}\beta_{j},\dots,\sum_{ij}\,\gamma_{ijn}\alpha_{i}\beta_{j})$.
Since $L$ is an elementary extension of $K$ (e.g. [4, A.5.1]), the resulting
“coded up” structure within $L$, namely $S$, is an elementary extension of
$R$; cf. [16, 1.1]. an example of an elementary extension of rings, meaning
that if $\sigma$ is a sentence (a formula without free variables) in the
language for rings and possibly naming parameters from $R$, then $\sigma$ is
true in $R$ iff it is true in $S$. In this situation we also say that $R$ is
an elementary subring of $S$ and write $R\prec S$. For instance, if $R\to S$
is an elementary extension, then $r\in R$ belongs to the Jacobson radical of
$R$ iff it belongs to the Jacobson radical of $S$, because we can take
$\sigma$ to be the sentence $\forall y\exists z\,((1+ry)z=1=z(1+ry))$. This is
a stronger condition than $R$ and $S$ being elementarily equivalent which
requires only that they satisfy the same sentences (without extra parameters
from $R$) in the language of rings.

In fact, for many of the results we require only that the embedding $R\to S$
be an elementary extension of $(R,R)$-bimodules (the sentences used in the
proofs will involve multiplications of elements of $R$ with elements of $S$
but not arbitary multiplications of elements of $S$).

######  Theorem 5.1.

(e.g. [4, 9.5.2] and then 5.4 below for the second statement) If $S$ is any
first-order structure (a module, a ring, …) and ${\cal U}$ is an ultrafilter
on an index set $I$, then the diagonal embedding $s\mapsto(s)_{i}/{\cal U}$
from $S$ to its ultrapower $S^{I}/{\cal U}$, is an elementary extension (of
modules, rings, …).

If $S\to S^{\prime}$ is an elementary embedding of structures, then
$S^{\prime}$ is an elementary substructure of some ultrapower of $S$.

If $R$ is a finite-dimensional algebra over a field $K$ and $S$ is an
elementary extension of $R$, then $S$ need not be a finite-dimensional algebra
over a field. For an example, take $R=Q\times Q$ where $Q$ is the field of
algebraic complex numbers. Then $Q$ is an elementary substructure of any
algebraically closed field of characteristic $0$ (e.g. [4, A.5.1]), in
particular $Q\to{\mathbb{C}}$ is an elementary embedding. By the Feferman-
Vaught Theorem (see [4, 9.6.5(b)]) $Q\times Q\to Q\times{\mathbb{C}}$ is an
elementary embedding but clearly the latter is not a finite-dimensional
algebra over any field. However, for elementary extensions that are
ultrapowers, the situation is nicer and we will make use of the above fact 5.1
that, if $R$ is a finite-dimensional algebra and $R\to S$ is an elementary
extension, then $S$ is an elementary subring of some ultrapower of $R$ and
those ultrapowers have a simple description.

######  Lemma 5.2.

Suppose that $R$ is a finite-dimensional algebra over a field $K$, with
$K$-basis $\beta_{1},\dots,\beta_{t}$. Let $R^{\ast}=R^{I}/{\cal U}$ be any
ultrapower of $R$. Then $R^{\ast}$ is an algebra over the field $K^{I}/{\cal
U}$ with basis $\beta_{1},\dots,\beta_{t}$ (and the algebraic relations
between $\beta_{1},\dots,\beta_{t}$ as elements of $R^{\ast}$ are generated by
the relations between them as elements of $R$).

Proof. Recall (cf. e.g. [11, 3.3.1]) that $R^{I}/{\cal U}$ can be constructed
as the factor ring of $R^{I}$ by the ideal of elements $(r_{i})_{i}$ with
$\\{i\in I:r_{i}=0\\}\in{\cal U}$. Every element $r^{\ast}$ of $R^{I}/{\cal
U}$ has the form $(r_{i})_{i}/{\cal U}$ where
$r_{i}=\sum_{j=1}^{t}\,\alpha_{ij}\beta_{j}$, so $r^{\ast}$ may be written as
$\sum_{j=1}^{t}\,(\alpha_{ij})_{i}/{\cal U}\cdot\beta_{j}$, where we are
identifying the elements $\beta_{j}\in R$ with their images
$(\beta_{j})_{i}/{\cal U}$ under the diagonal embedding of $R$ into
$R^{\ast}$. Thus (the images of) $\beta_{1},\dots,\beta_{t}$ in $R^{\ast}$
form a generating set for $R^{\ast}$ over $K^{\ast}=K^{I}/{\cal U}$. To see
that they are linearly independent over $K^{\ast}$, suppose that
$\sum_{i}^{t}\,\alpha^{\ast}_{j}\beta_{j}=0$ where
$\alpha_{j}^{\ast}=(\alpha_{ij})_{i}/{\cal U}$. Then the set of $i\in I$ such
that $\sum_{1}^{t}\,\alpha_{ij}\beta_{j}=0$ is in ${\cal U}$; since the
$\beta_{j}$ are linearly independent over $K$, the $\alpha_{ij}$ appearing in
those zero relations all are $0$; in particular, for each $j$, $\\{i\in
I:\alpha_{ij}=0\\}$ is in ${\cal U}$ and hence each $\alpha_{j}^{\ast}=0$, as
required. $\Box$

######  Remark 5.3.

Note that if $R$ is a finite-dimensional algebra over a field $K$, then the
bimodule ${}_{R}R_{R}$, that is, $R$ as a right module over the $K$-algebra
$R^{\rm op}\otimes_{K}R$, being of finite length as a $K$-module, has both acc
and dcc on pp-definable subgroups.

######  Theorem 5.4.

(Keisler-Shelah Theorem) [21] If $M$, $N$ are structures (for the same first
order, finitary language), then $M$ and $N$ are elementarily equivalent if and
only if they have isomorphic ultrapowers $M^{I}/{\cal U}\simeq N^{J}/{\cal V}$
for some sets $I$, $J$ and ultrafilters ${\cal U}$, ${\cal V}$ on
them666Indeed one may even require $I=J$ and ${\cal U}={\cal V}$..

Another useful result is the Downwards Löwenheim-Skolem Theorem which, for
rings, and in the countable version, is the following.

######  Theorem 5.5.

Suppose that $X$ is a countable subset of the ring $S$. Then there is a
countable elementary subring of $S$ which contains $X$.

The more general theorem is as follows.

######  Theorem 5.6.

Suppose that $S$ is a first-order ${\cal L}$-structure where the cardinality
of the language ${\cal L}$ is $\kappa$; suppose that $X$ is a subset of $S$.
Then there is an elementary substructure of $S$ which contains $X$ and is of
cardinality $\leq|X|+\kappa$.

By the cardinality of a language we mean $\aleph_{0}$ plus the number of extra
(constant, function, relation) symbols in the language. For instance the
cardinality of the language of rings is $\aleph_{0}$, the cardinality of the
language of $R$-modules is $\aleph_{0}+|R|$, and the cardinality of the
language for treating elementary extensions of a ring $R$ is $\aleph_{0}+|R|$
(because the sentences in the definition of elementary extension may use
parameters from $R$).

We recall the condition ([8, 8.10], see [11, 1.1.13]) for one pp formula to
imply another.

######  Theorem 5.7.

Suppose that $\psi(\overline{x})$ and $\phi(\overline{x})$ are pp formulas in
the language of $R$-modules; say $\psi(\overline{x})$ is
$\exists\overline{z}\,\big{(}(\overline{x}\,\overline{z})H_{\psi}=0\big{)}$
and $\phi$ is
$\exists\overline{y}\,\big{(}(\overline{x}\,\overline{y})H_{\phi}=0\big{)}$,
where $H_{\phi}$ and $H_{\psi}$ are matrices with entries in $R$. Then
$\psi\leq\phi$ iff there are matrices $G=\left(\begin{array}[]{c}G^{\prime}\\\
G^{\prime\prime}\end{array}\right)$ and $K$, with entries in $R$, such that

$\left(\begin{array}[]{cc}I&G^{\prime}\\\
0&G^{\prime\prime}\end{array}\right)H_{\phi}=H_{\psi}K$

(here $I$ is the $n\times n$ identity matrix where $n$ is the length of
$\overline{x}$, and $0$ denotes a zero matrix with $n$ columns).

We noted, at the beginning of Section 3, that if $R\to S$ is a morphism of
rings, then, for each $n$ there is an induced lattice homomorphism ${\rm
pp}^{n}_{R}\to{\rm pp}^{n}_{S}$, where ${\rm pp}^{n}_{R}$ is the lattice of
equivalence classes of pp formulas for right $R$-modules with $n$ free
variables. In the case of an elementary embedding, this is an embedding of
lattices.

######  Corollary 5.8.

([9, Prop. 3]) If $f:R\to S$ is an elementary embedding of rings (or just of
$(R,R)$-bimodules), then $\phi\to f_{\ast}\phi$ induces, for each $n$, an
embedding of the lattice ${\rm pp}^{n}_{R}$ into the lattice ${\rm
pp}^{n}_{S}$. In particular $\psi\leq\phi$ for $R$-modules iff
$f_{\ast}\psi\leq f_{\ast}\phi$ for $S$-modules.

Proof. The direction $\Rightarrow$ in the second sentence needs no assumptions
and is 3.2. For the other direction, suppose that $\phi$, $\psi$ are pp
formulas for $R$-modules and that $f_{\ast}\psi\leq f_{\ast}\phi$. Then there
are matrices $G$ and $K$ with entries in $S$ making the matrix equation in the
statement of 5.7 true. We can write that matrix equation as a conjunction,
$\theta(\overline{r},\overline{s})$, of equations in the language of rings and
with parameters $\overline{r}$ \- the elements of $R$ which appear in
$H_{\phi}$ and $H_{\psi}$ \- and parameters $\overline{s}$ \- the elements of
$S$ which appear in $G$. Existentially quantifying out the parameters
$\overline{s}$, we obtain
$\exists\overline{y}\,\theta(\overline{r},\overline{y})$ \- a sentence in the
language of rings, indeed a sentence in the language of $(R,R)$-bimodules,
with parameters from $R$. That sentence is true in $S$ so, since $R$ is an
elementary subring (or subbimodule) of $S$, it is true in $R$. That is, there
is a matrix $G^{\prime}$ with entries in $R$ which solves the matrix equation
in the statement of 5.7, and so $\psi\leq\phi$ for $R$-modules, as required.
$\Box$

######  Lemma 5.9.

([9, Prop. 18]) Suppose that $R\prec S$. If $M\in{\rm Mod}\mbox{-}R$, then the
canonical $R$-linear map $M\to M\otimes_{R}S$, given by $m\mapsto m\otimes
1_{S}$, is a pure embedding of $R$-modules. In particular, $M$ is in the
definable subcategory of ${\rm Mod}\mbox{-}R$ generated by
$M\otimes_{R}S_{R}$.

Proof. If the map $a\mapsto a\otimes 1_{S}$ from $M$ to $M\otimes_{R}S$ were
not an embedding, then there would be some $a_{1},\dots,a_{n}$ in $M$ and
$r_{1},\dots r_{n}\in R$ such that $\sum_{i=1}^{n}\,a_{i}r_{i}$ is nonzero in
$M$ but $\sum_{i=1}^{n}\,a_{i}\otimes r_{i}=0$ in $M\otimes S$. By Herzog’s
criterion, 4.4, there would then be a pp formula $\phi(\overline{x})$ for
$R$-modules such that ${M}\models\phi(\overline{a})$ and ${}_{R}S\models
D\phi(\overline{r})$. Since the embedding $R\to S$ of left $R$-modules is pure
(since it is elementary), it follows that ${}_{R}R\models D\phi(\overline{r})$
and hence, again by 4.4, $\sum_{i=1}^{n}\,a_{i}\otimes
r_{i}=\sum_{i=1}^{n}\,a_{i}r_{i}=0$ already in $R$. Therefore $M\to M\otimes
S$ given by $a\mapsto a\otimes 1_{S}$ is monic.

To see that, more generally, this map is a pure embedding of right
$R$-modules, suppose that
$M\otimes_{R}S\models\phi(\sum_{i=1}^{n}\,a_{i}\otimes r_{i})$ ($a_{i}$,
$r_{i}$ as above) where $\phi$ is a pp formula in the language of $R$-modules.
Then, by 4.8, we have $M\models\psi(\overline{a})$ and
${}_{R}S_{S}\models(\phi:\psi)(\overline{r})$ for some pp $\psi$ in the
language of $R$-modules. As noted before 4.11, since $\phi$, $\psi$ both are
in the language of $R$-modules, $(\phi:\psi)$ is in the language of
$(R,R)$-bimodules, so we have ${}_{R}S_{R}\models(\phi:\psi)(\overline{r})$
and hence, since $R\prec S$, we have
${}_{R}R_{R}\models(\phi:\psi)(\overline{r})$. By Herzog’s criterion again, we
deduce that already $M\simeq
M\otimes_{R}R\models\phi(\sum_{i=1}^{n}\,a_{i}\otimes r_{i})$, as required.
$\Box$

We note the following.

######  Proposition 5.10.

Suppose that $R\prec S$.

(1) If $R$ is right coherent, then ${}_{R}S$ is flat but not necessarily
Mittag-Leffler.

(2) If $R$ is right coherent and left perfect, then ${}_{R}S$ is projective
(and hence Mittag-Leffler).

Proof. Since $R\prec S$ implies that ${}_{R}S$ is elementarily equivalent to
${}_{R}R$, the positive statements follow by [20, Thms. 4, 5] (see [8, Chpt.
14]). An example for the negative statement in (1) is to take $R$ to be the
localisation ${\mathbb{Z}}_{(p)}$ of ${\mathbb{Z}}$ at a prime $p$ and
consider some nonprincipal ultrafilter ${\cal U}$ on the index set $\omega$
(start with the filter generated by all subsets of $\omega$ with finite
complement and apply Zorn’s Lemma to extend to an ultrafilter). Consider the
element $a=(p,p^{2},\dots,p^{n},\dots)/{\cal U}$ of $S=R^{\omega}/{\cal U}$.
Then $p^{n}|a$ for every $n$ (by Łos’ Theorem, but this follows directly from
the ultrapower construction) and so ${\rm pp}^{S}(a)$ is not finitely
generated and ${}_{R}S$ is not Mittag-Leffler, despite $S$ being an elementary
extension of $R$. $\Box$

We will concentrate on tensor extension but note the following regarding
direct extension.

If $R$ is an elementary subring of $S$ then, in general, there are many more
definable subcategories of ${\rm Mod}\mbox{-}S$ than of ${\rm Mod}\mbox{-}R$:
for example over tame hereditary algebras not of finite representation type,
there are series of definable subcategories parametrised by the projective
line over the underlying field (which field will be very large if $S$ is a
large ultrapower of $R$). If, however, these rings are of finite
representation type, then every definable subcategory of ${\rm Mod}\mbox{-}S$
is the direct extension of a definable subcategory of ${\rm Mod}\mbox{-}R$. To
see that, first recall that elementary equivalence of rings preserves finite
representation type.

######  Proposition 5.11.

([2, 2.5]) If $R$ and $S$ are elementarily equivalent rings and if $R$ is of
finite representation type, then so is $S$.

Indeed in [2] it is shown that much is preserved by elementary equivalence
between rings of finite representation type, including the Auslander-Reiten
quiver and so, in particular, the number of indecomposable modules. Since
those indecomposables are the points of the Ziegler spectrum and since the
topology is discrete in the case of finite representation type (and since the
(closed) subsets correspond bijectively to the definable subcategories), we
obtain the following corollary.

######  Corollary 5.12.

If $R$ is of finite representation type and $R\prec S$, then direct extension
of definable subcategories gives a bijection between the definable
subcategories of ${\rm Mod}\mbox{-}R$ and the definable subcategories of ${\rm
Mod}\mbox{-}S$.

In fact, the methods used in this paper give another proof of 5.11, see [9,
Cor. 8].

###  5.2 Tensor extension and restriction of definable subcategories

In this section, given $R\prec S$, we take a definable subcategory ${\cal D}$
of ${\rm Mod}\mbox{-}R$ and compare the tensor, ${\cal D}\otimes_{R}S$, and
direct, ${\cal D}^{S}$, extensions of ${\cal D}$ to ${\rm Mod}\mbox{-}S$. We
also compare the restriction of ${\cal D}\otimes_{R}S$ to ${\rm Mod}\mbox{-}R$
with the original subcategory ${\cal D}$. Again, our proofs work just assuming
that $R\to S$ is an elementary embedding of $(R,R)$-bimodules.

######  Proposition 5.13.

If $f:R\to S$ is an elementary extension of rings, then ${\rm
DefTr}^{S}_{R}={\rm Mod}\mbox{-}R$.

Proof. By assumption $R$ and $S$ are elementarily equivalent as $R$-rings
(that is, rings with the elements of $R$ named as parameters in the language),
so by 5.4 they have isomorphic ultrapowers: $\alpha:S^{J}/{\cal V}\simeq
R^{I}/{\cal U}$ for some index sets $I$, $J$ and ultrafilters ${\cal U}$ on
$I$ and ${\cal V}$ on $J$ such that $\alpha$ preserves the diagonal images of
$R$, meaning that, for each $r\in R$, $\alpha\big{(}(fr)_{j}/{\cal
V}\big{)}=(r)_{i}/{\cal U}$, where $(fr)_{j}/{\cal V}$ and $(r)_{i}/{\cal U}$
are the constant sequences $fr$, respectively $r$.

Let $M$ be any $R$-module. Note that $M^{I}/{\cal U}$ has a natural action as
a right $R^{I}/{\cal U}$-module, namely $(a_{i})_{i}/{\cal
U}\cdot(r_{i})_{i}/{\cal U}=(a_{i}r_{i})_{i}/{\cal U}$, so it also is a right
$S^{J}/{\cal V}$-module by $(a_{i})_{i}/{\cal U}\cdot(s_{j})_{j}/{\cal
V}=(a_{i})_{i}/{\cal U}\cdot\alpha\big{(}(s_{j})_{j}/{\cal V}\big{)}$. This
action of $S^{J}/{\cal V}$ extends the action of $R$ on $M^{I}/{\cal U}$ since
$(a_{i})_{i}/{\cal U}\cdot(fr)_{j}/{\cal V}=(a_{i})_{i}/{\cal
U}\cdot\alpha\big{(}(fr)_{j}/{\cal V}\big{)}=(a_{i})_{i}/{\cal
U}\cdot(r)_{i}/{\cal U}=(a_{i}r)_{i}/{\cal U}$. Hence the action of $S$ on
$M^{I}/{\cal U}$ via the diagonal embedding $S\to S^{I}/{\cal U}$ extends the
action of $R$ on $M^{I}/{\cal U}$. So $(M^{I}/{\cal U})_{R}\in{\rm
DefTr}^{S}_{R}$. The diagonal embedding of $M$ into $M^{I}/{\cal U}$ is an
elementary embedding of $R$-modules (5.1), in particular a pure embedding, so
$M\in{\rm DefTr}^{S}_{R}$ and ${\rm DefTr}^{S}_{R}={\rm Mod}\mbox{-}R$, as
required. $\Box$

######  Lemma 5.14.

Suppose that $f:R\to S$ is an elementary embedding of rings and let ${\cal D}$
be a definable subcategory of ${\rm Mod}\mbox{-}R$. Then $({\cal
D}^{S})|_{R}={\cal D}$.

Proof. This is immediate from 3.10 and 5.13. $\Box$

There are various conditions under which $M_{R}$ and $M\otimes_{R}S_{R}$
generate the same definable subcategory and hence are definably equivalent.
For instance, we have the following from [9].

######  Corollary 5.15.

([9, Prop. 21]) Suppose that $R\prec S$ and that $M$ is a right $R$-module.
Assume either that $M_{R}$ is Mittag-Leffler or that the bimodule
${}_{R}R_{R}$ has the ascending chain condition on pp-definable subgroups.
Then $M$ and $M\otimes_{R}S_{R}$ are definably equivalent

Proof. We recall the proofs. In view of 5.9 we have to show that
$M\otimes_{R}S_{R}$ is in the definable subcategory generated by $M_{R}$. So
suppose that $\sigma/\tau$ is a pp-pair open on $M\otimes_{R}S_{R}$, say
$\overline{a}\otimes\overline{s}$ is in $\sigma(M\otimes_{R}S_{R})$ but not in
$\tau(M\otimes_{R}S_{R})$.

First suppose that $M_{R}$ is Mittag-Leffler. Then there is a pp formula
$\phi$ such that ${\rm pp}^{M}(\overline{a})$ is generated by $\phi$ and
hence, by 4.7(2), such that ${}_{R}S_{R}\models(\sigma:\phi)(\overline{s})$
and also such that $\overline{s}\notin(\tau:\phi)(_{R}S_{R})$. Then
${}_{R}S_{R}$ satisfies the sentence (in the language of $(R,R)$-bimodules,
and where “$\neg$” means “not”)

$\exists\overline{y}\,\big{(}(\sigma:\phi)(\overline{y})\wedge\neg(\tau:\phi)(\overline{y})\big{)}.$

Since $R$ is an elementary subring of $S$, it also satisfies this sentence, so
choose $\overline{r}$ from $R$ in
$(\sigma:\phi)(_{R}R_{R})\setminus(\tau:\phi)(_{R}R_{R})$. So we have $M\simeq
M\otimes_{R}R_{R}\models\sigma(\overline{a}\otimes\overline{r})$ and also, by
choice of $\phi$ and 4.7(2), $\overline{a}\otimes\overline{r}$ is not in
$\tau(M)$. That is, $\sigma/\tau$ is open on $M$, as required.

In the second case, $M$ is arbitrary but we have the acc on pp-definable
subgroups of ${}_{R}R_{R}$. Consider the set of pp-definable subgroups of
${}_{R}R_{R}$ of the form $(\tau:\phi)(_{R}R_{R})$ with
$\overline{a}\in\phi(M)$. Since the set of such $\phi$ is closed under finite
$\wedge$ and since (from the definitions) we have
$(\tau:\phi\wedge\phi^{\prime})\geq(\tau:\phi)+(\tau:\phi^{\prime})$, the set
of those subgroups $(\tau:\phi)(_{R}R_{R})$ with $\phi\in{\rm
pp}^{M}(\overline{a})$ is upwards-directed so, by assumption, has a maximal
element $(\tau:\phi)(_{R}R_{R})$ say. Since
${}_{R}S_{R}\models\sigma(\overline{a}\otimes\overline{s})$, there is
$\phi^{\prime}\in{\rm pp}^{M}(\overline{a})$ with
${}_{R}S_{R}\models(\tau:\phi)(\overline{s})$. Replacing each of $\phi$ and
$\phi^{\prime}$ by $\phi\wedge\phi^{\prime}$, we may assume that
$\phi=\phi^{\prime}$. As before, we therefore obtain
$\overline{r}\in(\sigma:\phi)(_{R}R_{R})\setminus(\tau:\phi)(_{R}R_{R})$ and
so have $M\models\sigma(\overline{a}\cdot\overline{r})$ (by
$\overline{a}\cdot\overline{r}$ we mean $\sum_{i}\,a_{i}r_{i}$). If we had
$M=M\otimes_{R}R_{R}\models\tau(\overline{a}\otimes\overline{r})$, then there
would be $\psi\in{\rm pp}^{M}(\overline{a})$ with
$\overline{r}\in(\tau:\psi)(_{R}R_{R})$; yet
$\overline{r}\notin(\tau:\phi)(_{R}R_{R})$, contradicting maximality of
$(\tau:\phi)(_{R}R_{R})$. Therefore
$\overline{a}\cdot\overline{r}\notin\tau(M)$ and so $\sigma/\tau$ is open on
$M$, as required. $\Box$

The second hypothesis is satisfied if $R$ is finitely generated as a module
over a noetherian centre (since every pp-definable subgroup is a module over
the centre of $R$). We have the following corollary.

######  Corollary 5.16.

Suppose that ${}_{R}R_{R}$ has the ascending chain condition on pp-definable
subgroups and that $R\prec S$. Let ${\cal D}$ be a definable subcategory of
${\rm Mod}\mbox{-}R$. Then $({\cal D}\otimes_{R}S)|_{R}={\cal D}$, so ${\cal
D}\otimes_{R}S\subseteq{\cal D}^{S}$.

######  Corollary 5.17.

Suppose that $R$ is a finite-dimensional algebra over a field and that $R\prec
S$. Let ${\cal D}$ be a definable subcategory of ${\rm Mod}\mbox{-}R$. Then
$({\cal D}\otimes_{R}S)|_{R}={\cal D}$, so ${\cal D}\otimes_{R}S\subseteq{\cal
D}^{S}$.

Proof. If $S$ is a finite-dimensional algebra, in particular if $S$ is an
ultrapower of $R$ then, see 5.2, this follows from 5.16. In general $S$ is
just an elementary subring of an ultrapower $R^{\ast}=R^{I}/{\cal U}$ of $R$.
If, for such a ring $S$, there were ${}_{S}S_{S}$-bimodule pp formulas
$\phi_{0}(\overline{x}),\phi_{1}(\overline{x}),\dots,\phi_{n}(\overline{x}),\dots$
such that
$\phi_{0}(_{S}S_{S})<\phi_{1}(_{S}S_{S})<\dots<\phi_{n}(_{S}S_{S})<\dots$ were
an infinite ascending chain then, for each $n$, we would have that
${}_{S}S_{S}$ satisfies the sentence, from the proof of 5.8, in the language
of $(S,S)$-bimodules which says that $\phi_{n}\leq\phi_{n+1}$ but not
$\phi_{n+1}\leq\phi_{n}$. Since $S\prec R^{\ast}$, $R^{\ast}$ also would
satisfy each of these sentences and therefore
${}_{R^{\ast}}R^{\ast}_{R^{\ast}}$ would have a proper ascending chain of pp-
definable subgroups, in contradiction to it (and therefore $R^{\rm op}\otimes
R$) being a finite-dimensional algebra. $\Box$

The two different hypotheses of 5.15 allow a common generalisation, as
follows. The proof from 5.15 applies but we present it rather differently so
as to give a different perspective on the $R\prec S$ condition.

######  Proposition 5.18.

Suppose that $R\prec S$ and that $M_{R}$ is atomic with respect to
${}_{R}S_{R}$. Let ${\cal D}=\langle M\rangle$ be the definable subcategory of
${\rm Mod}\mbox{-}R$ generated by $M$. Then $({\cal D}\otimes_{R}S)|_{R}={\cal
D}$, so ${\cal D}\otimes_{R}S\subseteq{\cal D}^{S}$.

Proof. Since $R$ and $S$ have, by 5.4, isomorphic ultrapowers (as $R$-rings)
we have, by 5.1, that $S$ is an elementary subring of some ultrapower of $R$,
so first we consider the case that $S=R^{I}/{\cal U}$ for some set $I$ and
ultrafilter ${\cal U}$ on $I$.

Suppose that $\sigma/\tau$ is closed on ${\cal D}$; we show that $\sigma/\tau$
is closed on $M\otimes_{R}S$, allowing us to deduce that
$M\otimes_{R}S\in{\cal D}_{S}$.

So suppose that $M\otimes_{R}S\models\sigma(\overline{a}\otimes\overline{s})$
where $\overline{s}=(\overline{r}_{i})_{i}/{\cal U}$ and, noting that $\sigma$
is in the language of $R$-modules, we actually have
${}_{R}S_{R}\models(\sigma:\phi)((\overline{r}_{i})_{i}/{\cal U})$, that is
${}_{R}(R^{I}/{\cal U})_{R}=(_{R}R_{R})^{I}/{\cal
U}\models(\sigma:\phi)((\overline{r}_{i})_{i}/{\cal U})$ for some $\phi\in{\rm
pp}^{M}(\overline{a})$. So, for some $J\in{\cal U}$, we have
${}_{R}R_{R}\models(\sigma:\phi)(\overline{r}_{j})$ for every $j\in J$.
Therefore $M\simeq
M\otimes_{R}R_{R}\models\sigma(\overline{a}\otimes\overline{r}_{j})$ for every
$j\in J$. Since $\sigma/\tau$ is closed on $M$, we have, for every $j\in J$,
$M\otimes_{R}R_{R}\models\tau(\overline{a}\otimes\overline{r}_{j})$. Then, for
each $j\in J$, $M\models\phi_{j}(\overline{a})$ and
${}_{R}R_{R}\models(\tau:\phi_{j})(\overline{r}_{j})$ for some $\phi_{j}$ such
that $\phi_{j}\in{\rm pp}^{M}(\overline{a})$.

By assumption, we may choose $\psi$ such that $(\tau:\psi)(_{R}R_{R})$ is
maximal among subgroups of the form $(\tau:\psi)(_{R}R_{R})$ with $\psi\in{\rm
pp}^{M}(\overline{a})$ and hence such that
${}_{R}R_{R}\models(\tau:\psi)(\overline{r}_{j})$ for all $j\in J$. Therefore
we have ${}_{R}(R^{I}/{\cal U})_{R}\models(\tau:\psi)(\overline{s})$, so
$M\otimes_{R}(R^{I}/{\cal U})_{R}\models\tau(\overline{a}\otimes\overline{s})$
and $\sigma/\tau$ is indeed closed on $M\otimes(R^{I}/{\cal U})$.

If $S$ is an elementary subring of a ring of the form $R^{I}/{\cal U}$, then,
when we have ${}_{R}S_{R}\models(\sigma:\phi)(\overline{s})$, hence
${}_{R}(R^{I}/{\cal U})_{R}\models(\sigma:\phi)(\overline{s})$, by the
argument above we obtain ${}_{R}(R^{I}/{\cal
U})_{R}\models(\tau:\psi)(\overline{s})$ and so, since $S\prec R^{I}/{\cal
U}$, the same is true for $S$, therefore the argument continues as above.

To finish, we recall that if $\sigma/\tau$ is open on $M\in{\cal D}$ then it
is open on $M\otimes_{R}S_{R}$, by 5.9. $\Box$

######  Corollary 5.19.

With assumptions as in 5.18, $M\otimes_{R}S_{R}$ generates the same definable
subcategory of ${\rm Mod}\mbox{-}R$ as $M_{R}$.

######  Corollary 5.20.

Suppose that $R\prec S$. Suppose that the definable subcategory ${\cal D}$ of
${\rm Mod}\mbox{-}R$ is generated, as a definable subcategory, by the finitely
presented modules in it. Then $({\cal D}\otimes_{R}S)|_{R}={\cal D}$, so
${\cal D}\otimes_{R}S\subseteq{\cal D}^{S}$.

Proof. Take $M$ as in 5.18 to be the direct sum of one copy of each module in
${\cal D}\,\cap\,{\rm mod}\mbox{-}R$. $\Box$

######  Example 5.21.

It is not in general the case that ${\cal D}\otimes_{R}S={\cal D}^{S}$, even
if $R\prec S$. Indeed consider ${\cal D}={\rm Mod}\mbox{-}R$ where
$R=K\widetilde{A}_{1}$ is the path algebra of the Kronecker quiver
$\widetilde{A}_{1}$ over an algebraically closed field $K$ and
$S=L\widetilde{A}_{1}$ over some proper algebraically closed extension $L$ of
$K$. (In fact any proper elementary extension of $R$ has this form because the
field $K$ is definable as the centre of $R$.) By 5.13 (or just directly),
${\cal D}^{S}={\rm Mod}\mbox{-}S$ which, as a definable (sub)category, is
generated by the indecomposable finite-dimensional $R$-modules. The effect of
$-\otimes_{R}S_{S}$ is just to replace $K$ by $L$ and so the images of these
indecomposable modules under $-\otimes_{R}S_{S}$ are the preprojectives, the
preinjectives and the simple-regular modules with parameter coming from
${\mathbb{P}}^{1}(K)$ but, together, these generate a proper definable
subcategory of ${\rm Mod}\mbox{-}S$ \- proper since it does not contain any
simple-regular module with parameter in $L\setminus K$.

## References

  * [1] G. R. Goodearl, Distributing tensor products over direct product, Pacific J. Math., 43(1) (1972), 107-110.
  * [2] C. Herrmann, C. U. Jensen and H. Lenzing, Applications of model theory to representations of finite-dimensional algebras, Math. Z., 178(1) (1981), 83-98.
  * [3] I. Herzog, Elementary duality of modules, Trans. Amer. Math. Soc., 340(1) (1993), 37-69.
  * [4] W. Hodges, Model Theory, Encyclopedia of Mathematics, Vol. 42, Cambridge University Press, 1993.
  * [5] H. Lenzing, Endlich präsentierbare Moduln, Arch. Math., 20(3) (1969), 262-266.
  * [6] M. Makkai, A theorem on Barr-exact categories with an infinitary generalization, Ann. Pure Appl. Logic, 47 (1990), 225-268.
  * [7] M. Prest, Duality and pure-semisimple rings, J. London Math. Soc. (2), 38 (1988), 403-409.
  * [8] M. Prest, Model Theory and Modules, London Math. Soc. Lect. Note Ser., Vol. 130, Cambridge University Press, 1988.
  * [9] M. Prest, The representation theories of elementarily equivalent rings, J. Symbolic Logic, 63(2) (1998), 439-450.
  * [10] M. Prest, Tensor product and theories of modules, J. Symbolic Logic, 64(2) (1999), 617-628.
  * [11] M. Prest, Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, Vol. 121, Cambridge University Press, 2009.
  * [12] M. Prest, Multisorted modules and their model theory, pp. 115-151 in Model Theory of Modules, Algebras and Categories, Contemporary Mathematics, Vol. 730, Amer. Math. Soc., 2019.
  * [13] M. Prest, Strictly atomic modules in definable categories, arXiv:2109:13750.
  * [14] M. Prest, Definable categories and monoidal categories, in preparation
  * [15] M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Seconde partie, Invent. Math., 13(1-2) (1971), 52-89.
  * [16] B. I. Rose, On the model theory of finite-dimensional algebras, Proc. London Math. Soc., (3) 40 (1980), 21-29.
  * [17] Ph. Rothmaler, Mittag-Leffler Modules and Positive Atomicity, Habilitationsschrift, Universität Kiel, 1994.
  * [18] Ph. Rothmaler, Mittag-Leffler Modules, Ann. Pure Applied Logic, 88(2-3) (1997), 227-239.
  * [19] Ph. Rothmaler, Mittag-Leffler modules and definable subcategories, pp. 171-196 in Contemp. Math., 710, Amer. Math. Soc., 2019.
  * [20] G. Sabbagh and P. Eklof, Definability problems for rings and modules, J. Symbolic Logic, 36(4) (1971), 623-649.
  * [21] S. Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math., 10 (1971), 224-233.
  * [22] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic, 26(2) (1984), 149-213.