# Contravariant finiteness and iterated strong tilting

Birge Huisgen-Zimmermann  Department of Mathematics, University of California,
Santa Barbara
CA 93106
USA<EMAIL_ADDRESS>,  Zahra Nazemian  University of Graz
Heinrichstrasse 36
8010 Graz, Austria<EMAIL_ADDRESS>and  Manuel Saorín
Departamento de Matemáticas
Universidad de Murcia, Aptdo. 4021
30100 Espinardo, Murcia
SPAIN<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ be the category of finitely
generated left modules of finite projective dimension over a basic Artin
algebra $\Lambda$. We develop a widely applicable criterion that reduces the
test for contravariant finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$ in $\Lambda\mbox{\rm-mod}$ to corner algebras $e\Lambda e$ for suitable
idempotents $e\in\Lambda$. The reduction substantially facilitates access to
the numerous homological benefits entailed by contravariant finiteness of
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$. The consequences pursued here
hinge on the fact that this finiteness condition is known to be equivalent to
the existence of a strong tilting object in $\Lambda\mbox{\rm-mod}$. We
moreover characterize the situation in which the process of strongly tilting
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration: This occurs precisely
when, in the category $\mbox{\rm mod-}\widetilde{\Lambda}$ of right modules
over the strongly tilted algebra $\widetilde{\Lambda}$, the subcategory of
modules of finite projective dimension is in turn contravariantly finite; the
latter condition can, once again, be tested on suitable corners $e\Lambda e$
of the original algebra $\Lambda$. In the (frequently occurring) positive
case, the sequence of consecutive strong tilts, $\widetilde{\Lambda}$,
$\widetilde{\widetilde{\Lambda}}$,
$\widetilde{\widetilde{\widetilde{\Lambda}}},\dots$, is shown to be periodic
with period $2$ (up to Morita equivalence); moreover, any two adjacent
categories in the sequence $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})$,
$\mathcal{P}^{<\infty}(\widetilde{\widetilde{\Lambda}}\mbox{\rm-mod})$,
$\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\widetilde{\widetilde{\Lambda}}}),\dots$, alternating between
right and left modules, are dual via contravariant $\mathop{\rm
Hom}\nolimits$-functors induced by tilting bimodules which are strong on both
sides.

Our methods rely on comparisons of right
$\mathcal{P}^{<\infty}$-approximations in the categories $\Lambda\mbox{\rm-
mod}$, $e\Lambda e\mbox{\rm-mod}$ and the Giraud subcategory of
$\Lambda\mbox{\rm-mod}$ determined by $e$; these interactions hold interest in
their own right. In particular, they underlie our analysis of the
indecomposable direct summands of strong tilting modules.

The second named author would like to thank all members of Department of
Mathematics at the University of Murcia for their warm hospitality during her
stay as a post doc, supported by a grant of the research project 19880/GERM/15
of the Fundación Séneca of Murcia. The third named author has been supported
by the Grant PID2020-113206GB-I00 funded by MCIN/AEI/10.13039/501100011033

2020 Mathematics Subject Classification: 16D90, 16G10, 16E05, 16E30, 16S90

##  1\. Introduction

Co- and contravariant finiteness of a subcategory $\mathcal{A}$ of the
category $\Lambda\mbox{\rm-mod}$ of finitely generated modules over an Artin
algebra $\Lambda$ were first considered by Auslander and Smalø in [2] and [3]:
If $\mathcal{A}=\mathop{\rm add}\nolimits(\mathcal{A})$ is closed under
extensions, the combination of these two finiteness conditions implies the
existence of internal almost split sequences in $\mathcal{A}$. Here we focus
on the full subcategory $\mathcal{A}=\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$ consisting of the modules of finite projective dimension; in this
situation contravariant finiteness implies the dual property, covariant
finiteness [15]. (See Section 2 for notation and definitions of the italicized
terms.) Subsequently, it was found that contravariant finiteness of
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ entails a plethora of
additional homological benefits for $\Lambda\mbox{\rm-mod}$ and the
unrestricted module category $\Lambda\mbox{\rm-Mod}$. Namely, this finiteness
condition not only validates the finitistic dimension conjectures for left
$\Lambda$-modules, i.e., confirms that $\mathop{\rm
l.findim}\nolimits(\Lambda)=\text{l.Findim}(\Lambda)<\infty$ in that case, but
gives rise to an intrinsic description of the $\Lambda$-modules of finite
projective dimension; see [1] and [15].

Crucial to our present investigation are the following facts:
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ if and only if $\Lambda\mbox{\rm-mod}$ contains a
strong tilting module, i.e., a tilting module $T$ which is relatively
$\mathop{\rm Ext}\nolimits$-injective within the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$. Such a module $T$ is
alternatively characterized by the condition that the contravariant functor
$\mathop{\rm Hom}\nolimits_{\Lambda}(-,T)$ induces a duality between
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and a specifiable resolving
subcategory of $\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$,
where $\widetilde{\Lambda}=\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$ (see [13] and [14]). In this situation we say that $\mathop{\rm
mod}\nolimits\text{-}\widetilde{\Lambda}$ results from $\Lambda\mbox{\rm-mod}$
via strong tilting. Finally, we recall that, up to isomorphism,
$\Lambda\mbox{\rm-mod}$ has at most one basic strong tilting module; see [1].

In the present article we tackle the foremost obstacle that stands in the way
of applying the theory we sketched: namely, the notoriously difficult task of
deciding whether, for specific choices of $\Lambda$, the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. The driving goals of our investigation are to
significantly reduce this difficulty and to explore the possibility and effect
of iterating the process of strongly tilting $\Lambda\mbox{\rm-mod}$. In both
directions we make headway, next to retrieving known results in a simpler,
more uniform format.

In more detail: In the first of our main results we address an Artin algebra
$\Lambda$ for which $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is known to
be contravariantly finite in $\Lambda\mbox{\rm-mod}$. In general,
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ need not inherit
contravariant finiteness. However, whenever it does, the process of strongly
tilting $\Lambda\mbox{\rm-mod}$ may be iterated in the following sense:
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting if
there exists an infinite sequence of basic algebras $(\Lambda_{i})_{i\geq 0}$
with $\Lambda_{0}=\Lambda$ such that $\mathop{\rm
mod}\nolimits\text{-}\Lambda_{i+1}$ results from
$\Lambda_{i}\text{-}\mathop{\rm mod}\nolimits$ via strong tilting if $i$ is
even, and $\Lambda_{i+1}\text{-}\mathop{\rm mod}\nolimits$ results from
$\mathop{\rm mod}\nolimits\text{-}\Lambda_{i}$ via strong tilting if $i$ is
odd.

######  Theorem A.

(For a more complete version, see Theorem 3.1 and Corollary 3.4.) Suppose that
$\Lambda$ is a basic Artin algebra such that
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. Let ${}_{\Lambda}T$ be the corresponding basic strong
tilting module and $\widetilde{\Lambda}=\mathop{\rm
End}\nolimits_{\Lambda}(T)^{\mathop{\rm op}\nolimits}$.

If $\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ is
contravariantly finite in $\mbox{\rm mod-}\widetilde{\Lambda}$, the category
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting. The
resulting sequence of basic Artin algebras $\Lambda=\Lambda_{0}$,
$\Lambda_{1}$, $\Lambda_{2}$, $\Lambda_{3},\dots$ is ultimately periodic with
period $2$: $\Lambda_{i}\cong\Lambda_{j}$ whenever $i$ and $j$ are positive
integers with the same parity. Their categories of modules of finite
projective dimension are linked by dualities

$\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\Lambda_{1})\leftrightarrow\mathcal{P}^{<\infty}(\Lambda_{2}\text{\rm-
mod})\leftrightarrow\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\Lambda_{3})\leftrightarrow\cdots$

induced by the contravariant $\mathop{\rm Hom}\nolimits$-functors of the
corresponding basic strong tilting bimodules
${}_{\Lambda_{i}}(T_{i})_{\Lambda_{i+1}}$, resp.
${}_{\Lambda_{i+1}}(T_{i})_{\Lambda_{i}}$.

In general, $\Lambda_{0}$ need not be isomorphic to $\Lambda_{2}$ however.

To return to the problem of determining the contravariant finiteness status of
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ in the first place, we suppose
that $e_{1},\dots,e_{n}$ form a complete set of orthogonal primitive
idempotents of $\Lambda$ and that the subsum $e=e_{1}+\dots+e_{m}$ includes
all idempotents $e_{i}$ that give rise to simple left modules of infinite
projective dimension. Our second and third main results refer to this setting.

######  Theorem B.

(For a complete version, see Theorem 4.6.) Adopt the notation of the preceding
paragraph, and suppose that $e\Lambda(1-e)$ has finite projective dimension as
a left module over the corner algebra $e\Lambda e$. Then
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ if and only if $\mathcal{P}^{<\infty}(e\Lambda
e\mbox{\rm-mod})$ is contravariantly finite in $e\Lambda e\mbox{\rm-mod}$.

Theorem 4.6 also spells out how, in case of existence, the minimal $($right$)$
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of
$M\in\Lambda\mbox{\rm-mod}$ relates to the minimal
$\mathcal{P}^{<\infty}(e\Lambda e\text{-}\mathop{\rm
mod}\nolimits)$-approximation of $eM$. The ensuing possibility of shucking off
primitive idempotents in checking for contravariant finiteness of
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ not only provides effortless
access to most of the cases in which this property has previously been
established (formerly involving considerable effort), but yields contravariant
finiteness in far more general situations.

Our argument relies on an exploration of the torsion-torsionfree triple
$(\mathcal{C},\mathcal{T},\mathcal{F})$ which is associated to the idempotent
$e$, and on the Giraud subcategory $\mathcal{G}$ of $\Lambda\mbox{\rm-Mod}$
(in the sense of Gabriel [9]) corresponding to the hereditary torsion pair
$(\mathcal{T},\mathcal{F})$.

In order to extend Theorem B to an efficient test of whether the stronger
conclusions of Theorem A hold for $\Lambda$, we once more assume that
$\Lambda$ is a basic Artin algebra such that
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite. As
before, we let $e_{1},\dots,e_{n}\in\Lambda$ be a complete set of primitive
idempotents, $T\in\Lambda\mbox{\rm-mod}$ the basic strong tilting module, and
$\widetilde{\Lambda}=\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$. By $S_{i}$ we denote the simple left $\Lambda$-module
corresponding to $e_{i}$. It is well-known that the indecomposable direct
summands $T_{1},\dots,T_{n}$ of ${}_{\Lambda}T$ coincide, up to isomorphism,
with the distinct indecomposable direct summands of $\bigoplus_{1\leq i\leq
n}\mathcal{A}_{i}$, where $\mathcal{A}_{i}$ is the minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of the
indecomposable injective left module with socle $S_{i}$. Our analysis of the
$T_{i}$ will pave the road towards showing that the question of unlimited
iterability of strong tilting of $\Lambda\mbox{\rm-mod}$ can in turn be played
back to the corner algebra $e\Lambda e$ for any idempotent $e$ as specified in
Theorem B.

######  Theorem C.

Let $m\leq n$ be chosen such that $\mathop{\rm
pdim}\nolimits_{\Lambda}S_{j}<\infty$ for $j>m$. Set $e=e_{1}+\cdots+e_{m}$
and assume that $\mathop{\rm pdim}\nolimits_{e\Lambda e}e\Lambda(1-e)<\infty$.
Then:

(1) (Proposition 5.2.) The number of distinct indecomposable direct summands
of $\bigoplus_{1\leq i\leq m}\mathcal{A}_{i}$ is $m$. Denote them by
$T_{1},\dots,T_{m}$ and set $T^{\prime}=\bigoplus_{1\leq i\leq m}T_{i}$.

For $j\geq m+1$, the approximation $\mathcal{A}_{j}$ of $S_{j}$ decomposes in
the form $\mathcal{A}_{j}=T_{j}\oplus U_{j}$, where $T_{j}$ is indecomposable
with the property that $S_{j}$ is the only simple module of finite projective
dimension in $\mathop{\rm soc}\nolimits T_{j}$, and all indecomposable direct
summands of $U_{j}$ occur as direct summands of $T^{\prime}$.

(2) (For more detail, see Theorem 5.4, and Corollaries 5.5, 5.6.)
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting if
and only if the same is true for $e\Lambda e\text{-}\mathop{\rm
mod}\nolimits$.

Our proof of Theorem C is based on connections between $T\in\Lambda\mbox{\rm-
mod}$ and the strong tilting objects in $e\Lambda e$-mod and in the Giraud
subcategory $\mathcal{G}$.

Earlier results for truncated path algebras (i.e., for algebras of the form
$KQ/I$, where $K$ is a field, $Q$ a quiver, and $I$ the ideal generated by the
paths of length $L+1$ for some $L\geq 1$), showcase the effects of iterated
strong tilting and the associated dualities among the
$\mathcal{P}^{<\infty}$-categories encountered along the iterations [13];
these findings readily follow from the above theorems. In fact, the present
results yield substantial generalizations of the picture that arose in the
truncated case (Propositions 6.1, 6.2, Corollaries 6.3 and 6.4). From our
reduction technique it also follows that, for any left serial algebra
$\Lambda$, both $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda)$ are contravariantly finite in
$\Lambda\mbox{\rm-mod}$ and $\mbox{\rm mod-}\Lambda$, respectively (the former
fact had been known; see [6]). Further applications address Artin algebras
arising from Morita contexts, such as algebras of triangular matrix type
(Theorem 6.6, Corollary 6.7, Examples 6.9), and the elimination of simple
modules of low projective dimension in the test for contravariant finiteness
(Proposition 5.7).

Section 2 assembles minimal conceptual background and builds the tools
required for proving our main theorems. Section 3 provides a general
characterization of the situation in which $\Lambda\mbox{\rm-mod}$ allows for
unlimited iteration of strong tilting. The results targeting tests for
contravariant finiteness of $\Lambda\mbox{\rm-mod}$ and for the availability
of repeated strong tilts of $\Lambda\mbox{\rm-mod}$ are contained in Sections
4 and 5, respectively. In Section 6, we specify applications.

##  2\. Notation, background and auxiliaries

Throughout, $\Lambda$ will be a basic Artin algebra, and $J$ its Jacobson
radical. We point out that the restriction to basic algebras does not affect
the generality of our investigation; we adopt it for increased transparency of
the underlying ideas. $\Lambda\mbox{\rm-Mod}$ and $\Lambda\mbox{\rm-mod}$
stand for the categories of all (resp., all finitely generated) left
$\Lambda$-modules. Further, $S_{1},\dots,S_{n}$ will be isomorphism
representatives of the simple objects in $\Lambda\mbox{\rm-Mod}$, and
$\mathop{\rm top}\nolimits M$, $\mathop{\rm soc}\nolimits M$ will stand for
the top and socle of $M\in\Lambda\mbox{\rm-Mod}$, respectively. By
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-Mod})$ (resp.,
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$) we denote the full subcategory
of $\Lambda\mbox{\rm-Mod}$ (resp., $\Lambda\mbox{\rm-mod}$) having as objects
the modules of finite projective dimension. Moreover, for any finitely
generated $\Lambda$-module $M$, we denote by $\mathop{\rm add}\nolimits(M)$
the full subcategory of $\Lambda\mbox{\rm-mod}$ consisting of the direct
summands of finite direct sums of copies of $M$. The module $M$ is basic if it
has no indecomposable direct summands of multiplicity $>1$.

###  2.1. Contravariant finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$ and strong tilting modules

Following Miyashita [18], we call a left $\Lambda$-module $T$ a tilting module
in case (1) $T$ belongs to $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$, (2)
$\mathop{\rm Ext}\nolimits^{i}_{\Lambda}(T,T)=0$ for $i\geq 1$, and (3) there
exists an exact sequence $0\rightarrow{{}_{\Lambda}\Lambda}\rightarrow
T_{0}\rightarrow\cdots\rightarrow T_{m}\rightarrow 0$ with
$T_{j}\in\mathop{\rm add}\nolimits(T)$. It is well known that any basic
tilting module has precisely $n=$ rank $K_{0}(\Lambda)$ indecomposable direct
summands, and that any tilting module $T\in\Lambda\mbox{\rm-mod}$ gives rise
to a left-right symmetric situation as follows: If
$\widetilde{\Lambda}=\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$, then the right $\widetilde{\Lambda}$-module
$T_{\widetilde{\Lambda}}$ is in turn a tilting module and $\mathop{\rm
End}\nolimits_{\widetilde{\Lambda}}(T)$ is canonically isomorphic to
$\Lambda$. This justifies the reference to a tilting bimodule
${{}_{\Lambda}T_{\widetilde{\Lambda}}}$. Strong tilting modules were first
considered by Auslander and Reiten in [1]. We introduce them via a
characterization equivalent to the original definition (see [1]): Namely, we
call a tilting module $T$ strong if it satisfies the following relative
injectivity condition in $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$: (4)
$\mathop{\rm Ext}\nolimits^{i}(M,T)=0$ for all
$M\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and $i\geq 1$. It was shown
by Auslander and Reiten [loc.cit.] that $\Lambda\mbox{\rm-mod}$ has a strong
tilting module if and only if the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$, a property which we will recall next. Moreover,
according to [loc.cit], in case of existence, there is precisely one basic
strong tilting module in $\Lambda\mbox{\rm-mod}$, up to isomorphism.

The concept of contravariant finiteness of subcategories of $\Lambda\mbox{\rm-
mod}$ has its roots in work of Auslander and Smalø [2]: Namely, the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ provided that for each object $M\in\Lambda\mbox{\rm-
mod}$, the functor $\mathop{\rm
Hom}\nolimits_{\Lambda}(-,M)_{|_{\mathcal{P}^{<\infty}(\Lambda\text{-mod})}}$
is finitely generated in the category of additive contravariant functors
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})\rightarrow\mathbf{Ab}$. This
condition translates into the following requirement for arbitrary
$M\in\Lambda\mbox{\rm-mod}$: There is an object
$A\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$, together with a map
$\phi\in\mathop{\rm Hom}\nolimits_{\Lambda}(A,M)$, such that each map in
$\mathop{\rm Hom}\nolimits_{\Lambda}(\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod}),M)$ factors through $\phi$. Any such pair $(A,\phi)$ is called a (right)
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M$. Since we
will only consider right approximations, we will frequently omit the qualifier
“right”. By a mild abuse of terminology, we will moreover refer to the domain
$A$ of $\phi$ as a $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation of $M$. Suppose $M$ has a
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation, say
$\phi:A\rightarrow M$. As was shown by Auslander-Smalø in [loc.cit.], up to
isomorphism, there is only one approximation $\phi:\mathcal{A}(M)\rightarrow
M$ such that $\mathcal{A}(M)$ has minimal length. It is alternatively
characterized by the condition that any endomorphism $u$ of $\mathcal{A}(M)$
which satisfies $\phi\circ u=\phi$ is an automorphism; we say that the map
$\phi$ is $($right$)$ minimal if this implication holds. Since it is unlikely
to cause misunderstandings, we will also refer to $\mathcal{A}(M)$ as “the”
minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M$
whenever convenient.

The mentioned existence result by Auslander-Reiten will be crucial in the
sequel. We state it for easy reference.

######  Theorem 2.1.

(1) [1] There exists a strong tilting module $T\in\Lambda\mbox{\rm-mod}$ if
and only if $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly
finite in $\Lambda\mbox{\rm-mod}$. In the positive case, the basic strong
tilting module is unique up to isomorphism: it is the direct sum of the
distinct indecomposable modules $C\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$ which satisfy $\mathop{\rm
Ext}\nolimits^{i}_{\Lambda}(\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod}),C)=0$
for $i\geq 1$.

(2) [13, Supplement II in Section 2.A] A more explicit description of the
indecomposable direct summands of a strong tilting module $T$, when it exists,
can be obtained from the following: Let $\mathcal{A}$ be the minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of the minimal
injective cogenerator of $\Lambda\mbox{\rm-Mod}$. Then $\mathop{\rm
add}\nolimits(T)=\mathop{\rm add}\nolimits(\mathcal{A})$.

###  2.2. The TTF-triple associated to an idempotent $e$ in $\Lambda$

We fix an idempotent element $e\in\Lambda$. By Jans [17] (see also [20,
Section VI.8]), $e$ defines a TTF triple
$(\mathcal{C}_{e},\mathcal{T}_{e},\mathcal{F}_{e})$ in the category
$\Lambda\mbox{\rm-Mod}$; this means that the pairs
$(\mathcal{C}_{e},\mathcal{T}_{e})$ and $(\mathcal{T}_{e},\mathcal{F}_{e})$
are both torsion pairs. The torsion, resp. torsionfree, classes are as
follows:

(a) $\mathcal{C}_{e}$ consists of the $\Lambda$-modules $C$ generated by
$\Lambda e$, i.e., the modules of the form

$C=\Lambda eC$.

(b) $\mathcal{T}_{e}$ consists of the left $\Lambda$-modules annihilated by
$e$.

(c) $\mathcal{F}_{e}$ consists of the $\Lambda$-modules $F$ with the property
that the annihilator

$\mathop{\rm ann}\nolimits_{F}(e\Lambda)$ of $e\Lambda$ in $F$ is zero.

Observe that the torsion pair $(\mathcal{T}_{e},\mathcal{F}_{e})$ is
hereditary, whence the corresponding torsion radical is left exact (see [20,
Proposition VI.3.1]). On the other hand, $(\mathcal{C}_{e},\mathcal{T}_{e})$
fails to be hereditary in general.

Further notation: Since we will keep the idempotent $e$ fixed, we will more
briefly write

$(\mathcal{C},\mathcal{T},\mathcal{F})$ for
$(\mathcal{C}_{e},\mathcal{T}_{e},\mathcal{F}_{e})$.

The torsion radicals $\nabla$ and $\Delta$ associated to the pairs
$(\mathcal{C},\mathcal{T})$ and $(\mathcal{T},\mathcal{F})$ are the idempotent
subfunctors of the identity functor on $\Lambda\mbox{\rm-Mod}$ given by:

$\nabla(M)=\Lambda eM\ \ \ \text{and}\ \ \ \Delta(M)=\mathop{\rm
ann}\nolimits_{M}(e\Lambda)=\\{x\in M\mid e\Lambda x=0\\},$

respectively. Note that $\nabla(M)$ is the largest submodule of $M$ with the
property that all simple modules in the top of $\nabla(M)$ belong to
$\mathcal{F}$ $\bigl{(}$equivalently, $\mathop{\rm
top}\nolimits\bigl{(}\nabla(M)\bigr{)}$ belongs to ${\mathop{\rm
add}\nolimits}(\Lambda e/Je)\bigr{)}$, whereas $\Delta(M)$ is the largest
submodule of $M$ which is annihilated by $e$ $\bigl{(}$equivalently, all
simple subfactors of $\Delta(M)$ belong to ${\mathop{\rm
add}\nolimits}\bigl{(}\Lambda(1-e)/J(1-e)\bigr{)}\bigr{)}$.

A third functor $\Lambda\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-Mod}$ will
serve to render the constructions in Sections 4 and 5 more transparent. It
assigns to each $\Lambda$-module $M$ the following subfactor of $M$:

$\mathfrak{core}(M)\ =\ \nabla(M)/\Delta\bigl{(}\nabla(M)\bigr{)}\ =\ \Lambda
eM/\Delta(\Lambda eM)$.

If $M$ is finitely generated, the core of $M$ has maximal length among the
subfactors $V/U$ of $M$ such that the top and socle of $V/U$ belong to
$\mathcal{F}$, i.e., such that $\mathop{\rm top}\nolimits(V/U),\mathop{\rm
soc}\nolimits(V/U)\in\mathop{\rm add}\nolimits(\Lambda e/Je)$. In fact, it can
easily be seen that this maximality property determines $\mathfrak{core}(M)$
up to isomorphism. Observe moreover, that $\mathcal{C}\cap\mathcal{F}$
consists of the $\Lambda$-modules which coincide with their cores.

######  Example 2.2.

To illustrate these endofunctors, we let $\Lambda=KQ/\langle R\rangle$, where
$K$ is a field, $Q$ the quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{a}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{d}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$

and $R=\\{ab,cd,dc,\beta_{1}c,\beta_{2}c,\\}\cup\\{\alpha_{k}b\beta_{l}\text{:
}k,l\in\\{1,2\\}\\}$.

We consider the TTF triple $(\mathcal{C},\mathcal{T},\mathcal{F})$ associated
to $e=e_{1}+e_{2}$ and apply the functors $\nabla$, $\Delta$,
$\mathfrak{core}$ to the indecomposable injective left $\Lambda$-modules shown
below:

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{2}$$\textstyle{3}$$\textstyle{4}$

  1. (1)

$\Delta(I_{j})=0$ $($i.e. $I_{j}\in\mathcal{F}$$)$ for $j=1,2$. The submodules
$\Delta(I_{3})$ of $I_{3}$ and $\Delta(I_{4})$ of $I_{4}$ are depicted by the
following diagrams:

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$$\textstyle{3}$

  2. (2)

The submodules $\nabla(I_{j})$, for $j=1,...,4$, are depicted by the diagrams

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{2}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$$\textstyle{3}$

  3. (3)

Finally, $\mathfrak{core}(I_{j})=\nabla(I_{j})$, for $i=1,2$, and
$\mathfrak{core}(I_{3})\cong S_{2}\oplus S_{2}\cong\mathfrak{core}(I_{4})$.

We point out that, in general, the core of a module may also have simple
composition factors annihilated by $e$; for a broader spectrum of examples,
see [13], [14].

###  2.3. The adjoint pair $(\Lambda e\otimes_{e\Lambda e}-,\,\bf{e})$

By $\bf{e}$ we denote the functor $\Lambda\mbox{\rm-Mod}\rightarrow e\Lambda
e\text{-Mod}$ which sends $M$ to $eM$. The two functors of the title thus play
the role of induction and restriction in the exchange of information between
$e\Lambda e$-modules on one hand and $\Lambda$-modules on the other.
Accordingly, $\Lambda e\otimes_{e\Lambda e}-:e\Lambda
e\text{-Mod}\rightarrow\Lambda\mbox{\rm-Mod}$ is left adjoint to $\bf{e}$.

The unit corresponding to this adjunction is the obvious natural
transformation

$\eta:1_{e\Lambda e\text{-Mod}}\ \longrightarrow\ {\bf{e}}\circ(\Lambda
e\otimes_{e\Lambda e}-),\quad U\longmapsto e(\Lambda e\otimes_{e\Lambda e}U).$

Clearly, $\eta$ is an isomorphism of functors, whence $\Lambda
e\otimes_{e\Lambda e}-$ is fully faithful (see [12, Proposition II.7.5]).

The counit $\epsilon:(\Lambda e\otimes_{e\Lambda e}-)\circ{\bf{e}}\
\rightarrow\ 1_{\Lambda\mbox{\rm-Mod}}$ of the adjunction is given by the
family $(\epsilon_{M})_{M\in\Lambda\mbox{\rm-Mod}}$, where
$\epsilon_{M}:\Lambda e\otimes_{e\Lambda e}eM\rightarrow M$ sends $a\otimes x$
to $ax$. Note that the image $\mathop{\rm Im}\nolimits(\epsilon_{M})$ equals
$\nabla(M)$.

We briefly explore the functor “restriction followed by induction”,
$\Lambda\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-Mod}$, which sends
$M\in\Lambda\mbox{\rm-Mod}$ to $M^{\ddagger}:=\Lambda e\otimes_{e\Lambda
e}eM$. Evidently, this functor sees only the core of a $\Lambda$-module $M$;
indeed, $M^{\ddagger}$ is naturally isomorphic to
$\mathfrak{core}(M)^{\ddagger}$. On the other hand, it preserves this core, as
the following lemma shows.

######  Lemma 2.3.

The functor $\mathfrak{core}:\Lambda\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-
Mod}$ is naturally equivalent to the functor

$\bigl{(}\Lambda e\otimes_{e\Lambda e}e(-)\bigr{)}/\Delta\bigl{(}\Lambda
e\otimes_{e\Lambda e}e(-)\bigr{)}.$

More precisely, the counit $\epsilon$ of the adjunction induces a family of
isomorphisms

$\overline{\epsilon}_{M}:\mathfrak{core}(M^{\ddagger})=(\Lambda
e\otimes_{e\Lambda e}eM)/\Delta(\Lambda e\otimes_{e\Lambda e}eM)\ \ \cong\ \
\mathfrak{core}(M),\ \ \ M\in\Lambda\mbox{\rm-Mod},$

which is natural in $M$.

###### Proof.

Clearly, $\epsilon$ gives rise to a family of epimorphisms

${\rho_{M}:\Lambda e\otimes_{e\Lambda
e}eM\rightarrow\nabla(M)/\Delta(\nabla(M))=\mathfrak{core}(M),}$

and $\mathop{\rm Ker}\nolimits(\rho_{M})$ contains $\Delta(\Lambda
e\otimes_{e\Lambda e}eM)$. That, conversely, $\mathop{\rm
Ker}\nolimits(\rho_{M})$ is contained in $\Delta(\Lambda e\otimes_{e\Lambda
e}eM)$ follows from the facts that ${\bf{e}}(\rho_{M}):e\Lambda
e\otimes_{e\Lambda e}eM\rightarrow e\bigl{(}\mathfrak{core}(M)\bigr{)}=eM$ is
an isomorphism in $e\Lambda e\mbox{\rm-mod}$ and $\Delta(\Lambda
e\otimes_{e\Lambda e}eM)$ is the largest submodule of $\Lambda
e\otimes_{e\Lambda e}eM$ which is annihilated by $e$. We conclude that the
maps $\overline{\epsilon}_{M}$ are indeed isomorphisms. ∎

###  2.4. The Giraud subcategory of $\Lambda\mbox{\rm-Mod}$ corresponding to
the torsion pair $(\mathcal{T},\mathcal{F})$

We apply well-known facts about localization with respect to a hereditary
torsion class $\mathcal{T}$ to the specialized situation where
$\mathcal{T}=\mathcal{T}_{e}$. We refer the reader to [10] and [20, Chapter
IX] for detail.

Recall that the Giraud subcategory $\mathcal{G}$ of $\Lambda\mbox{\rm-Mod}$
relative to the hereditary torsion pair $(\mathcal{T},\mathcal{F})$ is a
realization, inside $\Lambda\mbox{\rm-Mod}$, of the quotient category
$\Lambda\mbox{\rm-Mod}/\mathcal{T}$; here we identify the torsion class
$\mathcal{T}$ with the (full) localizing subcategory of $\Lambda\mbox{\rm-
Mod}$ that has object class $\mathcal{T}$. In particular, $\mathcal{G}$ is a
Grothendieck category. Concretely, $\mathcal{G}$ is the full subcategory of
$\Lambda\mbox{\rm-Mod}$ whose objects are the torsionfree $\Lambda$-modules
$F$ with the following restricted injectivity property: $\mathop{\rm
Ext}\nolimits_{\Lambda}^{1}(X,F)=0$ for all (cyclic) torsion modules
$X\in\mathcal{T}$.

It is well known that the fully faithful inclusion functor
$\iota:\mathcal{G}\rightarrow\Lambda\mbox{\rm-Mod}$ has an exact left adjoint
$\sigma:\Lambda\mbox{\rm-Mod}\rightarrow\mathcal{G}$, which is identifiable
with the quotient functor $\Lambda\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-
Mod}/\mathcal{T}$; in particular, the pair $(\mathcal{G},\sigma)$ has the
universal property of such a quotient. We will write $M_{\sigma}$ for
$\sigma(M)$. In parallel, $f_{\sigma}=\sigma(f)$ is the map
$M_{\sigma}\rightarrow N_{\sigma}$ induced by $f\in\mathop{\rm
Hom}\nolimits_{\Lambda}(M,N)$.

To describe $M_{\sigma}$ up to isomorphism, we abbreviate $M/\Delta(M)$ by
$\overline{M}$ and let $E(\overline{M})$ be an injective envelope of
$\overline{M}$. Then $M_{\sigma}$ is the preimage under the canonical map
$E(\overline{M})\rightarrow E(\overline{M})/\overline{M}$ of the torsion
submodule $\Delta\bigl{(}E(\overline{M})/\overline{M}\bigr{)}$ of the
quotient. In particular, $\sigma$ is the identity on the objects of
$\mathcal{G}$.

The explicit description of $\sigma:\Lambda\mbox{\rm-
Mod}\rightarrow\Lambda\mbox{\rm-Mod}$ reveals that this functor, in turn, sees
only the core of a $\Lambda$-module M, i.e.,
$M_{\sigma}=\mathfrak{core}(M)_{\sigma}$, and that it preserves cores, meaning
that $\mathfrak{core}(M_{\sigma})$ is canonically isomorphic to
$\mathfrak{core}(M)$; in fact, $M_{\sigma}$ is an essential extension of
$\mathfrak{core}(M)$ which is maximal relative to the requirement that this
core be preserved. The following alternative incarnations of the category
$\mathcal{G}$ will be helpful in Sections 4 and 5.

######  Lemma 2.4.

Suppose $e=e_{1}+\cdots+e_{m}$ is a decomposition of $e$ into primitive
idempotents.

1. [20, Proposition XI.8.6] The categories $\mathcal{G}$ and $e\Lambda e\mbox{\rm-Mod}$ are equivalent. Quasi-inverse equivalences send $F\in\mathcal{G}$ to $eF$ in one direction, and send $X\in e\Lambda e\mbox{\rm-Mod}$ to $(\Lambda e\otimes_{e\Lambda e}X)_{\sigma}$ in the other.

The indecomposable projective objects of $\mathcal{G}$ are $(\Lambda
e_{i})_{\sigma}$ for $1\leq i\leq m$, and the indecomposable injectives are
$\bigl{(}E(S_{i})\bigr{)}_{\sigma}=E(S_{i})$ for $1\leq i\leq m$.

2. [9] The subcategory $\mathcal{C}\cap\mathcal{F}$ of $\Lambda\mbox{\rm-Mod}$ is in turn equivalent to $\mathcal{G}$. Quasi-inverse equivalences send $M=\mathfrak{core}(M)$ to $M_{\sigma}$; in reverse, $F\in\mathcal{G}$ is sent to $\mathfrak{core}(F)=\nabla(F)$. In particular, the functors $\sigma$ and $\sigma\circ\mathfrak{core}$ from $\Lambda\mbox{\rm-Mod}$ to $\Lambda\mbox{\rm-Mod}$ are naturally isomorphic.

The indecomposable projective objects of $\mathcal{C}\cap\mathcal{F}$ are
$\mathfrak{core}(\Lambda e_{i})=\Lambda e_{i}/\Delta(\Lambda e_{i})$ for
$1\leq i\leq m$, and the indecomposable injectives are
$\mathfrak{core}\bigl{(}E(S_{i})\bigr{)})=\nabla\bigl{(}E(S_{i})\bigr{)}$ for
$1\leq i\leq m$.

We add some notation: The unit of the adjunction $(\sigma,\iota)$ is the
natural transformation

$\mu=(\mu_{M}):1_{\Lambda\mbox{\rm-Mod}}\longrightarrow\iota\circ\sigma,\ \
\text{with}\ \mu_{M}\in\mathop{\rm Hom}\nolimits_{\Lambda}(M,M_{\sigma})\
\text{canonical}.$

If $M$ is torsionfree, we identify $\mu_{M}$ with the inclusion map
$M\hookrightarrow M_{\sigma}\subseteq E(M)$. Clearly, $\mu_{M}$ is an
isomorphism precisely when $M$ belongs to $\mathcal{G}$.

######  Remark 2.5.

We point out that the mentioned functors linking the subcategories
$\mathcal{A}$ of $\Lambda\mbox{\rm-Mod}$ $($resp. of $e\Lambda e\mbox{\rm-
Mod}$$)$ introduced in subsections 2.2-2.4 restrict to functors connecting the
pertinent intersections $\mathcal{A}\cap\Lambda\mbox{\rm-mod}$, resp.,
$\mathcal{A}\cap e\Lambda e\mbox{\rm-mod}$ and $\mathcal{A}\cap\mathcal{G}$.

######  Example 2.6.

(Return to Example 2.2.) In this instance, $(I_{j})_{\sigma}=I_{j}$ for
$j=1,2$, because $I_{1}$ and $I_{2}$ are injective objects of $\mathcal{F}$,
and consequently of $\mathcal{G}$. Now let $j\in\\{3,4\\}$. In either case,
$I_{j}/\Delta(I_{j})\cong S_{2}\oplus S_{2}$, whence
$E_{j}:=E(I_{j}/\Delta(I_{j}))\cong I_{2}\oplus I_{2}$. Since the torsion
submodule of $(I_{2}\oplus I_{2})/(S_{2}\oplus S_{2})$ is zero, we obtain
$(I_{j})_{\sigma}=S_{2}\oplus S_{2}$, and the map
$\mu_{I_{j}}:I_{j}\longrightarrow(I_{j})_{\sigma}=S_{2}\oplus S_{2}$ is the
obvious projection.

###  2.5. The poset of essential $\Delta$-extensions of a morphism

In constructing $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximations
of $\Lambda$-modules $M$ from $\mathcal{P}^{<\infty}(e\Lambda
e\text{-mod})$-approximations of $eM$, passage to maximal extensions of the
type described in this subsection will be crucial. Throughout we refer to the
torsion theory $(\mathcal{T},\mathcal{F})$ introduced in 2.2.

######  Definition.

Given a morphism $f:M\longrightarrow Y$ in $\Lambda\mbox{\rm-mod}$, we
consider the following _eligible extensions of $f$_: These are the pairs
$(L,g)$, where $L$ is an essential extension of $M$ with the additional
property that $L/M\in\mathcal{T}$ and $g\in\mathop{\rm
Hom}\nolimits_{\Lambda}(L,Y)$ satisfies $g_{|M}=f$.

We say that two eligible pairs $(L,g)$ and $(L^{\prime},g^{\prime})$ are
_isomorphic_ if there is an isomorphism $\psi:L\rightarrow L^{\prime}$ such
that $g^{\prime}\circ\psi=g$. An _essential $\Delta$-extension of $f$_ is the
isomorphism class $[(L,g)]$ of an eligible pair $(L,g)$.

The set $\mathcal{E}_{f}$ of all essential $\Delta$-extensions $[(L,g)]$ of
$f$ is a poset under the following partial order:
$[(L,g)]\preceq[(L^{\prime},g^{\prime})]$ in case there is a monomorphism
$\psi:L\longrightarrow L^{\prime}$ such that $g^{\prime}\circ\psi=g$.

We comment on the legitimacy of the final definition: Welldefinedness of the
relation $\preceq$ is clear. To check that it is antisymmetric, it suffices to
observe that the existence of monomorphisms $L\longrightarrow L^{\prime}$ and
$L^{\prime}\longrightarrow L$ forces the finitely generated $\Lambda$-modules
$L$ and $L^{\prime}$ to have the same length, whence monomorphisms between
them are isomorphisms.

######  Proposition 2.7.

Suppose that $M$ and $Y$ are finitely generated torsionfree left
$\Lambda$-modules, and let $f\in\mathop{\rm Hom}\nolimits_{\Lambda}(M,Y)$.
Then the poset $(\mathcal{E}_{f},\preceq)$ has a maximum.
$\bullet$ In case $Y\in\mathcal{G}$, this maximum is
$[(M_{\sigma},f_{\sigma}{:M_{\sigma}\longrightarrow Y_{\sigma}=Y})]$.
$\bullet$ For a general torsionfree module $Y$, the maximum of
$\mathcal{E}_{f}$ is $[(N,\phi)]$, where $\phi:N\longrightarrow Y$ results
from the pullback of $(f_{\sigma},\mu_{Y})$:

$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\chi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{Y}}$$\textstyle{M_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\sigma}}$$\textstyle{Y_{\sigma}}$

###### Proof.

In light of the hypothesis $\Delta(M)=0$, we have $M\subseteq
M_{\sigma}\subseteq E(M)$ for an injective envelope $E(M)$ of $M$, where the
first inclusion coincides with the map $\mu_{M}:M\hookrightarrow M_{\sigma}$
of 2.3. Due to the equality $M_{\sigma}/M=\Delta(E(M)/M)$, we find
$M_{\sigma}$ to be the largest among the submodules $M^{\prime}$ of $E(M)$
which contain $M$ and satisfy $M^{\prime}/M\in\mathcal{T}$. Consequently, all
elements of $\mathcal{E}_{f}$ are represented by pairs $(L,g)$ with
$M\subseteq L\subseteq M_{\sigma}$, and we may restrict our focus to such
representatives. Conversely, whenever $h:N\rightarrow Y$ is a homomorphism
with $M\subseteq N\subseteq M_{\sigma}$ and $h_{|M}=f$, the pair $(N,h)$
represents a class in $\mathcal{E}_{f}$.

To address our first claim, we note that the lengths of increasing sequences
in $\mathcal{E}_{f}$ are bounded from above by the composition length of
$E(M)$. In particular, each element of $\mathcal{E}_{f}$ is majorized by a
maximal element in this set. Therefore, we only need to prove that any two
elements $[(L,g)]$ and $[(N,h)]$ of $\mathcal{E}_{f}$ have a common upper
bound. Consider the restrictions $g_{|X},h_{|X}:X\longrightarrow Y$ to the
intersection $X=L\cap N$. Since the difference $g_{|X}-h_{|X}$ vanishes on
$M$, it induces a homomorphism $X/M\longrightarrow Y$, which in turn vanishes
because $X/M\in\mathcal{T}$ and $Y\in\mathcal{F}$. The resulting equality
$g_{|X}=h_{|X}$ guarantees that the assignment $\rho:L+N\longrightarrow Y$
with $\rho(l+m)=g(l)+h(m)$ is a well-defined map in $\mathop{\rm
Hom}\nolimits_{\Lambda}(L+N,Y)$. In light of $L+N\subseteq M_{\sigma}$, the
pair $(L+N,\rho)$ represents an element in $\mathcal{E}_{f}$ which majorizes
both $[(L,g)]$ and $[(N,h)]$.

Now suppose that $Y\in\mathcal{G}$. Then $f_{\sigma}:M_{\sigma}\rightarrow
Y_{\sigma}\cong Y$ gives rise to an essential $\Delta$-extension
$[(M_{\sigma},f_{\sigma})]$. That this extension is maximal in
$\mathcal{E}_{f}$ is immediate from the description of $M_{\sigma}$.

To prove the final claim, we note that the map $\chi$ in the pullback diagram
is an injection since $\mu_{Y}$ is. Thus the submodule $\mu_{M}^{-1}(\chi(N))$
of $M_{\sigma}$ is isomorphic to $N$. On replacing $N$ by this copy and
adjusting $\phi$ and $\chi$ accordingly, we obtain another pullback diagram
for $(f_{\sigma},\mu_{Y})$ with $M\subseteq N\subseteq M_{\sigma}$, where
$\chi$ is now replaced by the inclusion map. This shows that $(N,\phi)$ is an
eligible pair in the sense of the above definition. To check maximality of
$[(N,\phi)]$ in $\mathcal{E}_{f}$, suppose that this $\Delta$-extension is
majorized by $[(N^{\prime},\phi^{\prime})]$, where $M\subseteq
N^{\prime}\subseteq M_{\sigma}$; let $\psi:N\hookrightarrow N^{\prime}$ be an
embedding with $\phi=\phi^{\prime}\circ\psi$. Since $\psi$ extends to an
automorphism of $E(M)$, we do not lose generality in viewing $\psi$ as a set
inclusion. We thus obtain a commutative diagram, in which $\kappa$ denotes the
inclusion map $N^{\prime}\hookrightarrow M_{\sigma}$:

$\textstyle{N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{v}$$\scriptstyle{\kappa}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$$\scriptstyle{\chi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{Y}}$$\textstyle{M_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\sigma}}$$\textstyle{Y_{\sigma}}$

Then $f_{\sigma}\circ\kappa\circ\psi=\mu_{Y}\circ\phi^{\prime}\circ\psi$,
whence the restrictions of $f_{\sigma}\circ\kappa$ and
$\mu_{Y}\circ\phi^{\prime}$ to the domain $N$ of $\psi$ coincide. Since the
epimorphic image $N^{\prime}/N$ of $N^{\prime}/M$ is a torsion module, while
$Y_{\sigma}$ is torsionfree, the argumentation backing the first claim shows
that $f_{\sigma}\circ\kappa=\mu_{Y}\circ\phi^{\prime}$. The universal property
of the pullback therefore yields a map $v\in\mathop{\rm
Hom}\nolimits_{\Lambda}(N^{\prime},N)$ with $\phi^{\prime}=\phi\circ v$ and
$\kappa=\chi\circ v$. In particular, $v$ is a monomorphism. We infer that $N$
and $N^{\prime}$ have the same length and conclude that $\psi$ is an
isomorphism. This proves $[(N^{\prime},\phi^{\prime})]=[(N,\phi)]$ as
required. ∎

##  3\. Iteration of strong tilting

When can the process of strongly tilting $\Lambda\mbox{\rm-mod}$ to a category
of right modules, $\mbox{\rm mod-}\widetilde{\Lambda}$, be iterated? In the
positive case, how do the resulting sequences of strongly tilted module
categories behave? The main purpose of this section is to answer these
questions. More specifically, we will show that, whenever $\Lambda\mbox{\rm-
mod}$ can be strongly tilted twice, the process can be iterated arbitrarily
and turns periodic after the initial step. Roughly speaking: In case
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ is in turn
contravariantly finite in $\mbox{\rm mod-}\widetilde{\Lambda}$, the initial
transition from $\Lambda$ to $\widetilde{\Lambda}$ increases the homological
symmetry by increasing the number of simple modules of finite projective
dimension. This symmetrization makes the subsequent sequence $\mbox{\rm
mod-}\widetilde{\Lambda}\rightsquigarrow\widetilde{\widetilde{\Lambda}}\mbox{\rm-
mod}\rightsquigarrow\cdots$ periodic with period $2$.

The statement of the following theorem is based on part (1) of Theorem 2.1.

######  Theorem 3.1.

Let $\Lambda$ be a basic Artin algebra such that
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. Moreover, let ${{}_{\Lambda}T}$ be the corresponding
basic strong tilting module, and $\widetilde{\Lambda}=\mathop{\rm
End}\nolimits_{\Lambda}(T)^{\mathop{\rm op}\nolimits}$. Suppose that
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ is in turn
contravariantly finite in $\mbox{\rm mod-}\widetilde{\Lambda}$, thus giving
rise to a basic tilting bimodule
${{}_{\widetilde{\widetilde{\Lambda}}}\widetilde{T}}_{\widetilde{\Lambda}}$
which is strong in $\mbox{\rm mod-}\widetilde{\Lambda}$. Then $\widetilde{T}$
is strong on both sides.

In particular, the process of strongly tilting $\Lambda\mbox{\rm-mod}$ then
allows for unlimited iteration, yielding a sequence of basic Artin algebras
$\Lambda=\Lambda_{0},\Lambda_{1},\Lambda_{2},\Lambda_{3},\dots$ with the
property that $\Lambda_{i}$ and $\Lambda_{j}$ are isomorphic whenever $i$ and
$j$ are positive integers with the same parity.

Moreover, the algebras $\Lambda_{0}$ and $\Lambda_{2}$ are isomorphic
precisely when the tilting bimodule ${}_{\Lambda}T_{\widetilde{\Lambda}}$ is
strong on both sides, i.e., when $T=\widetilde{T}$.

######  Comment 3.2.

If, in the hypotheses of Theorem 3.1, the qualifiers “basic” are dropped, one
obtains Morita equivalent pairs of algebras $(\Lambda_{i},\Lambda_{i+2})$ for
all $i\geq 1$.

###### Proof.

By a result of Auslander and Green (see [1, Proposition 6.5] and [7,
Proposition 7.1] for a short argument), a tilting bimodule ${}_{A}U_{B}$ over
Artin algebras $A$ and $B$ is strong in $A\text{-mod}$ if and only if all
simple right $B$-modules embed into $U_{B}$. In light of the hypothesis that
${}_{\Lambda}T$ is a strong tilting module, we thus find that all simple right
$\widetilde{\Lambda}$-modules are contained in $\mathop{\rm soc}\nolimits
T_{\widetilde{\Lambda}}$, up to isomorphism. Moreover, we deduce that
strongness of ${}_{\widetilde{\widetilde{\Lambda}}}\widetilde{T}$ as a tilting
object in $\widetilde{\widetilde{\Lambda}}\mbox{\rm-mod}$ will follow if we
can show that all simple right $\widetilde{\Lambda}$-modules embed into
$\widetilde{T}_{\widetilde{\Lambda}}$.

To realize such an embedding, let
$(\widetilde{\Lambda}/\widetilde{J})_{\widetilde{\Lambda}}=\widetilde{S}_{1}\oplus\cdots\oplus\widetilde{S}_{n}$,
where the $\widetilde{S}_{i}$ are simple. Further denote by
$E(\widetilde{S}_{1}\oplus\cdots\oplus\widetilde{S}_{n})$ an injective
envelope, and let $\widetilde{\mathcal{A}}$ be a minimal
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$-approximation of
the latter. In light of our hypothesis that
$\widetilde{T}_{\widetilde{\Lambda}}$ is a strong tilting object in $\mbox{\rm
mod-}\widetilde{\Lambda}$, the categories $\mathop{\rm
add}\nolimits(\widetilde{T}_{\widetilde{\Lambda}})$ and $\mathop{\rm
add}\nolimits(\widetilde{\mathcal{A}})$ coincide; see Theorem 2.1(2). Embed
$T_{\widetilde{\Lambda}}$ into an injective right
$\widetilde{\Lambda}$-module, say
$\iota:T_{\widetilde{\Lambda}}\hookrightarrow\bigoplus_{1\leq i\leq
n}E(\widetilde{S}_{i})^{m_{i}}$. For a suitable exponent $m$, we obtain a
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$-approximation
$\phi:\widetilde{\mathcal{A}}^{m}\rightarrow\bigoplus_{1\leq i\leq
n}E(\widetilde{S}_{i})^{m_{i}}$. Since $T_{\widetilde{\Lambda}}$ belongs to
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$, the embedding
$\iota$ factors through $\phi$, say $\iota=\phi\circ f$ for some
$f\in\mathop{\rm
Hom}\nolimits_{\widetilde{\Lambda}}(T,\widetilde{\mathcal{A}}^{m})$. Then $f$
is injective, and we deduce that all simple right
$\widetilde{\Lambda}$-modules embed into $\widetilde{\mathcal{A}}^{m}$.
Clearly, this implies that all simples in $\mbox{\rm mod-}\widetilde{\Lambda}$
embed into $\mathop{\rm soc}\nolimits\widetilde{\mathcal{A}}$. Invoking the
fact that $\mathop{\rm add}\nolimits(\widetilde{\mathcal{A}})=\mathop{\rm
add}\nolimits(\widetilde{T}_{\widetilde{\Lambda}})$, we conclude that all
$\widetilde{S}_{i}$ occur in $\mathop{\rm
soc}\nolimits\widetilde{T}_{\widetilde{\Lambda}}$ as well, which proves two-
sided strongness of
${{}_{\widetilde{\widetilde{\Lambda}}}\widetilde{T}}_{\widetilde{\Lambda}}$.

The final assertion follows from the preceding argument. ∎

The result is sharp in the sense that the existence of a strong tilting module
in $\Lambda\mbox{\rm-mod}$ does not, by itself, provide sufficient homological
symmetry to allow for iteration of strong tilting.

######  Example 3.3.

This example is due to Igusa-Smalø-Todorov [16]. Let $\Lambda=KQ/I$, where $Q$
is the quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$

and $I\subseteq KQ$ is the ideal generated by
$\beta\gamma,\gamma\alpha,\gamma\beta$. Clearly the right socle of $\Lambda$
contains both simple right $\Lambda$-modules, whence
${\rm{l.findim}}(\Lambda)=0$ (see [4]). Hence the basic strong tilting object
in $\Lambda\mbox{\rm-mod}$ is $T={{}_{\Lambda}\Lambda}$, and
$\widetilde{\Lambda}$ is isomorphic to $\Lambda$. However, the category
$\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})=\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda)$ fails
to be contravariantly finite in $\mbox{\rm mod-}\Lambda$ (see [16]).

For a first application, we combine Theorem 3.1 with the fact that strong
tilting modules induce contravariant equivalences of categories of modules of
finite projective dimension (see [13, Reference Theorem III and Theorem 1] or
[14, Theorems 7,8]). This yields

######  Corollary 3.4.

Retain the notation and hypotheses of Theorem 3.1. Then the pairs of functors
$\bigl{(}\mathop{\rm Hom}\nolimits_{\Lambda}(-,T)$, $\mathop{\rm
Hom}\nolimits_{\widetilde{\Lambda}}(-,T)\bigr{)}$ and $\bigl{(}\mathop{\rm
Hom}\nolimits_{\widetilde{\Lambda}}(-,\widetilde{T}),\mathop{\rm
Hom}\nolimits_{\widetilde{\widetilde{\Lambda}}}(-,\widetilde{T})\bigr{)}$
induce dualities

$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})\ \longleftrightarrow\
\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})\cap{{}^{\perp}(T_{\widetilde{\Lambda}}})\ \ \
\subseteq\ \ \ \mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})\
\longleftrightarrow\
\mathcal{P}^{<\infty}(\widetilde{\widetilde{\Lambda}}\text{-}\mathop{\rm
mod}\nolimits),$

where ${{}^{\perp}(T_{\widetilde{\Lambda}})}$ is the full subcategory of
$\mbox{\rm mod-}\widetilde{\Lambda}$ whose objects are the modules
$\widetilde{M}$ with $\mathop{\rm
Ext}\nolimits^{i}_{\widetilde{\Lambda}}(\widetilde{M},T)=0$ for $i\geq 1$.
Moreover, ${{}^{\perp}(T_{\widetilde{\Lambda}})}$ contains
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ if and only if
${}_{\Lambda}T_{\widetilde{\Lambda}}$ is two-sided strong, i.e., precisely
when $T\cong\widetilde{T}$.

In case the process of strongly tilting ${\Lambda\mbox{\rm-mod}}$ permits for
iteration, periodicity of the sequence ${\Lambda\mbox{\rm-
mod}}\rightsquigarrow\mbox{\rm mod-}\widetilde{\Lambda}\rightsquigarrow\cdots$
may set in with delay. In fact, if $\Lambda$ is a truncated path algebra one
always obtains a sequence of basic strong tilts, ${\Lambda\mbox{\rm-
mod}}\rightsquigarrow\mbox{\rm mod-}\widetilde{\Lambda}\rightsquigarrow\cdots$
(see [13, Theorem 19]); for this class of algebras, $\Lambda\mbox{\rm-
mod}\approx\widetilde{\widetilde{\Lambda}}\text{-mod}$ if and only if $Q$ does
not have a precyclic source (see [7, Corollary 7.2]). We refer to [13] and
[14] for examples based on quivers with precyclic sources where the transition
from the tilting module $T_{\widetilde{\Lambda}}$ to the basic strong tilting
module $\widetilde{T}_{\widetilde{\Lambda}}$ in $\mbox{\rm
mod-}\widetilde{\Lambda}$ is displayed.

We conclude the section with a variant of the criterion of Auslander and Green
for a strong tilting module $T\in\Lambda\mbox{\rm-mod}$ to be strong also as a
tilting module over $\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$; it was implicitly used in the proof of Theorem 3.1. The
upcoming version of the criterion does not rely on structural information
regarding ${}_{\Lambda}T$ and provides additional motivation for the detection
of twosided strongness.

######  Observation 3.5.

Suppose $\Lambda$ is an Artin algebra such that
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. Then the following statements are equivalent:

(1) The basic strong tilting module $T\in\Lambda\mbox{\rm-mod}$ is strong also
in $\mbox{\rm mod-}\widetilde{\Lambda}$, where
$\widetilde{\Lambda}=\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$.

(2) Every simple left $\Lambda$-module of infinite projective dimension embeds
into a $\Lambda$-module of finite projective dimension.

(3) There exists a duality $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})\longleftrightarrow\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})$.

If (1)-(3) are satisfied, then $\mathop{\rm Hom}\nolimits_{\Lambda}(-,T)$ and
$\mathop{\rm Hom}\nolimits_{\widetilde{\Lambda}}(-,T)$ induce $($unique up to
isomorphism$)$ quasi-inverse dualities
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})\ \longleftrightarrow\
\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$.

###### Proof.

“(1)$\implies$(2)” is immediate from the criterion of Auslander and Green
quoted in the proof of Theorem 3.1. “(2)$\implies$(1)”: By (2), every simple
left $\Lambda$-module $S$ embeds into a module $M=M(S)$ of finite projective
dimension. Consider an embedding $\iota:M\hookrightarrow E$ of $M$ into an
injective module $E$, and let $\phi:\mathcal{A}\rightarrow E$ be a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation. Then $\iota$
factors through $\phi$, which shows $M$ to be contained in $\mathcal{A}$ up to
isomorphism. Hence so is $S$. Since $\mathop{\rm
add}\nolimits(\mathcal{A})\subseteq\mathop{\rm add}\nolimits(T)$ by hypothesis
(cf. the proof of Theorem 3.1), we conclude that $S$ embeds into $T$. Thus an
application of [1, Proposition 6.5] yields (1).

The equivalence “(1)$\iff$(3)” follows from [13, Reference Theorem III and
Theorem 1] or [14, Theorems 7,8]. For the supplementary statement, see [14,
Corollary 9]. ∎

##  4\. Testing for contravariant finiteness of the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$

Our main objective in this section is to show that, in exploring whether the
subcategory $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly
finite in $\Lambda\mbox{\rm-mod}$, a comparatively mild hypothesis (4.1(ii)
below) permits us to eliminate primitive idempotents which correspond to
simple left $\Lambda$-modules of finite projective dimension. This reduces the
problem to a more manageable corner $e\Lambda e$ of $\Lambda$.

###  4.1. The setting

######  Setting 4.1 (Blanket hypotheses).

Throughout this section, we will assume that the idempotent $e\in\Lambda$
satisfies the following two conditions:
(i) The semisimple left $\Lambda$-module $\Lambda(1-e)/J(1-e)$ has finite
projective dimension.
(ii) $e\Lambda(1-e)$ has finite projective dimension as a left $e\Lambda
e$-module.

######  Remarks 4.2.

$\bullet$ Suppose that $e=e_{1}+\cdots+e_{m}$ and $1-e=e_{m+1}+\cdots+e_{n}$,
where the $e_{i}$ are orthogonal primitive idempotents of $\Lambda$. Then the
conditions under 4.1 amount to finiteness of the projective dimensions of the
simple $\Lambda$-modules $S_{i}=\Lambda e_{i}/Je_{i}$ for $i\geq m+1$ and
those of the $e\Lambda e$-modules $e\Lambda e_{i}$ for $i\geq m+1$. In
particular, 4.1(i) implies that, for all $M\in\Lambda\mbox{\rm-mod}$,

$\mathop{\rm pdim}\nolimits_{\Lambda}M<\infty\iff\mathop{\rm
pdim}\nolimits_{\Lambda}\mathfrak{core}(M)<\infty\iff\mathop{\rm
pdim}\nolimits_{\Lambda}M_{\sigma}<\infty$.

With regard to 4.1(ii), we mention that, in case $\Lambda$ is a path algebra
modulo relations, the quiver and relations of $e\Lambda e$ are available from
those of $\Lambda$, making this condition computationally accessible.

$\bullet$ Suppose that $S_{1},\dots,S_{m}$ are precisely the simple left
$\Lambda$-modules of infinite projective dimension; in particular, this means
that 4.1(i) is satisfied. Even over monomial algebras, condition 4.1(ii) is
not automatic in this situation, as is witnessed by an example of Fuller and
Saorín [8, Example 4.2]. However, if all of the supplementary simples
$S_{m+1},\dots,S_{n}$ have projective dimension at most $1$, then 4.1(ii) does
always hold in this scenario [loc.cit.].

In the present setting, the adjoint pair $(\Lambda e\otimes_{e\Lambda e}-,\
{\bf{e}})$ is particularly useful towards the transfer of homological
information between the categories $\Lambda\mbox{\rm-mod}$ and $e\Lambda
e\mbox{\rm-mod}$. Namely:

######  Lemma 4.3.

Suppose $\Lambda$ satisfies the conditions under 4.1. Then the functors
${\bf{e}}:\Lambda\mbox{\rm-mod}\rightarrow e\Lambda e\text{-mod}$ and its left
adjoint $\Lambda e\otimes_{e\Lambda e}-:e\Lambda e\mbox{\rm-
mod}\rightarrow\Lambda\mbox{\rm-mod}$ preserve and reflect finite projective
dimension.

###### Proof.

By [8, Lemma 1.2(b)], the functor ${\bf{e}}:\Lambda\mbox{\rm-mod}\rightarrow
e\Lambda e\text{-mod}$ preserves finite projective dimension. In light of the
canonical isomorphism ${\bf{e}}\circ(\Lambda e\otimes_{e\Lambda
e}-)\cong\mathop{\rm id}\nolimits_{e\Lambda e\mbox{\rm-mod}}$ this shows that
$\Lambda e\otimes_{e\Lambda e}-$ reflects finite projective dimension.

To see that $\Lambda e\otimes_{e\Lambda e}-$ preserves finite projective
dimension, let $X\in\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$, and let

${\mathbb{X}}:\ \
0\rightarrow\,Q_{m}\,{\overset{f_{m}}{\longrightarrow}}\,\cdots\,{\overset{f_{2}}{\longrightarrow}}\;Q_{1}\,{\overset{f_{1}}{\longrightarrow}}\,Q_{0}\,{\overset{f_{0}}{\longrightarrow}}\,X\,\rightarrow
0$

be a projective resolution in $e\Lambda e\mbox{\rm-mod}$. Then all terms
$\Lambda e\otimes_{e\Lambda e}Q_{j}$ of the complex $\Lambda
e\otimes_{e\Lambda e}{\mathbb{X}}$ are projective over $\Lambda$; indeed, the
$Q_{j}$ belong to $\mathop{\rm add}\nolimits(_{e\Lambda e}e\Lambda e)$, whence
the modules $\Lambda e\otimes_{e\Lambda e}Q_{j}$ belong to the category
$\mathop{\rm add}\nolimits(\Lambda e\otimes_{e\Lambda e}e\Lambda
e)=\mathop{\rm add}\nolimits({{}_{\Lambda}\Lambda e})$. Moreover, all homology
modules of $\Lambda e\otimes_{e\Lambda e}{\mathbb{X}}$ have finite projective
dimension in $\Lambda\mbox{\rm-mod}$; indeed, due to the natural isomorphism
${\bf{e}}(\Lambda e\otimes_{e\Lambda e}{\mathbb{X}})\,\cong\,{\mathbb{X}}$,
they are all annihilated by ${\bf{e}}$, whence their simple composition
factors are direct summands of $\Lambda(1-e)/J(1-e)$. Writing $(F_{i})$ for
the differential $(\Lambda e\otimes_{e\Lambda e}f_{i})$ of $\Lambda
e\otimes_{e\Lambda e}{\mathbb{X}}$, we find that $\mathop{\rm
Ker}\nolimits(F_{m})$ has finite projective dimension. In view of projectivity
of $\Lambda e\otimes_{e\Lambda e}Q_{m}$, so does $\mathop{\rm
Im}\nolimits(F_{m})$, whence we obtain finiteness of $\mathop{\rm
pdim}\nolimits_{\Lambda}\mathop{\rm Ker}\nolimits(F_{m-1})$ from that of
$\mathop{\rm pdim}\nolimits_{\Lambda}\mathop{\rm
Ker}\nolimits(F_{m-1})/\mathop{\rm Im}\nolimits(F_{m})$. An obvious induction
thus shows that $\mathop{\rm Im}\nolimits(F_{0})\cong\Lambda\otimes_{e\Lambda
e}X$ has finite projective dimension over $\Lambda$.

That the functor ${\bf{e}}$ reflects finite projective dimension is now seen
as follows: Given $M\in\Lambda\mbox{\rm-mod}$ such that ${\bf{e}}(M)$ has
finite projective dimension over $e\Lambda e$, we apply the conclusion of the
preceding paragraph and Lemma 2.3 to deduce that $\mathop{\rm
pdim}\nolimits_{\Lambda}\mathfrak{core}(M)<\infty$. Once more invoking 4.1(i),
we conclude that $\mathop{\rm pdim}\nolimits_{\Lambda}M<\infty$. ∎

The following consequences of Lemma 4.3 will facilitate applications of the
main result of this section.

First we combine Lemma 4.3 with the equivalences $e\Lambda
e\text{-Mod}\approx\mathcal{G}\approx{\mathcal{C}\cap\mathcal{F}}$ of Lemma
2.4, to find that finiteness of projective dimensions transfers smoothly among
the categories singled out in Section 2.4. Namely: If
$M_{\sigma}\in\Lambda\mbox{\rm-Mod}$ is an object of the Giraud subcategory
$\mathcal{G}$ of $\Lambda\mbox{\rm-Mod}$, then $M_{\sigma}$ has finite
projective dimension in $\mathcal{G}$ if and only if $\mathop{\rm
pdim}\nolimits_{e\Lambda e}eM_{\sigma}<\infty$, if and only if $\mathop{\rm
pdim}\nolimits_{\Lambda}M_{\sigma}<\infty$. Analogously: If
$M=\mathfrak{core}(M)$ is an object of ${\mathcal{C}\cap\mathcal{F}}$, then
$M$ has finite projective dimension in this subcategory if and only if
$\mathop{\rm pdim}\nolimits_{\Lambda}M<\infty$. This leads to

######  Lemma 4.4.

Let $M\in\Lambda\mbox{\rm-mod}$, suppose that $eM$ has a
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$-approximation, and let
$q:X\rightarrow eM$ be a minimal such approximation. Then $(\Lambda
e\otimes_{e\Lambda e}q)_{\sigma}:(\Lambda e\otimes_{e\Lambda
e}X)_{\sigma}\rightarrow(\Lambda e\otimes_{e\Lambda e}eM)_{\sigma}\cong
M_{\sigma}$ is a minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation of $M_{\sigma}$.

###### Proof.

Referring to the explicit category equivalences of Lemma 2.4, we see that
$(\Lambda e\otimes_{e\Lambda e}q)_{\sigma}$ is a minimal approximation of
$M_{\sigma}$ with respect to the subcategory
$\mathcal{P}^{<\infty}\bigl{(}\mathcal{G}\cap(\Lambda\mbox{\rm-mod})\bigr{)}$
of $\mathcal{G}\cap(\Lambda\mbox{\rm-mod})$. In light of the preceding
remarks, the domain of this map has finite projective dimension also in
$\Lambda\mbox{\rm-mod}$. Right minimality of $(\Lambda e\otimes_{e\Lambda
e}q)_{\sigma}$ in $\Lambda\mbox{\rm-mod}$ is clear. To see that $(\Lambda
e\otimes_{e\Lambda e}q)_{\sigma}$ is actually a
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M_{\sigma}$,
let $U\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and $f\in\mathop{\rm
Hom}\nolimits_{\Lambda}(U,M_{\sigma})$. Since $M_{\sigma}$ is torsionfree and
$U/\Delta(U)$ again belongs to $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$,
we do not lose generality in assuming that $U$ is torsionfree. Consequently,
$\mu_{U}:U\rightarrow U_{\sigma}$ is an embedding with the property that
$U_{\sigma}/U\in\mathcal{T}$. Now the restricted injectivity property of
$M_{\sigma}$ yields an extension of $f$ to $f^{*}:U_{\sigma}\rightarrow
M_{\sigma}$, i.e., $f=f^{*}\circ\mu_{U}$. Since $U_{\sigma}$ has finite
projective dimension in $\mathcal{G}\cap(\Lambda\mbox{\rm-mod})$, we further
obtain a factorization $f^{*}=(\Lambda e\otimes_{e\Lambda e}q)_{\sigma}\circ
g$ for some $g\in\mathop{\rm Hom}\nolimits_{\Lambda}(U_{\sigma},(\Lambda
e\otimes_{e\Lambda e}X)_{\sigma})$. This clearly yields a factorization of $f$
through $(\Lambda e\otimes_{e\Lambda e}q)_{\sigma}$. ∎

The next observation depends only on Condition 4.1(i).

######  Lemma 4.5.

Let $M\in\Lambda\mbox{\rm-mod}$, and let $p:\mathcal{A}(M)\rightarrow M$ be a
minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation. Suppose
$p$ factors in the form $p=\tau\circ\rho$ with $\rho\in\mathop{\rm
Hom}\nolimits_{\Lambda}(\mathcal{A}(M),N)$ and $\tau\in\mathop{\rm
Hom}\nolimits_{\Lambda}(N,M)$.

If $\mathop{\rm pdim}\nolimits N<\infty$ and $\rho$ is an epimorphism, then
$\rho$ is an isomorphism. In particular: If $M\in\mathcal{F}$, then also
$\mathcal{A}(M)\in\mathcal{F}$.

###### Proof.

For the first implication it suffices to observe that, under the given
premise, $\tau:N\rightarrow M$ is in turn a
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M$.
Minimality of $p$ thus shows that
$\text{length}(\mathcal{A}(M))\leq\text{length}(N)$. Therefore the epimorphism
$\rho$ is an isomorphism.

To derive the final implication, assume that $M$ is torsionfree, and apply the
preceding implication to the canonical map
$\rho:\mathcal{A}(M)\rightarrow\mathcal{A}(M)/\Delta\bigl{(}\mathcal{A}(M)\bigr{)}$.
∎

###  4.2. The main theorem

The upcoming theorem shows that, in the situation of 4.1, contravariant
finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ in
$\Lambda\mbox{\rm-mod}$ is equivalent to contravariant finiteness of
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ in $e\Lambda e\mbox{\rm-
mod}$. In fact, we obtain sharper information, relating minimal
$\mathcal{P}^{<\infty}$-approximations in $\Lambda\mbox{\rm-mod}$ to minimal
$\mathcal{P}^{<\infty}$-approximations in $e\Lambda e\mbox{\rm-mod}$.

As before, $(\mathcal{T},\mathcal{F})=({\mathcal{T}}_{e},{\mathcal{F}}_{e})$
is the torsion pair of subsection 2.2. Recall that the class $\mathcal{T}$
consists of the $\Lambda$-modules all of whose composition factors belong to
$\mathop{\rm add}\nolimits\bigl{(}\Lambda(1-e)/J(1-e)\bigr{)}$, while
$\mathcal{F}$ consists of the modules $F$ with $\mathop{\rm soc}\nolimits
F\in\mathop{\rm add}\nolimits(\Lambda e/Je)$.

######  Theorem 4.6.

Let $\Lambda$ be a basic Artin algebra, and let $e\in\Lambda$ be an idempotent
such that the left $\Lambda$-module $\Lambda(1-e)/J(1-e)$ has finite
projective dimension and $e\Lambda(1-e)$ has finite projective dimension as a
left $e\Lambda e$-module.

Then $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite
in $\Lambda\mbox{\rm-mod}$ if and only if $\mathcal{P}^{<\infty}(e\Lambda
e\mbox{\rm-mod})$ is contravariantly finite in $e\Lambda e\text{-}{\mathop{\rm
mod}\nolimits}$.

In more detail: The following two implications hold for every finitely
generated left $\Lambda$-module $F\in\mathcal{F}$.

  1. (1)

If $p:M\longrightarrow F$ is a minimal right
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $F$, then
$p_{|eM}:eM\longrightarrow eF$ is a minimal right
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$-approximation of $eF$.

  2. (2)

Suppose that $q:X\longrightarrow eF$ is a minimal right
$\mathcal{P}^{<\infty}(e\Lambda e\text{-}\mathop{\rm
mod}\nolimits)$-approximation of $eF$, and let $f$ be the composition

$\frac{\Lambda e\otimes_{e\Lambda e}X}{\Delta(\Lambda e\otimes_{e\Lambda
e}X)}\ \ \ \overset{\overline{1\otimes q}}{\longrightarrow}\ \ \ \frac{\Lambda
e\otimes_{e\Lambda e}eF}{\Delta(\Lambda e\otimes_{e\Lambda e}eF)}\ \
\overset{\overline{\epsilon}_{F}}{\longrightarrow}\ \ F,$

where $\overline{\epsilon}_{F}$ is induced by the natural map
$\epsilon_{F}:\Lambda e\otimes_{e\Lambda e}eF\rightarrow F$. Then any
representative of the unique maximal element of the poset $\mathcal{E}_{f}$ of
essential $\Delta$-extensions of $f$ (cf. Proposition 2.7 for existence) is a
minimal right $\mathcal{P}^{<\infty}(\Lambda\text{-}\mathop{\rm
mod}\nolimits)$-approximation of $F$.

######  Remark 4.7.

By Lemma 2.3, the map $\overline{1\otimes q}$ in statement (2) of the theorem
is naturally isomorphic to the map $\mathfrak{core}(1\otimes
q):\mathfrak{core}(\Lambda e\otimes_{e\Lambda
e}X)\rightarrow\mathfrak{core}(F)$ and, modulo the canonical isomorphism
$\mathfrak{core}(F)\cong\frac{\Lambda e\otimes_{e\Lambda e}eF}{\Delta(\Lambda
e\otimes_{e\Lambda e}eF)}$, the map
$\overline{\epsilon}_{F}:\mathfrak{core}(F)=\nabla(F)\rightarrow F$ is the
embedding of $\mathfrak{core}(F)$ into $F$.

###### Proof.

We start by showing that the first assertion follows from (1) and (2). Indeed,
the correspondence $S\leftrightarrow eS$ is a bijection between the
isomorphism classes of simple $\Lambda$-modules in $\mathop{\rm
add}\nolimits(\Lambda e/Je)$ and those of the simple left $e\Lambda
e$-modules. In light of [1, Proposition 3.7] and the fact that the simples in
$\mathop{\rm add}\nolimits\bigl{(}\Lambda(1-e)/J(1-e)\bigr{)}$ have finite
projective dimension, it thus suffices to show that any simple module
$S\in\mathop{\rm add}\nolimits(\Lambda e/Je)$ has a
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation precisely when
$eS$ has a $\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$-approximation.
Clearly, the simple summands $S$ of $\Lambda e/Je$ satisfy $S=\Lambda
eS\in\mathcal{F}$, and thus the required equivalence arises from (1) and (2)
as a special case.

(1) We assume that $p:M\rightarrow F$ is a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $F$, and set
$p^{\prime}=p_{|eM}:eM\longrightarrow eF$. Moreover, we abbreviate $\Lambda
e\otimes_{e\Lambda e}Y$ to $Y^{\dagger}$ for $Y\in{e\Lambda e\mbox{\rm-mod}}$
and $\Lambda e\otimes_{e\Lambda e}eN$ to $N^{\ddagger}$ for
$N\in\Lambda\mbox{\rm-Mod}$; thus $(-)^{\dagger}$ and $(-)^{\ddagger}$ are
functors $e\Lambda e\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-Mod}$ and
$\Lambda\mbox{\rm-Mod}\rightarrow\Lambda\mbox{\rm-Mod}$, respectively, such
that $N^{\ddagger}=(eN)^{\dagger}$. Let $\alpha\in\mathop{\rm
Hom}\nolimits_{e\Lambda e}(Z,eF)$ for some $Z\in\mathcal{P}^{<\infty}(e\Lambda
e\mbox{\rm-mod})$. By Lemma 4.3,
$Z^{\dagger}\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$, and hence the
composition $\epsilon_{F}\circ(1\otimes\alpha):\Lambda e\otimes_{e\Lambda
e}Z=Z^{\dagger}\rightarrow F$ belongs to $\mathop{\rm
Hom}\nolimits_{\Lambda}(\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod}),F)$. By
hypothesis, we thus obtain a map $\beta\in\mathop{\rm
Hom}\nolimits_{\Lambda}(Z^{\dagger},M)$ such that
$\epsilon_{F}\circ(1\otimes\alpha)=p\circ\beta$. Since the functor
${\bf{e}}\circ(\Lambda e\otimes_{e\Lambda e}-)={\bf{e}}\circ(-)^{\dagger}$ is
naturally equivalent to $1_{{\bf{e}}\Lambda e\text{-Mod}}$, application of
${\bf{e}}$ to this equality shows that $\beta$ factors through $p^{\prime}$.
Therefore $p^{\prime}$ is a $\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-
mod})$-approximation of $eF$.

To prove that $p^{\prime}$ is (right) minimal, it suffices to show that no
nonzero direct summand of $eM$ is contained in the kernel of $p^{\prime}$. So
suppose $eM=X\oplus Y$ with $p^{\prime}(Y)=0$, and let $\pi:eM\rightarrow X$
be the projection onto $X$ along $Y$. The latter gives rise to a corresponding
projection ${\rm{pr}}:M^{\ddagger}/\Delta(M^{\ddagger})\rightarrow
X^{\dagger}/\Delta(X^{\dagger})$. Bearing in mind that $M\in\mathcal{F}$ by
Lemma 4.5, we obtain the following commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\ddagger}/\Delta(M^{\ddagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$pr$\scriptstyle{\overline{\epsilon}_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathop{\rm
Coker}\nolimits(\overline{\epsilon}_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\dagger}/\Delta(X^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Coker}\nolimits(u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the left-hand square is the pushout of $\overline{\epsilon}_{M}$ and pr.
In particular, $\rho$ is an epimorphism and $u$ is a monomorphism. Both of the
flanking terms of the bottom exact sequence have finite projective dimension:
For $X^{\dagger}/\Delta(X^{\dagger})$ this follows from Lemma 4.3, for the
righthand term it follows from $\mathop{\rm Coker}\nolimits(u)\cong\mathop{\rm
Coker}\nolimits(\overline{\epsilon}_{M})\cong M/\nabla(M)$. Consequently, also
$N$ has finite projective dimension.

On defining $q$ to be the restriction of
$\overline{\epsilon}_{F}\circ(\overline{1\otimes p^{\prime}})$ to
$X^{\dagger}/\Delta(X^{\dagger})$, one readily checks that the following
diagram commutes:

$\textstyle{M^{\ddagger}/\Delta(M^{\ddagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$pr$\scriptstyle{\overline{\epsilon}_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X^{\dagger}/\Delta(X^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{F}$

The universal property of the pushout therefore yields a map $v\in\mathop{\rm
Hom}\nolimits_{\Lambda}(N,F)$ such that $v\circ\rho=p$. By Lemma 4.5, $\rho$
is an isomorphism, whence so is pr, meaning that
$Y^{\dagger}/\Delta(Y^{\dagger})=0$. Thus $Y^{\dagger}\in\mathcal{T}$, and we
conclude that $Y\cong{\bf{e}}(Y^{\dagger})=0$. This confirms minimality of
$p^{\prime}$.

(2) Let $q:X\longrightarrow eF$ and $f$ be as in the assertion, and suppose
that the pair $(M,p)$ represents the maximum of the set $\mathcal{E}_{f}$. To
check that $p:M\rightarrow F$ is a $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation of $F$, let $h\in\mathop{\rm
Hom}\nolimits_{\Lambda}(N,F)$ with
$N\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$. Since $\Delta(F)=0$ and
$N/\Delta(N)$ again belongs to $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$,
we may assume that $\Delta(N)=0$. In light of Lemma 4.3, the restriction
$h^{\prime}=h|_{eN}:eN\rightarrow eF$ is a morphism in $\mathop{\rm
Hom}\nolimits_{e\Lambda e}\bigl{(}\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-
mod}),eF\bigr{)}$, whence it factors through $q$, say $h^{\prime}=q\circ\eta$
with $\eta\in\mathop{\rm Hom}\nolimits_{e\Lambda e}(eN,X)$. Again referring to
Lemma 4.3, we moreover see that the map
$\overline{1\otimes\eta}:N^{\ddagger}/\Delta(N^{\ddagger})\rightarrow
X^{\dagger}/\Delta(X^{\dagger})$ induced by $\eta$ is a morphism in
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$. In order to suitably extend
$\overline{1\otimes\eta}$ to a homomorphism with domain $N$ via the
monomorphism
$\overline{\epsilon}_{N}:N^{\ddagger}/\Delta(N^{\ddagger})\rightarrow N$, we
consider the pushout diagram of the pair
$(\overline{\epsilon}_{N},\overline{1\otimes\eta})$, as shown in the diagram
below:

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{\ddagger}/\Delta(N^{\ddagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{1\otimes\eta}}$$\scriptstyle{\overline{\epsilon}_{N}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{h}$$\textstyle{\mathop{\rm
Coker}\nolimits(\overline{\epsilon}_{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\dagger}/\Delta(X^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{1\otimes
q}}$$\scriptstyle{\lambda}$$\scriptstyle{f}$$\textstyle{N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Coker}\nolimits(\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{F^{\ddagger}/\Delta(F^{\ddagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\epsilon}_{F}}$$\textstyle{F}$

Since the image of $\overline{\epsilon}_{N}$ is $\nabla(N)$, we may use the
argumentation in the proof of (1) to infer that
$N^{\prime}\in\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$. By the naturality
of $\overline{\epsilon}$ (see Lemma 2.3),
$\overline{\epsilon}_{F}\circ\overline{1\otimes h^{\prime}}$ coincides with
$h\circ\overline{\epsilon}_{N}$, which yields
$f\circ(\overline{1\otimes\eta})=h\circ\overline{\epsilon}_{N}$; in other
words, the above diagram fully commutes. Thus the universal property of the
pushout provides us with a map $\phi\in\mathop{\rm
Hom}\nolimits_{\Lambda}(N^{\prime},F)$ such that $\phi\circ\lambda=f$ and
$\phi\circ g=h$. Given that $F$ is torsionfree, $\phi$ factors through the
canonical map $\pi:N^{\prime}\rightarrow N^{\prime}/\Delta(N^{\prime})$;
denote the induced map $N^{\prime}/\Delta(N^{\prime})\rightarrow F$ by
$\overline{\phi}$, and note that the composition
$\overline{\lambda}:=\pi\circ\lambda$ is still an embedding, since
Im$(\lambda)\cap\Delta(N^{\prime})=0$; it is thus harmless to view
$\overline{\lambda}$ as a set inclusion. We will ascertain that the pair
$\bigl{(}N^{\prime}/\Delta(N^{\prime}),\,\overline{\phi}\,\bigr{)}$ gives rise
to a class in $\mathcal{E}_{f}$: Indeed, the cokernel of $\overline{\lambda}$
belongs to $\mathcal{T}$, since it is an epimorphic image of $N/\nabla(N)$.
Torsionfreeness of $N^{\prime}/\Delta(N^{\prime})$ moreover guarantees that
$\overline{\lambda}$ is an essential extension. That $\overline{\phi}$ extends
$f$, is immediate from our construction. The inequality
$[\bigl{(}N^{\prime}/\Delta(N^{\prime}),\,\overline{\phi}\,\bigr{)}]\preceq[(M,p)]$
now yields a monomorphism $v:N^{\prime}/\Delta(N^{\prime})\rightarrow M$ such
that $p\circ v=\overline{\phi}$. It is straightforward to deduce that $h$
factors through $p$ as required.

To see that $p$ is a right minimal morphism, consider a decomposition
$p=\begin{pmatrix}p_{1}&0\end{pmatrix}:M=U\oplus V\longrightarrow F$, where
$p_{1}$ is right minimal. Due to the fact that $[(M,p)]\in\mathcal{E}_{f}$,
the domain $M$ of $p$ is an essential extension of
$X^{\dagger}/\Delta(X^{\dagger})$ such that the quotient of $M$ modulo
$X^{\dagger}/\Delta(X^{\dagger})$ belongs to $\mathcal{T}$. This means that
$eM=e\bigl{(}X^{\dagger}/\Delta(X^{\dagger})\bigr{)}$, whence the induced map
$p^{\prime}=p_{|eM}:eM\rightarrow eF$ coincides with the restriction
$f^{\prime}$ of $f$ to $e\bigl{(}X^{\dagger}/\Delta(X^{\dagger})\bigr{)}$. In
light of the natural equivalence ${\bf{e}}\circ(\Lambda e\otimes_{e\Lambda
e}-)\cong 1_{e\Lambda e\mbox{\rm-mod}}$, this identifies $f^{\prime}$ with the
minimal approximation $q:X\rightarrow eF$. Due to the matrix decomposition
$p^{\prime}=\begin{pmatrix}p_{1}^{\prime}&0\end{pmatrix}:X\cong eU\oplus
eV\rightarrow eF$, where $p_{1}^{\prime}=p_{1}{{}_{|eU}}$, right minimality of
$q$ thus yields $eV=0$, so that $V\in\mathcal{T}$. On the other hand,
$V\in\mathcal{F}$; this follows from the fact that $M\in\mathcal{F}$, because
the torsionfree module $X^{\dagger}/\Delta(X^{\dagger})$ is an essential
submodule of $M$. Consequently, $V=0$, which proves minimality of $p$ as
claimed. ∎

We apply Theorem 4.6 to the situation where the
$\mathcal{P}^{<\infty}$-categories of $e\Lambda e\mbox{\rm-mod}$ and
(equivalently) $\Lambda\mbox{\rm-mod}$ are contravariantly finite in the
corresponding ambient module categories. The proposition below picks up the
theme of Lemma 4.4 and Remark 4.7. It reinforces our understanding of the
links among the minimal approximations of objects $M\in\Lambda\mbox{\rm-mod}$
and those of the corresponding $\Lambda$-modules $M/\Delta(M)$, $M_{\sigma}$,
and $\mathfrak{core}(M)$. These connections underlie the upcoming exploration
of the basic algebra $\widetilde{\Lambda}$ that results from strongly tilting
$\Lambda\mbox{\rm-mod}$.

######  Proposition 4.8.

Continue to adopt conditions 4.1(i) and (ii), and suppose that
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ is contravariantly finite in
$e\Lambda e\mbox{\rm-mod}$. $\bigl{(}$By Theorem 4.6, this implies
contravariant finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ in
$\Lambda\mbox{\rm-mod}$.$\bigr{)}$ For $M\in\Lambda\mbox{\rm-mod}$, denote by
$\mathcal{A}(M)$ the minimal right $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation of $M$.

(a) If $M\in\mathcal{G}$, then $\mathcal{A}(M)\in\mathcal{G}$.

(b) If $M$ is torsionfree, then $\mathcal{A}(M)$ naturally embeds into
$\mathcal{A}(M_{\sigma})$, and the core of $\mathcal{A}(M)$ coincides with
that of $\mathcal{A}(M_{\sigma})$.

(c) For arbitrary $M\in\Lambda\mbox{\rm-mod}$,
$\mathcal{A}\bigl{(}M/\Delta(M)\bigr{)}\cong\mathcal{A}(M)/\Delta(\mathcal{A}(M))$,
and the torsion submodule $\Delta(\mathcal{A}(M))$ is isomorphic to
$\Delta(M)$. In particular, there is a natural homomorphism
$\rho:\mathcal{A}(M)\rightarrow\mathcal{A}(M_{\sigma})$ such that $\mathop{\rm
Ker}\nolimits(\rho)$ and $\mathop{\rm Coker}\nolimits(\rho)$ belong to
$\mathcal{T}$.

###### Proof.

Part (a) is a consequence of Lemma 4.4. We re-encounter this fact in a
notational setup that is convenient towards parts (b) and (c).

Suppose that $M\in\Lambda\mbox{\rm-mod}$ is torsionfree, and fix an injective
envelope $E$ of $\mathcal{A}(M)$. Since $\mathcal{A}(M)$ is again torsionfree
(Lemma 4.5), so is $\mathop{\rm soc}\nolimits\bigl{(}\mathcal{A}(M)\bigr{)}$,
which implies that $\mathop{\rm soc}\nolimits\bigl{(}\mathcal{A}(M)\bigr{)}$
is contained in the submodule $\mathfrak{core}\bigl{(}\mathcal{A}(M)\bigr{)}$
of $\mathcal{A}(M)$. Therefore $E$ is also an injective envelope of
$\mathfrak{core}\bigl{(}\mathcal{A}(M)\bigr{)}$. Moreover, let
$p:\mathcal{A}(M)\rightarrow M$ be a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation, and set
$X=e\mathcal{A}(M)$. By Theorem 4.6 and Remark 4.7, the restriction ${\bf
e}(p):e\mathcal{A}(M)\rightarrow eM$ is then a minimal
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$-approximation of $eM$, and
$p$ is maximal in the set $\mathcal{E}_{f}$ of essential $\Delta$-extensions
of the composition

($\bullet$) $f:\ \frac{\Lambda e\otimes_{e\Lambda e}X}{\Delta(\Lambda
e\otimes_{e\Lambda e}X)}\,\cong\,\,\mathfrak{core}(\mathcal{A}(M))\
\longrightarrow\,\mathfrak{core}(M)\hookrightarrow\ M.$

(a) Suppose $M=M_{\sigma}$. In view of the fact that ${\bf
e}(p):X=e\mathcal{A}(M)\rightarrow eM$ is a minimal
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$-approximation of $eM$, Lemma
4.4 guarantees that $(\Lambda e\otimes_{e\Lambda e}X)_{\sigma}$ is a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of
$M_{\sigma}=M$, i.e., $\mathcal{A}(M)\cong\bigl{(}\mathcal{A}(M))_{\sigma}$ as
claimed.

(b) In light of (a), $(\Lambda e\otimes_{e\Lambda
e}X)_{\sigma}\cong\bigl{(}\frac{\Lambda e\otimes_{e\Lambda e}X}{\Delta(\Lambda
e\otimes_{e\Lambda e}X)}\bigr{)}_{\sigma}$ is a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M_{\sigma}$.
Hence we infer from $(\bullet)$ that

$\mathcal{A}(M_{\sigma})\cong\bigl{(}\mathfrak{core}\mathcal{A}(M)\bigr{)}_{\sigma}\cong\mathcal{A}(M)_{\sigma}.$

We may thus identify $\mathcal{A}(M_{\sigma})$ with a submodule of $E$
containing $\mathcal{A}(M)$ such that
$\mathcal{A}(M_{\sigma})/\mathcal{A}(M)\in\mathcal{T}$. The latter condition
implies that
$\mathfrak{core}\bigl{(}\mathcal{A}(M_{\sigma})\bigr{)}=\mathfrak{core}\bigl{(}\mathcal{A}(M)\bigr{)}$.

(c) Now let $M$ be arbitrary, and let
$\pi:M\twoheadrightarrow\overline{M}=M/\Delta(M)$ be the canonical map.
Moreover, suppose that $q:\mathcal{A}(\overline{M})\rightarrow\overline{M}$ is
a minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of
$\overline{M}$. On the model of previous arguments, it is straightforward to
check that the map $q^{\prime}:N\rightarrow M$ in the following pullback is a
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of $M$.

$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q^{\prime}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{A}(\overline{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\overline{M}}$

The fact that $\mathop{\rm Ker}\nolimits(\pi^{\prime})\cong\mathop{\rm
Ker}\nolimits(\pi)$ shows that $\mathop{\rm Ker}\nolimits(\pi^{\prime})$ is a
torsion module, necessarily equal to $\Delta(N)$ because
$\mathcal{A}(\overline{M})$ is torsionfree; hence $q^{\prime}$ induces an
isomorphism $\Delta(q^{\prime}):\Delta(N)\cong\Delta(M)$. To check right
minimality of $q^{\prime}$, suppose $u\in\mathop{\rm
End}\nolimits_{\Lambda}(N)$ satisfies $q^{\prime}\circ u=q^{\prime}$. On
applying the functors $\Delta$ and $(1:\Delta)$, the latter acting as
$M\rightarrow(1:\Delta)(M):=\overline{M}$, we find that
$\Delta(u)\in\mathop{\rm End}\nolimits_{\Lambda}\bigl{(}\Delta(N)\bigr{)}$ is
an isomorphism, since $\Delta(q^{\prime})$ is an isomorphism; further
$(1:\Delta)(u)\in\mathop{\rm
End}\nolimits_{\Lambda}\bigl{(}N/\Delta(N)\bigr{)}\cong\mathop{\rm
End}\nolimits_{\Lambda}\bigl{(}\mathcal{A}(\overline{M})\bigr{)}$ is an
isomorphism, since $(1:\Delta)(q^{\prime})\cong
q:\mathcal{A}(\overline{M})\rightarrow\overline{M}$ was chosen right minimal.
Consequently, also $u$ is an isomorphism, which proves $q$ to be right minimal
as claimed.

We conclude that $q^{\prime}$ coincides with $p$ up to isomorphism, whence
$\mathcal{A}(M)/\Delta(\mathcal{A}(M))\linebreak{\cong\mathcal{A}(\overline{M})}$
and $\Delta\bigl{(}\mathcal{A}(M)\bigr{)}\cong\Delta(M)$. In particular, the
composition

$\rho:\mathcal{A}(M)\ \twoheadrightarrow\ \mathcal{A}(M)/\Delta(M)\ \cong\
\mathcal{A}(\overline{M})\ \hookrightarrow\
\mathcal{A}(\overline{M}_{\sigma})=\mathcal{A}(M_{\sigma})$

has the postulated properties. ∎

######  Remark 4.9.

Analyzing the proof of the preceding proposition, one observes the following:
If $\pi_{X}:X\longrightarrow\overline{X}=X/\Delta(X)$ and
$p_{X}:\mathcal{A}(X)\longrightarrow X$ denote the canonical projection and
the minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of a
module $X$, then one has a composition of pullbacks

$\textstyle{\mathcal{A}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{A}(M)}}$$\scriptstyle{p_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{M}}$$\textstyle{\mathcal{A}(\overline{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\overline{M}}}$$\textstyle{\overline{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}(M_{\sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{M_{\sigma}}}$$\textstyle{M_{\sigma}}$

such that the compositions of the vertical arrows are isomorphic to the
canonical maps
$\mu_{\mathcal{A}(M)}:\mathcal{A}(M)\longrightarrow\mathcal{A}(M)_{\sigma}$
and $\mu_{M}:M\longrightarrow M_{\sigma}$, respectively.

In applying Theorem 4.6, we typically decompose $e$ and $1-e$ into primitive
idempotents: $e=e_{1}+\cdots+e_{m}$ and $1-e=e_{m+1}+\cdots+e_{n}$. Since part
(2) of Theorem 4.6 aims at reducing the contravariant finiteness test for
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ to that for
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$, we are interested in making
$e\Lambda e$ as “small” as possible. Hence the situation where all simple left
modules $S_{i}$ of finite projective dimension correspond to idempotents
$e_{i}$ for $i\geq m+1$ is of particular interest. A modification of an
example by Fuller-Saorín (see [8, Example 4.2]) shows that, even for monomial
algebras $\Lambda$, neither of the conditions 4.1(i) nor 4.1(ii) in the
hypothesis of Theorem 4.6 is dispensable.

######  Example 4.10.

Let $\Lambda$ be the monomial algebra based on the quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\gamma}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$

which is defined by the graphs of its indecomposable projective left
$\Lambda$-modules, namely

|  |
---|---|---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\varepsilon}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{2}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{2}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{2}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{2}$$\textstyle{2}$

Clearly, the simple left $\Lambda$-modules $S_{1}$, $S_{2}$ have infinite
projective dimension, whereas $\mathop{\rm pdim}\nolimits_{\Lambda}S_{3}=2$.
Let $e=e_{1}+e_{2}$. Then condition 4.1(i) is satisfied. On the other hand,
the indecomposable projective left $e\Lambda e$-modules $e\Lambda e_{i}$ for
$i=1,2$ and the $e\Lambda e$-module $e\Lambda(1-e)=\Lambda e_{3}$ have the
following graphs:

$\begin{matrix}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
20.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 40.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 58.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
17.5pt\raise-30.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{2}$}}}}}}}{\hbox{\kern
40.5pt\raise-30.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
58.5pt\raise-30.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
81.5pt\raise-30.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}{\hbox{\kern-5.5pt\raise-60.88885pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}{\hbox{\kern
20.0pt\raise-60.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.5pt\raise-60.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
58.5pt\raise-60.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}\ignorespaces}}}}\ignorespaces&\qquad&\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}\ignorespaces}}}}\ignorespaces\cr\begin{matrix}\hbox{the
indecomposable}\cr\hbox{projective left}\cr e\Lambda
e\hbox{-modules}\end{matrix}&&\begin{matrix}\hbox{the}\ e\Lambda
e\hbox{-module}\cr e\Lambda(1-e)\end{matrix}\end{matrix}$

This shows that the left $e\Lambda e$-module $e\Lambda(1-e)$ has infinite
projective dimension. To see that the equivalence of Theorem 4.6 fails,
observe that $\mathop{\rm l.findim}\nolimits e\Lambda e=0$, whence
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ is contravariantly finite in
$e\Lambda e\mbox{\rm-mod}$. Yet $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$
is not contravariantly finite in $\Lambda\mbox{\rm-mod}$, since the simple
module $S_{1}$ does not have a $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation. Indeed, consider the family $(M_{n})_{n\in\mathbb{N}}$
of objects in $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ shown below:

---
$\textstyle{\overset{\displaystyle
x_{1}\smash[t]{\vphantom{\int}}}{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{\displaystyle
x_{2}\smash[t]{\vphantom{\int}}}{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\overset{\displaystyle
x_{n}\smash[t]{\vphantom{\int}}}{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{2}$$\textstyle{2}$$\textstyle{2}$$\textstyle{\cdots}$$\textstyle{2}$$\textstyle{2}$

With the aid of Criterion 10 of [11], one readily checks that no homomorphism
$\phi\in\mathop{\rm
Hom}\nolimits_{\Lambda}\bigl{(}\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod}),S_{1}\bigr{)}$ allows for factorization of all of the following maps
$f_{n}\in\mathop{\rm Hom}\nolimits_{\Lambda}(M_{n},S_{1})$ in the form
$f_{n}=\phi\circ g_{n}$; here $f_{n}(x_{1})=\overline{e_{1}}\in\Lambda
e_{1}/Je_{1}$ and $f_{n}(x_{i})=0$ for $i>1$.

We mention that, in the presence of condition 4.1(ii), condition 4.1(i) in the
hypothesis of Theorem 4.6 is not superfluous either. Instances attesting to
this are ubiquitous. In the above example, take $e=e_{1}$, and note that this
choice makes $e\Lambda e$ semisimple.

##  5\.  The basic strong tilting object in
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and its endomorphism algebra

In this section, we focus on the situation where
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. Let $\widetilde{\Lambda}=\mathop{\rm
End}\nolimits_{\Lambda}(T)^{\mathop{\rm op}\nolimits}$, where ${}_{\Lambda}T$
is the basic strong tilting object in $\Lambda\mbox{\rm-mod}$. The guiding
question is this: When is $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})$ in turn contravariantly finite in $\mbox{\rm
mod-}\widetilde{\Lambda}$? In light of Theorem 3.1, this amounts to the
question of when the process of strongly tilting $\Lambda\mbox{\rm-mod}$ can
be repeated arbitrarily. As witnessed by Example 3.3, the possibility of
iteration is not automatic in case $\Lambda\mbox{\rm-mod}$ can be strongly
tilted once.

######  Setting 5.1 (Upgraded blanket hypothesis).

Throughout this section, let $\Lambda$ be a basic Artin algebra and
$e\in\Lambda$ an idempotent satisfying conditions (i) and (ii) of Setting 4.1,
namely $\mathop{\rm pdim}\nolimits_{\Lambda}\Lambda(1-e)/J(1-e)<\infty$ and
$\mathop{\rm pdim}\nolimits_{e\Lambda e}e\Lambda(1-e)<\infty$, next to the
additional condition that

(iii) $\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ is contravariantly
finite in $e\Lambda e\mbox{\rm-mod}$.

By Theorem 4.6, $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is
contravariantly finite in $\Lambda\mbox{\rm-mod}$ in this setting. We will
introduce an idempotent $\widetilde{e}$ in $\widetilde{\Lambda}$ which
naturally corresponds to $e$. Our objective is to show that the blanket
hypotheses 4.1 of the previous section carry over to $\mbox{\rm
mod-}\widetilde{\Lambda}$, meaning that the right $\widetilde{\Lambda}$-module
$(1-\widetilde{e})\widetilde{\Lambda}/(1-\widetilde{e})\widetilde{J}$ in turn
has finite projective dimension, and the corner
$(1-\widetilde{e})\widetilde{\Lambda}\widetilde{e}$ has finite projective
dimension as a right $\widetilde{e}\widetilde{\Lambda}\widetilde{e}$-module.
In light of Theorem 4.6, this will then allow us to deduce contravariant
finiteness of $\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ from
that of $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{e}\widetilde{\Lambda}\widetilde{e})$.

To this end, we will first assemble some information about the
$\Lambda$-module structure of the basic strong tilting module ${}_{\Lambda}T$.

From Section 2.1 we know that $\mathop{\rm add}\nolimits T=\mathop{\rm
add}\nolimits\mathcal{A}$, where $\mathcal{A}$ is a minimal
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximation of an injective
cogenerator of $\Lambda\mbox{\rm-mod}$. In particular, any such minimal
approximation $\mathcal{A}$ is a strong tilting module. To pin down a
candidate for $\mathcal{A}$, we decompose both $e$ and $1-e$ into sums of
primitive idempotents of $\Lambda$, say $e=e_{1}+\cdots+e_{m}$ and
$1-e=e_{m+1}+\cdots+e_{n}$. Moreover, we let $S_{i}=\Lambda e_{i}/Je_{i}$ be
the corresponding simple modules, and choose $\mathcal{A}_{i}$ to be the
minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$-approximations of their
injective envelopes $E(S_{i})$. Clearly, $\mathcal{A}:=\bigoplus_{1\leq i\leq
n}\mathcal{A}_{i}$ is then as required. Since the injective objects of the
Giraud subcategory $\mathcal{G}$ of $\Lambda\mbox{\rm-mod}$ relative to the
torsion pair $(\mathcal{T},\mathcal{F})$ (see 2.4) are precisely the
torsionfree injective $\Lambda$-modules, the subsum $\bigoplus_{1\leq i\leq
m}E(S_{i})$ is an injective cogenerator in $\mathcal{G}$. Moreover, according
to Lemma 2.2, left multiplication by $e$ induces an equivalence of categories
$\mathcal{G}\cap\Lambda\mbox{\rm-mod}\cong(e\Lambda e)\text{-mod}$, and
$\bigoplus_{1\leq i\leq m}eE(S_{i})$ is an injective cogenerator in $e\Lambda
e\mbox{\rm-mod}$. We separately explore the direct sums $\bigoplus_{1\leq
i\leq m}\mathcal{A}_{i}$ and $\bigoplus_{m+1\leq i\leq n}\mathcal{A}_{i}$.

######  Proposition 5.2.

Assume the hypotheses 5.1, and let $T\in\Lambda\mbox{\rm-mod}$ be the basic
strong tilting module. Then $\mathop{\rm add}\nolimits(\bigoplus_{1\leq i\leq
m}\mathcal{A}_{i})$ contains precisely $m$ isomorphism classes of
indecomposable modules, represented by $T_{1},\dots,T_{m}$ say. If
$T^{\prime}=\bigoplus_{1\leq i\leq m}T_{i}$, then the direct summand
$T^{\prime}$ of $T$ is an object of $\mathcal{G}$, and $eT^{\prime}$ is the
basic strong tilting object in $e\Lambda e\mbox{\rm-mod}$, up to isomorphism.

Supplementary direct summands $T_{m+1},\dots,T_{n}$ of $T$ such that
$T=T^{\prime}\,\oplus\,\bigoplus_{m+1\leq i\leq n}T_{i}$ are as follows: For
$m+1\leq i\leq n$, the minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation $\mathcal{A}_{i}$ of $E(S_{i})$ decomposes in the form
$\mathcal{A}_{i}=T_{i}\oplus U_{i}$ with $\Delta(\mathop{\rm soc}\nolimits
T_{i})=S_{i}$ and $U_{i}\in\mathop{\rm add}\nolimits(T^{\prime})$.

In particular, $\Delta(\mathop{\rm soc}\nolimits T)=S_{m+1}\oplus\cdots\oplus
S_{n}$.

###### Proof.

By Theorem 4.6(1), $\bigoplus_{1\leq i\leq m}e\mathcal{A}_{i}$ is a
$\mathcal{P}^{<\infty}(e\Lambda e\text{-mod})$-approximation of the injective
cogenerator $\bigoplus_{1\leq i\leq m}eE(S_{i})$ in $e\Lambda e\mbox{\rm-
mod}$, which shows that $\bigoplus_{1\leq i\leq m}e\mathcal{A}_{i}$ is a
strong tilting object in $e\Lambda e\mbox{\rm-mod}$. Since the rank of
$K_{0}(e\Lambda e)$ equals $m$, this implies that $\mathop{\rm
add}\nolimits(\bigoplus_{1\leq i\leq m}e\mathcal{A}_{i})$ contains precisely
$m$ isomorphism classes of indecomposable $e\Lambda e$-modules (see Section
2.1). According to Proposition 4.8, the sum $\bigoplus_{1\leq i\leq
m}\mathcal{A}_{i}$ in turn belongs to $\mathcal{G}$, and therefore the
category equivalence $\mathcal{G}\cap\Lambda\mbox{\rm-mod}\cong e\Lambda
e\text{-mod}$ guarantees that $\mathop{\rm add}\nolimits(\bigoplus_{1\leq
i\leq m}\mathcal{A}_{i})$ contains the same number of isomorphism classes of
indecomposable objects. If these are represented by $T_{1},\dots,T_{m}$, then
$e(\bigoplus_{1\leq i\leq m}T_{i})$ is the basic strong tilting $e\Lambda
e$-module by construction. This proves the first claim.

For the final claims, let $i\geq m+1$. We observe that $S_{i}=\mathop{\rm
soc}\nolimits E(S_{i})$ is the only simple torsion module which embeds into
$\mathcal{A}_{i}$; indeed, this is immediate from the fact that
$\Delta(\mathcal{A}_{i})\cong\Delta\bigl{(}E(S_{i})\bigr{)}$ by Proposition
4.8(c). Hence precisely one of the indecomposable direct summands of
$\mathcal{A}_{i}$ contains a copy of $S_{i}$ in its socle; say
$\mathcal{A}_{i}=T_{i}\oplus U_{i}$ with $T_{i}$ indecomposable and
$S_{i}=\Delta(\mathop{\rm soc}\nolimits T_{i})$, while $\Delta(U_{i})=0$. Then
$\bigoplus_{m+1\leq i\leq n}T_{i}$ consists of $n-m$ pairwise nonisomorphic
direct summands in $\mathop{\rm add}\nolimits(\mathcal{A})$, none of which
occurs among $T_{1},\dots,T_{m}$, and we conclude that $T=\bigoplus_{1\leq
i\leq n}T_{i}$ up to isomorphism. Since the $U_{i}$ are torsionfree and hence
do not belong to $\mathop{\rm add}\nolimits(\bigoplus_{m+1\leq i\leq
n}T_{i})$, they belong to $\mathop{\rm add}\nolimits(\bigoplus_{1\leq i\leq
m}T_{i})$.

In view of the fact that $T^{\prime}$ is torsionfree, the final claim follows.
∎

######  Definition.

We refer to Proposition 5.2. Let $\widetilde{e}_{i}\in\widetilde{\Lambda}$ be
the projection $T\rightarrow T_{i}\subseteq T$ with respect to the
decomposition $T=\bigoplus_{1\leq i\leq n}T_{i}$. Viewing $\widetilde{e}_{i}$
as an endomorphism of $T$, we thus obtain a primitive idempotent in
$\widetilde{\Lambda}$. We define $\widetilde{e}:=\sum_{1\leq i\leq
m}\widetilde{e}_{i}\in\widetilde{\Lambda}$. Thus $\widetilde{e}$ is the
projection $T\rightarrow T^{\prime}\subseteq T$ along
$T^{\prime\prime}:=\bigoplus_{m+1\leq i\leq n}T_{i}$.

######  Example 5.3.

(Return to Example 2.2.) Let $\Lambda$ be the algebra of Example 2.2. It is
readily seen that $e=e_{1}+e_{2}$ satisfies the conditions of Setting 5.1,
that $e\Lambda e$ is a self-injective algebra and that the indecomposable
injective modules $I_{1}$ and $I_{2}$ have finite projective dimension. Thus
$T^{\prime}=T_{1}\oplus T_{2}=I_{1}\oplus I_{2}$. Recall from Proposition 5.2
that $T_{3}$ and $T_{4}$ are the indecomposable summands of
$\mathcal{A}_{3}\oplus\mathcal{A}_{4}$ which are not in $\mathcal{F}$. Let us
fix $j\in\\{3,4\\}$ in the sequel. By Proposition 4.8 and Remark 4.9, in order
to calculate $\mathcal{A}_{j}$, we need to identify the minimal
$\mathcal{P}^{<\infty}(\Lambda\mathop{\rm mod}\nolimits)$-approximation of
$(I_{j})_{\sigma}=S_{2}\oplus S_{2}$ (see Example 2.6). Then, by Lemma 4.4,
such an approximation is of the form $(1\otimes q)_{\sigma}\oplus(1\otimes
q)_{\sigma}$, where $q:X\longrightarrow eS_{2}$ is the minimal
$\mathcal{P}^{<\infty}(e\Lambda e\mathop{\rm mod}\nolimits)$-approximation,
whence the projective cover $e\Lambda e_{2}\longrightarrow eS_{2}$ since
$e\Lambda e$ is self-injective. It easily follows that $(1\otimes q)_{\sigma}$
gets identified with the canonical projection $\rho:(\Lambda
e_{2})_{\sigma}=I_{1}\longrightarrow S_{2}$. By Proposition 4.8 and Remark 4.9
again, the $\Lambda$-module $\mathcal{A}_{j}$ is the upper left corner of the
pullback of $\rho\oplus\rho:I_{1}\oplus I_{1}\longrightarrow S_{2}\oplus
S_{2}$ and the projection
$\mu_{I_{j}}:I_{j}\longrightarrow(I_{j})_{\sigma}=S_{2}\oplus S_{2}$. It then
follows that $T_{3}=\mathcal{A}_{3}$ is given by the following diagram, and
$T_{4}=\mathcal{A}_{4}$ is obtained from $T_{3}$ by factoring out the copy of
$S_{3}$ in the socle of $T_{3}$.

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{3}=\mathcal{A}_{3}:}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{3}$

######  Theorem 5.4.

Let $\Lambda$ be a basic Artin algebra. Assume the hypotheses 5.1, and let $T$
be the basic strong tilting module in $\Lambda\mbox{\rm-mod}$. If
$\widetilde{\Lambda}=\mathop{\rm End}\nolimits_{\Lambda}(T)^{\mathop{\rm
op}\nolimits}$, the corner algebra
$\widetilde{e}\widetilde{\Lambda}\widetilde{e}$ is isomorphic to
$\bigl{(}\mathop{\rm End}\nolimits_{e\Lambda
e}(eT^{\prime})\bigr{)}^{\mathop{\rm op}\nolimits}$, where $eT^{\prime}$ is
the basic strong tilting module in $e\Lambda e\mbox{\rm-mod}$ specified in
Proposition 5.2.

The pair $(\widetilde{\Lambda},\widetilde{e})$ satisfies the right-hand
versions of conditions 4.1: Namely, $(i)$ the right
$\widetilde{\Lambda}$-module
$(1-\widetilde{e})\widetilde{\Lambda}/(1-\widetilde{e})\widetilde{J}$ has
finite projective dimension, and $(ii)$ the right
$\widetilde{e}\widetilde{\Lambda}\widetilde{e}$-module
$(1-\widetilde{e})\widetilde{\Lambda}\widetilde{e}$ has finite projective
dimension.

###### Proof.

The isomorphism $\widetilde{e}\widetilde{\Lambda}\widetilde{e}\cong\mathop{\rm
End}\nolimits_{\Lambda}(T^{\prime})^{\mathop{\rm
op}\nolimits}\cong\bigl{(}\mathop{\rm End}\nolimits_{e\Lambda
e}(eT^{\prime})\bigr{)}^{\mathop{\rm op}\nolimits}$ is immediate from
Proposition 5.2 and the comments that precede it. As above, suppose that
$e=e_{1}+\cdots+e_{m}$ and $1-e=e_{m+1}+\cdots+e_{n}$, where the $e_{i}$ are
primitive.

Regarding condition (i) of the claim: By strongness of ${}_{\Lambda}T$, the
functor $\mathop{\rm Hom}\nolimits_{\Lambda}(-,T):\Lambda\mbox{\rm-
mod}\rightarrow\mbox{\rm mod-}\widetilde{\Lambda}$ induces a contravariant
equivalence between $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and a
certain subcategory of $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})$, whence $\mathop{\rm
pdim}\nolimits_{\widetilde{\Lambda}}\mathop{\rm
Hom}\nolimits_{\Lambda}(S_{i},T)<\infty$ for all $i\geq m+1$ (see [13,
Reference Theorem III and Theorem 1] or [14, Theorems 7,8]). So we only need
to show that the right $\widetilde{\Lambda}$-module $\mathop{\rm
Hom}\nolimits_{\Lambda}(S_{i},T)$ is isomorphic to
$\widetilde{e}_{i}\widetilde{\Lambda}/\widetilde{e}_{i}\widetilde{J}$ for
$m+1\leq i\leq n$. Let $i\geq m+1$. In light of Proposition 5.2,
$S_{i}=\Delta(\mathop{\rm soc}\nolimits T_{i})$ is the only occurrence of
$S_{i}$ in the socle of $T$, and therefore any homomorphism $S_{i}\rightarrow
T$ maps $S_{i}$ onto $\Delta(\mathop{\rm soc}\nolimits T_{i})$. If
$\text{in}_{i}:S_{i}\hookrightarrow T_{i}\subseteq T$ is an embedding, we thus
find that $\mathop{\rm
Hom}\nolimits_{\Lambda}(S_{i},T)\cong\text{in}_{i}\,\widetilde{\Lambda}$. One
checks that the latter module is annihilated by the radical $\widetilde{J}$ of
$\widetilde{\Lambda}$, but not annihilated by $\widetilde{e}_{i}$, and
concludes that $\mathop{\rm
Hom}\nolimits_{\Lambda}(S_{i},T)\cong\widetilde{e}_{i}\widetilde{\Lambda}/\widetilde{e}_{i}\widetilde{J}$
as postulated.

To verify condition (ii), namely finiteness of $\mathop{\rm
pdim}\nolimits_{\widetilde{e}\widetilde{\Lambda}\widetilde{e}}\,(1-\widetilde{e})\widetilde{\Lambda}\widetilde{e}$,
we again use Proposition 5.2. The isomorphism
$\widetilde{e}\widetilde{\Lambda}\widetilde{e}\cong\mathop{\rm
End}\nolimits_{\Lambda}(T^{\prime})^{\mathop{\rm op}\nolimits}$ shows that our
claim amounts to finite projective dimension of $\mathop{\rm
Hom}\nolimits_{\Lambda}(T^{\prime\prime},T^{\prime})$ over $\Gamma=\mathop{\rm
End}\nolimits_{\Lambda}(T^{\prime})^{\mathop{\rm op}\nolimits}$. Since
$\mathop{\rm add}\nolimits(T^{\prime\prime})\linebreak\subseteq\mathop{\rm
add}\nolimits\bigl{(}\bigoplus_{m+1\leq i\leq n}\mathcal{A}_{i}\bigr{)}$ by
the proposition, it suffices to show that $\mathop{\rm
pdim}\nolimits_{\Gamma}\,\mathop{\rm
Hom}\nolimits_{\Lambda}(\mathcal{A}_{i},T^{\prime})\linebreak<\infty$ for
$i\geq m+1$. Fix $i\geq m+1$ in the following. Towards another reduction step,
we apply Proposition 4.8 to a minimal $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$-approximation $\widehat{\mathcal{A}}_{i}$ of $E(S_{i})_{\sigma}$. Part
(c) of 4.8 provides us with a map
$\rho:\mathcal{A}_{i}\rightarrow\widehat{\mathcal{A}}_{i}$ such that both
$\mathop{\rm Ker}\nolimits(\rho)$ and $\mathop{\rm Coker}\nolimits(\rho)$
belong to $\mathcal{T}$. Since the restriction of the functor $\mathop{\rm
Hom}\nolimits_{\Lambda}(-,T)$ to $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})$ is exact, so is $\mathop{\rm
Hom}\nolimits_{\Lambda}(-,T^{\prime})_{|_{\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-
mod})}}$, and consequently torsionfreeness of $T^{\prime}$ shows $\mathop{\rm
Hom}\nolimits_{\Lambda}(\rho,T^{\prime}):\mathop{\rm
Hom}\nolimits_{\Lambda}(\widehat{\mathcal{A}}_{i},T^{\prime})\rightarrow\mathop{\rm
Hom}\nolimits_{\Lambda}(\mathcal{A}_{i},T^{\prime})$ to be an isomorphism of
right $\Gamma$-modules. Lemma 4.3 moreover ensures that
$e\widehat{\mathcal{A}}_{i}$ has finite projective dimension in $e\Lambda
e\mbox{\rm-mod}$. Returning to the category equivalence
$\mathcal{G}\cap(\Lambda\mbox{\rm-mod})\approx e\Lambda e\mbox{\rm-mod}$ which
sends $M$ to $eM$ (cf. Lemma 2.4), we thus find finiteness of $\mathop{\rm
pdim}\nolimits_{\Gamma}\mathop{\rm
Hom}\nolimits_{\Lambda}(\widehat{\mathcal{A}}_{i},T^{\prime})$ to be
equivalent to finiteness of the projective dimension of $\mathop{\rm
Hom}\nolimits_{e\Lambda e}(e\widehat{\mathcal{A}}_{i},eT^{\prime})$ over
$\mathop{\rm End}\nolimits_{e\Lambda e}(eT^{\prime})^{\mathop{\rm
op}\nolimits}$ alias $\widetilde{e}\widetilde{\Lambda}\widetilde{e}$. Given
that $\mathop{\rm Hom}\nolimits_{e\Lambda e}(-,eT^{\prime})$ takes objects in
$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ to objects in
$\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{e}\widetilde{\Lambda}\widetilde{e})$ [loc.cit., applied to
$e\Lambda e$ in place of $\Lambda$], condition (ii) follows. ∎

Theorem 5.4 ensures that, with a “duplicate” in $\widetilde{\Lambda}$ of the
original idempotent $e\in\Lambda$, the test provided by Theorem 4.6 is again
available towards deciding whether $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{\Lambda})$ is contravariantly finite in $\mbox{\rm
mod-}\widetilde{\Lambda}$.

######  Corollary 5.5.

Adopt the hypotheses and notation of Theorem 5.4. Then contravariant
finiteness of $\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ in
$\mbox{\rm mod-}\widetilde{\Lambda}$ is equivalent to contravariant finiteness
of $\mathcal{P}^{<\infty}(\mbox{\rm
mod-}\widetilde{e}\widetilde{\Lambda}\widetilde{e})$ in $\mbox{\rm
mod-}\widetilde{e}\widetilde{\Lambda}\widetilde{e}$.

In light of the fact that the basic strong tilt $\widetilde{e\Lambda e}$ of
$e\Lambda e$ coincides with $\widetilde{e}\widetilde{\Lambda}\widetilde{e}$,
we thus obtain: Not only is existence of a strong tilting object in
$\Lambda\mbox{\rm-mod}$ equivalent to existence of a strong tilting object in
$e\Lambda e\mbox{\rm-mod}$ under conditions 4.1, but the same hypothesis
implies that $\Lambda\mbox{\rm-mod}$ allows for arbitrary repetitions of
strong tilting precisely when this is true for $e\Lambda e$-mod. This
conclusion compiles information from Theorems 3.1, 4.6 and Corollary 5.5.

######  Corollary 5.6.

Let $\Lambda$ be a basic Artin algebra and $e\in\Lambda$ an idempotent
satisfying the hypotheses $4.1$ $($i.e., all simple left $\Lambda$-modules of
infinite projective dimension belong to $\mathop{\rm add}\nolimits(\Lambda
e/Je)$, and $\mathop{\rm pdim}\nolimits_{e\Lambda e}e\Lambda(1-e)<\infty$$)$.
Then the category $\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of
strong tilting if and only if the same is true for the category $e\Lambda
e\mbox{\rm-mod}$.

A first straightforward application of our techniques shows that, in testing
for contravariant finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$
or iterability of strong tilting, we may automatically discard the idempotents
corresponding to the simple left $\Lambda$-modules of projective dimension at
most $1$. Namely:

######  Proposition 5.7.

Let $\Lambda$ be a basic Artin algebra on which we fix the complete set of
primitive idempotents $\\{e_{1},...,e_{r},e_{r+1},...,e_{n}\\}$, ordered in
such a way that $\mathop{\rm pdim}\nolimits(\Lambda e_{i}/Je_{i})\leq 1$ for
$i>r$. For $e=\sum_{i=1}^{r}e_{i}$, the following assertions are equivalent:

  1. (1)

$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$ $($resp., $\Lambda\mbox{\rm-mod}$ allows for unlimited
iteration of strong tilting$)$;

  2. (2)

$\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ is contravariantly finite in
$e\Lambda e\mbox{\rm-mod}$ $($resp., $e\Lambda e\mbox{\rm-mod}$ allows for
unlimited iteration of strong tilting$)$.

###### Proof.

From the proof of [8, Proposition 2.1], we know that $e\Lambda(1-e)$ is
projective as a left $e\Lambda e$-module. Thus the pair $(\Lambda\mbox{\rm-
mod},e)$ satisfies the blanket hypotheses of Setting 4.1. Assertions 1 and 2
are thus consequences of Theorem 4.6 and Corollary 5.6, respectively. ∎

##  6\.  Applications and examples

By way of the equivalences established in the previous sections, we can now
easily secure contravariant finiteness of
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ and
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\widetilde{\Lambda})$ in cases in which
this originally required a considerable effort. Moreover, the unifying reasons
behind these results become more transparent through the reduction and permit
us to expand the settings to which they apply.

###  6.1. Precyclic/postcyclic vertices and normed Loewy lengths

An instance in which the simplification gained by reduction to corner algebras
stands out is that of truncated path algebras and their strong tilts (see [13]
and [14]); without a reduction technique, it is challenging to confirm that
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting in
this case.

In light of Theorem 4.6, the first step, namely to confirm contravariant
finiteness of $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ for truncated
$\Lambda$, has now been trivialized. To generalize it, recall that, given any
path algebra modulo relations, $\Lambda=KQ/I$, a vertex $e_{i}$ of $Q$
(systematically identified with a primitive idempotent of $\Lambda$) is called
precyclic if there exists a path of length $\geq 0$ which starts in $e_{i}$
and ends on an oriented cycle; the attribute postcyclic is dual, and $e_{i}$
is called critical if it is both pre- and postcyclic. Clearly, all vertices
which give rise to simple modules of infinite projective dimension are among
the precyclic ones; moreover $e_{i}\Lambda e_{j}=0$ whenever $e_{i}$ is
precyclic, but $e_{j}$ is not. Theorem 4.6 thus yields

######  Proposition 6.1.

Let $\Lambda=KQ/I$ be an arbitrary path algebra modulo relations,and let $e$
be the sum of the primitive idempotents corresponding to the precyclic
vertices of $Q$. If $\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-mod})$ is
contravariantly finite in $e\Lambda e\mbox{\rm-mod}$, e.g., if $\mathop{\rm
l.findim}\nolimits e\Lambda e=0$, then
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$.

For truncated $\Lambda$, the final condition concerning the left finitistic
dimension of $e\Lambda e$ is clearly satisfied since all indecomposable
projective left $e\Lambda e$-modules $e\Lambda e_{i}$ for precyclic $e_{i}$
have the same Loewy length; this is not necessarily true for the
indecomposable injective left $e\Lambda e$-modules, but it is for those whose
socles correspond to critical vertices; namely, if $e^{\prime}$ is the sum of
the critical vertices of $Q$, then the indecomposable injective left
$e^{\prime}\Lambda e^{\prime}$-modules also have coinciding Loewy lengths in
the truncated case. The combination of these two conditions which norm the
Loewy lengths of certain projective or injective modules is, in fact, all that
is needed to guarantee that $\Lambda\mbox{\rm-mod}$ allows for iterated strong
tilting.

######  Proposition 6.2.

Again, let $\Lambda=KQ/I$ be a path algebra modulo relations and $e$ the sum
of the primitive idempotents corresponding to the precyclic vertices of $Q$.
Moreover, let $e^{\prime}$ be the sum of those idempotents which correspond to
the critical vertices.

Suppose that all indecomposable projective left $e\Lambda e$-modules have the
same Loewy length, and that the analogous equality holds for the Loewy lengths
of the indecomposable injective left $e^{\prime}\Lambda e^{\prime}$-modules.
Then $\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting,
thus giving rise to a sequence of related module categories $\Lambda\mbox{\rm-
mod}\,\rightsquigarrow\,\mbox{\rm
mod-}\widetilde{\Lambda}\,\rightsquigarrow\,\widetilde{\widetilde{\Lambda}}\mbox{\rm-
mod}\,\rightsquigarrow\cdots$. Starting with the first strong tilt, $\mbox{\rm
mod-}\widetilde{\Lambda}$, the Morita equivalence classes of these categories
repeat periodically with period $2$.

###### Proof.

Suppose that $e=e_{1}+\cdots+e_{m}$ and $e^{\prime}=e_{1}+\cdots+e_{r}$ for
some $r\leq m$. Since $\mathop{\rm l.findim}\nolimits e\Lambda e=0$ due to the
first condition on Loewy lengths, $\mathcal{P}^{<\infty}(e\Lambda e\mbox{\rm-
mod})$ is contravariantly finite in $e\Lambda e\mbox{\rm-mod}$. Proposition
6.1 thus guarantees that $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is
contravariantly finite in $\Lambda\mbox{\rm-mod}$.

As we already pointed out above, the conditions (i) and (ii) of Setting 4.1
are satisfied for the pair $(\Lambda\mbox{\rm-mod},e)$, whence, by Theorem
5.4, we only need to show that $e\Lambda e\text{-mod}$ allows for unlimited
iteration of strong tilting. Set $\Lambda^{\prime}=e\Lambda e$, and let
$Q^{\prime}$, resp. $J^{\prime}$, be the quiver and Jacobson radical of
$\Lambda^{\prime}$, respectively. We already know that the category
$\mathcal{P}^{<\infty}(\Lambda^{\prime}\mbox{\rm-mod})$ is contravariantly
finite in $\Lambda^{\prime}\mbox{\rm-mod}$, due to the vanishing of the left
finitistic dimension of $\Lambda^{\prime}$. The latter in fact entails that
the basic strong left $\Lambda^{\prime}$-module $T^{\prime}$ is a copy of the
left regular module ${}_{\Lambda^{\prime}}\Lambda^{\prime}$. Consequently, the
strongly tilted algebra $\widetilde{\Lambda^{\prime}}=\widetilde{e\Lambda e}$,
i.e., the opposite of $\mathop{\rm
End}\nolimits_{\Lambda^{\prime}}(T^{\prime})$, coincides with
$\Lambda^{\prime}$. By Theorem 4.6, it therefore suffices to check that
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda^{\prime})$ is contravariantly
finite in $\mbox{\rm mod-}\Lambda^{\prime}$. To confirm this, we observe that
the pair $(\mbox{\rm mod-}\Lambda^{\prime},e^{\prime})$ in turn satisfies the
hypotheses 4.1: Indeed, the precyclic vertices of $(Q^{\prime})^{\mathop{\rm
op}\nolimits}$ are precisely $e_{1},\dots,e_{r}$, whence the simple right
$\Lambda^{\prime}$-modules of infinite projective dimension are among the
quotients $e_{i}\Lambda^{\prime}/e_{i}J^{\prime}$ for $i\leq r$, and
$(e-e^{\prime})\Lambda^{\prime}e^{\prime}=0$. The right finitistic dimension
of $e^{\prime}\Lambda^{\prime}e^{\prime}=e^{\prime}\Lambda e^{\prime}$ is in
turn zero, because all indecomposable projective right $e^{\prime}\Lambda
e^{\prime}$-modules have the same Loewy length; indeed, this follows by
duality from the second of our two hypotheses. Consequently, another
application of Proposition 6.1 yields contravariant finiteness of
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda^{\prime})$ as required.

The concluding statements are part of Theorem 3.1. ∎

We deduce Theorem D of [13] as a special case.

######  Corollary 6.3.

Suppose that $\Lambda=KQ/I$ is a truncated path algebra. Then
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting.

The proof of the following generalization of Proposition 6.2 is immediate from
that of the latter.

######  Corollary 6.4.

Let $\Lambda=KQ/I$, and let $e$ and $e^{\prime}$ be as in the statement of
6.2. If $\,\mathop{\rm l.findim}\nolimits e\Lambda e=0$ and
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}e^{\prime}\Lambda e^{\prime})$ is
contravariantly finite in $\mbox{\rm mod-}e^{\prime}\Lambda e^{\prime}$, then
$\Lambda\mbox{\rm-mod}$ allows for unlimited iteration of strong tilting.

Recall that an Artin algebra $\Lambda$ is said to be left serial if all
indecomposable projective left $\Lambda$-modules are uniserial. The algebras
which are left and right serial are also called Nakayama algebras. In [6] it
was shown that, for any split left serial algebra $\Lambda$, the category
$\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is contravariantly finite in
$\Lambda\mbox{\rm-mod}$. The conclusion actually carries over to the category
$\mbox{\rm mod-}\Lambda$ of right $\Lambda$-modules. Namely:

######  Proposition 6.5.

Suppose that $\Lambda$ is a path algebra modulo relations. If $\Lambda$ is
left serial, then $\mathcal{P}^{<\infty}(\Lambda\mbox{\rm-mod})$ is
contravariantly finite in $\Lambda\mbox{\rm-mod}$ and
$\mathcal{P}^{<\infty}(\mbox{\rm mod-}\Lambda)$ is contravariantly finite in
$\mbox{\rm mod-}\Lambda$.

###### Proof.

The assertion for left modules was proved in [6]. We only address right
$\Lambda$-modules.

By hypothesis, $\Lambda\cong KQ/I$ is left serial. In particular, this means
that $Q$ is free of double arrows. Without loss of generality, we assume that
$Q$ is a connected quiver; we may further assume that it is not a tree, since
otherwise $\Lambda$ has finite global dimension, which renders the
contravariant finiteness claim trivial. By left seriality, $Q$ then contains a
single oriented cycle such that all off-cycle vertices are pre- but not
postcyclic. Consequently, any vertex of $Q^{\mathop{\rm op}\nolimits}$ either
belongs to said cycle or else is post- but not precyclic. On letting $e$ be
the sum of the precyclic vertices of $Q^{\mathop{\rm op}\nolimits}$, i.e., the
vertices located on the oriented cycle in the present situation, we thus
obtain a Nakayama algebra $e\Lambda e$. Hence the
$\mathcal{P}^{<\infty}$-categories in both $e\Lambda e\mbox{\rm-mod}$ and
$\mbox{\rm mod-}e\Lambda e$ are contravariantly finite in the corresponding
ambient module categories by [6, Theorem 5.2]. (The latter may alternatively
be deduced from the fact that Nakayama algebras have finite representation
type.) On combining this with Proposition 6.1, we obtain the claim. ∎

###  6.2. Applications to Morita contexts

In this subsection we shall see that Morita contexts provide a tool to
construct examples of $\mathcal{P}^{<\infty}$-contravariant finiteness and
iteration of strong tilting. Recall that a _Morita context_ (over a ground
commutative ring $K$) consists of a sextuple $(A,B,M,N,\varphi,\psi)$, where
$A$ and $B$ are $K$-algebras, $M$ and $N$ are an $A$-$B$\- and a
$B$-$A$-bimodule, which we always assume with the same action of $K$ on the
left and on the right, and $\varphi:M\otimes_{B}N\longrightarrow A$ and
$\psi:N\otimes_{A}M\longrightarrow B$ are morphisms of $A$-$A$\- and
$B$-$B$-bimodules, respectively, satisfying certain compatibiliity conditions
(see [19]) which are exactly the ones that make $\Lambda=\begin{pmatrix}A&M\\\
N&B\end{pmatrix}$ into a $K$-algebra with the obvious multiplication. Recall
that $\tau_{A}=\text{Im}(\varphi)$ and $\tau_{B}=\text{Im}(\psi)$ are two-
sided ideals of $A$ and $B$, respectively, called the _trace ideals_ of the
Morita context.

The Morita contexts in which we are interested have some additional
properties. We assume that $K$ is artinian and $A$, $B$, $M$ and $N$ are
finitely generated as $K$-modules. Such a Morita context will be called a
_basic Morita context of Artin algebras_ if $\Lambda$ is basic or,
equivalently, if $A$ and $B$ are basic, $\tau_{A}\subseteq J(A)$ and
$\tau_{B}\subseteq J(B)$, where $J(-)$ denotes the Jacobson radical.