Ziegler closures of some unstable tubes

0.3 MB·

by  Lorna Gregory

ZIEGLER CLOSURES OF SOME UNSTABLE TUBES
7 LORNAGREGORY
1
0
2

n
a Abstract. We describe the modules in the Ziegler closure of ray and
J coray tubes in module categories over finite-dimensional algebras. We

1 improveslightlyonKrause’sresultforstabletubesbyshowingthatthe
1 inverse limit along acoray in a ray or coray tubeis indecomposable, so
inparticular,theinverselimitalongacorayinastabletubeisindecom-

] posable. In order to do all this we first describe the finitely presented
T
modulesoverandtheZieglerspectraofiteratedone-pointextensionsof
R valuation domains. Finally we give a sufficient condition for the k-dual
. of a Σ-pure-injectivemodule over a k-algebra to be indecomposable.
h
t Ray and coray tubes frequently occur in Auslander-Reiten quivers of
a
m finite-dimensional algebras. After stable tubes (also referred to as smooth
[ tubes) they are the simplest form of Ringel and D’Este’s coherent tubes.

Generalising Krause’s definition, [Kra98b, pg 20], of a generalised tube
1
we introduce the notions of a generalised coray tube and a generalised ray
v
4 tube. Our aim here is to use these notions to describe the Ziegler closures
8 of unstable tubes containing no projective modules (dually, unstable tubes
1
containing no injective modules).
3
0 As in the case of stable tubes, we show that every ray tube has finitely
. many non-finitely presented indecomposable pure-injectives in its Ziegler
1
0 closure each of which is either a direct limit along a ray, an inverse limit
7 along a coray or a generic module. Improving slightly on Krause’s results
1
we show that in any ray or coray tube (thus also in any stable tube) the
:
v inverse limit along a coray is indecomposable. This result was claimed in
i
X [Pre09, 15.1.10] but no proof is given and the proof indicated there does not
r work.
a
In section 2, we describe the finitely presented modules over iterated one-
point extensions of discrete valuation domains. This allows us, see section
3, to describe the indecomposable pure-injectives over iterated one-point
extensions of discrete valuation domains.
Using these results, we describe a functor from the module category of
the iterated one-point extension of k[[x]] to a module category over a finite-
dimensional k-algebra containing a generalised ray tube such that every
Date: January 13, 2017.
The author acknowledges the support of EPSRC through Grant EP/K022490/1 and
thanksIvoHerzog and Mike Prest for useful discussions. This work was completed while
theauthor was at theUniversity of Manchester.
1
2 LORNAGREGORY
module in the closure of the generalised ray tube is a direct summand of
some module in the image of this functor.
In section 5 we introduce short embeddings. These are embeddings f :
M → L between finite-length modules over an artin algebra such that for
all a ∈ M, if ϕ generates the pp-type of a in M and ψ generates the pp-
type of f(a) in L then the interval [ψ,ϕ] in the pp-1-lattice is finite-length.
Equivalently, an embedding is short if the cokernel of (f,−) : (L,−) →
(M,−) in (mod−R,Ab)fp is finite-length.
Shortembeddingsallowustoinvestigatetheendomorphismringsofdirect
limits along rays in ray and coray tubes. Using results in 5, we are able to
showthatdirectlimitsalongraysinrayandcoray tubesareindecomposable
and moreover that the canonical embedding of k into their endomorphism
rings factored out by the radical is an isomorphism. In section 7 we will
show that this implies that their k-duals are indecomposable.
Inthefinalsection weputall thistogether todescribetheZiegler closures
of ray and coray tubes for finite-dimensional algebras. We end by discussing
some open questions.
Throughout this paper we switch freely between the more concrete pp-
formula formalism and the more abstract functor category formalism.
Throughoutthispaper,ifRisaringthenMod−R(respectivelyR−Mod)
denotes the category of right R-modules (respectively left R-modules) and
mod−R (respectively R−mod) denotes that category of finitely presented
right R-modules (respectively left R-modules).
1. Preliminaries
Let R be a ring.
A pp-n-formula is a formula in the language LR = (0,+,(·r)r∈R) of
(right) R-modules of the form
∃y(x,y)H = 0
wherexisan-tupleofvariables andH isanappropriately sizedmatrix with
entries in R. If ϕ is a pp-formula and M is a right R-module then ϕ(M)
denotes the set of all elements m ∈ Mn such that ϕ(m) holds. Note that
for any module M, ϕ(M) is a subgroup of Mn equipped with the addition
induced by addition in M. A pair of pp-n-formulas ϕ/ψ is called a pp-pair
if for all R-modules M, ϕ(M) ⊇ ψ(M).
If we weaken our definition of a pp-formulato include all formulas (in one
variable) in the language of (right) R-modules, L , which are equivalent
R
over the theory of R-modules, T , to a pp-formula then the T -equivalence
R R
classes of pp-n-formulas become a lattice under implication with the join of
two formulas ϕ,ψ given by
(ϕ+ψ)(x) := ∃y,z(x = y+z∧ϕ(y)∧ψ(z))
andthemeetgiven byϕ∧ψ. Givena(right)R-moduleM,wewritepp1(M)
R
for the lattice of pp-definable subgroups of M.
ZIEGLER CLOSURES OF SOME UNSTABLE TUBES 3
Let M be an R-modules and m ∈ Mn. The pp-type of m in M, denoted
ppM(m), is the set all pp-n-formulas ϕ such that m ∈ ϕ(M).
R
If M is finitely presented module and m ∈ Mn then there is a pp-n-
formula ϕ which generates the pp-type of m in M, that is, for all pp-
formulas ψ, ψ ≥ ϕ if and only if m ∈ ψ(M). Conversely, if ϕ is a pp-
n-formula, then there exists a finitely presented module M and m ∈ Mn
such that ϕ generates the pp-type of m in M. See [Pre09, Section 1.2.2] for
proofs.
Any pp-pair ϕ/ψ, gives rise to a finitely presented functor F : mod−
ϕ/ψ
R → Ab which sends a right R-module M to ϕ(M)/ψ(M) and conversely,
any finitely presented functor is isomorphic to one of the form F , [Pre09,
ϕ/ψ
10.2.30].
Given a lattice L, let ∼ be the congruence relation generated by the
simple intervals in L. We define what it means for a lattice to have finite
m-dimension. For the more general ordinal valued dimension, see [Pre09,
section 7.2]. We say that L has m-dimension 0 if L/ ∼ is the one point
lattice. We say that L has m-dimension n+1 if L/∼ has m-dimension L.
We say that a module has m-dimension n if its lattice of pp-definable
subgroups has m-dimension n.
AnembeddingofR-modulesf :M → N ispureifforevery pp-1-formula
ϕ, f(ϕ(M)) = ϕ(N)∩f(M). A right R-module M is pure-injective if it
is injective over all pure-embeddings.
In section 7, we will use that a module is pure-injective if and only if
it is algebraically compact [Pre09, 4.3.11]. An R-module M is said to be
algebraically compact if every system of equations over R in arbitrary
manyvariables withparameters inM suchthat every finitesubsystemhasa
solution in M, has a solution in M. Equivalently, a module is algebraically
compact if every collection of cosets of pp-definable subgroups which has
the finite intersection property has non-empty intersection.
We say that a module is Σ-pure-injective if it has no descending chain
of pp-definable subgroups. Every Σ-pure-injective module is pure-injective.
The (right) Ziegler spectrum of a ring R is a topological space with set
of points, pinj , the isomorphism classes of indecomposable pure-injectives
R
and basis of open sets given by
(ϕ/ψ) := {N ∈ pinj | ϕ(N) 6= ψ(N)}
R
where ϕ/ψ is a pp-pair.
Our descriptions of Ziegler closures and Ziegler spectras in this paper are
all based on the Ziegler spectrum of a discrete valuation domain V.
Theindecomposablepure-injectives over V aretheindecomposablefinite-
length modules V/ml for l ∈ N, the Pru¨fer module E(V/m) (that is, the
injective hullof V/m), thecompletion V of V andQ(V)thefield of fractions
of V. See for instance [Zie84, 5.1].
b
A subset X of Zg is closed if the following two properties hold:
V
4 LORNAGREGORY
(1) If X contains infinitely many finite-length modules then X contains
E(V/m), V and Q(V).
(2) If X contains V or E(V/m) then X contains Q(V).
b
This description of the topology can be extracted from [Pre09, section
b
5.2.1].
A definable subcategory of Mod −R is a full subcategory closed under
directlimits, productsandpure-submodules. IfD isadefinablesubcategory
then there exists pp-pairs ϕ /ψ , λ ∈ I such that M ∈ D if and only if
λ λ
ϕ (M) = ψ (M) for all λ ∈ I and conversely, all subcategories of this form
λ λ
are definable subcategories.
Adefinablesubcategory of Mod−R is determined by theindecomposable
pure-injective it contains [Pre09, 5.1.4]. Thus there is a bijective correspon-
dencebetween closed subsetof Zg and definablesubcategories of Mod−R.
R
Definition 1.1. Let ϕ be a pp-n-formula in the language of right R-modules
of the form ∃y¯(x¯,y¯)H = 0 where x¯ is a tuple of n variables, y¯ is a tuple of l
variables, H = (H′ H′′)T and H′ (respectively H′′) is a n×m (respectively
l×m)matrixwithentriesinR. ThenDϕisthe pp-n-formulainthe language
I H′ x¯
of left R-modules ∃z¯ = 0.
(cid:18) 0 H′′ (cid:19)(cid:18) z¯ (cid:19)
Similarly, let ϕ be a pp-n-formula in the language of left R-modules of
x¯
the form ∃y¯H = 0 where x¯ is a tuple of n variables, y¯ is a tuple of
(cid:18) y¯ (cid:19)
l variables, H = (H′ H′′) and H′ (respectively H′′) is a m×n (respectively
m×l)matrixwithentriesinR. ThenDϕisthe pp-n-formulainthe language
I 0
of right R-modules ∃z¯(x¯,z¯) = 0.
(cid:18) H′ H′′ (cid:19)
Note that the pp-formula a|x for a ∈ R is mapped by D to a formula
equivalent with respect to T to xa = 0 and the pp-formula xa = 0 for
R
a ∈ R is mapped by D to a formula equivalent with respect to T to a|x.
R
Theorem 1.2. [Pre88, Chapter 8][Pre09, 1.3.1] The map ϕ → Dϕ induces
an anti-isomorphism between the lattice of right pp-n-formulae and the lat-
tice of left pp-n-formulae. In particular, if ϕ,ψ are pp-n-formulae then
D(ϕ+ψ) is equivalent to Dϕ∧Dψ and D(ϕ∧ψ) is equivalent to Dϕ+Dψ.
Let R be a k-algebra. We denote that standard dual Hom(−,k) of an
R-module M, respectively a morphism f, by M∗, respectively f∗.
Lemma 1.3. Let R be a k-algebra and M an R-module. If ϕ(M) ≤ ψ(M)
then Dψ(M∗)≤ Dϕ(M∗).
Thus if M is an R-module with dcc (respectively acc) on pp-definable
subgroups then M∗ has acc (respectively dcc) on pp-definable subgroups.
So, in particular, if the pp-1-lattice of M is finite-length then so is the pp-
1-lattice of M∗.
ZIEGLER CLOSURES OF SOME UNSTABLE TUBES 5
Another key tool in this paper is the use of additive functors F : Mod−
R → Mod −S which commute with direct limits and arbitrary products.
Such functors are called interpretation functors. These are exactly the
functorswhicharefinitelypresentedwhencomposedwiththeforgetfulfunc-
tor to Ab and thus given by pp-pairs. This is where there name comes from
[Pre97]. See [Kra98a, 7.2] and [Pre11, 12.9] for the equivalence.
We will need the following facts about interpretation functors F : Mod−
R → Mod−S.
(1) There exists an n ∈ N, such that for all M ∈ Mod−R, there is a
lattice embedding pp1(FM) ֒→ ppn(M).
S R
(2) If M is pure-injective then FM is pure-injective. See [Pre97, 3.16]
or [Kra98a, 6.1].
(3) If M is Σ-pure-injective (respectively has the acc on pp-definable
subgroups)then FM is Σ-pure-injective (respectively has the acc on
pp-definable subgroups). Follows from (1).
(4) If pp1(M) has m-dimension α then pp1(FM) has m-dimension less
R S
than or equal to α. Follows from (1) plus the fact that the m-
dimension of pp1(M) is equal to the m-dimension of ppn(M).
R R
(5) If D is a definable subcategory of Mod−R then after closing under
direct summands and isomorphism, FD is a definable subcategory.
See [Pre12, 3.8].
(6) If the m-dimension of pp1 is α then the m-dimension of the smallest
R
definable subcategory containing the image of F is less that or equal
to α.
2. Iterated one-point extensions of discrete valuation domains
LetRbearing,kafieldandLak−R-bimodule. Theone-pointextension
of R by L is the ring
R 0
R[L]:= .
(cid:18) L k (cid:19)
RightmodulesoverR[L]maybeviewedastriples(M ,M ,Γ )whereM
0 1 M 0
is a k-vector space, M is a right R-module and Γ : M → Hom (L,M )
1 M 0 R 1
is a k-homomorphism. A morphism between two triples (N ,N ,Γ ) and
0 1 N
(M ,M ,Γ ) is given by a pair (f ,f ) where f : N → M is a k-vector
0 1 M 0 1 0 0 0
space homomorphism, f : N → M is an R-module homomorphism and
1 1 1
the following diagram commutes
N ΓN // Hom (L,N ) .
0 R 1
f0 Hom(L,f1)
M(cid:15)(cid:15) ΓM // Hom ((cid:15)(cid:15)L,M )
0 R 1
There are two full and faithful embeddings of Mod−R into Mod−R[L]:
F :Mod−R → Mod−R[L] M 7→ (0,M,0)
0
6 LORNAGREGORY
and
F :Mod−R → Mod−R[L] M 7→ (Hom(L,M),M,Id ).
1 Hom(L,M)
The forgetful functor
r : Mod−R[L]→ R (M ,M ,Γ) 7→ M
0 1 1
is right adjoint to F and left adjoint to F .
0 1
Each of these additive functors commute with direct limits and arbitrary
products and thus are interpretation functors.
Throughout the rest of this section, let V be a discrete valuation domain
with maximal ideal m and let R := V, L = V/m =:k,
0 0
R 0
R := n
n+1 (cid:18) Ln k (cid:19)
and L = F L .
n+1 1 n
Finally, for each n ∈ N, let T := (k,0,0) ∈Mod−R .
n n
The category of finitely presented modules over a discrete valuation do-
mainV is Krull-Schmidtandtheindecomposablefinitely presentedmodules
are V and V/mn where n ∈ N.
Our goal for the rest of this section is to prove the following theorem and
thus classify the finitely presented right R -modules.
n
Theorem 2.1. The category of finitely presented right modules over R is
n
Krull-Schmidt. The indecomposable finitely presented modules over R are
n
of the form T , FmFn−mN where 0 ≤ m ≤ n and N is an indecomposable
n 0 1
finitely presented module over V and Fn−k−lFlT where k + l ≤ n and
0 1 k
0≤ k,l.
Lemma 2.2. If M = (M ,M ,Γ) is finitely presented then M is finitely
0 1 1
presented and M is finite-dimensional.
0
Proof. SupposeRl −→f Rm → M → 0 is a presentation for M. Applying
n+1 n+1
the exact functor r we get that rRl −r→f rRm → M → 0 is a presenta-
n+1 n+1 1
tion for M . Since rR is finitely presented, M is finitely presented.
1 n+1 1
AsamoduleoveritselfR is(k,R ⊕L ,Γ)whereΓ : k → Hom(L ,R ⊕
n+1 n n n n
L )take 1 ∈ k tothehomomorphismwhichsendsl ∈ L to (0,l) ∈R ⊕L .
n n n n
Thus kl −f→0 km → M → 0 is exact and M is finite-dimensional.
0 0
(cid:3)
In order to prove 2.1, we prove the following two conditions by induction.
Note that 2.1 follows from B by induction on n.
n
A : If M ∈ mod−R , M a finite-dimensional k-vector space and Γ :
n n 0
M → Hom(L ,M) is aninjective k-vector spacehomomorphismthenthere
0 n
is a basis v ,...,v for M and orthogonal idempotent endomorphisms e of
1 n 0 i
M such that e Γ(v ) = Γ(v ) and e M is indecomposable.
i i i i
ZIEGLER CLOSURES OF SOME UNSTABLE TUBES 7
B : All finitely presented modules over R are direct sums of modules of
n n
the form F M,F M and T where M is a finitely presented indecomposable
1 0 n
module over Rn−1 and dimHom(Ln,N) ≤ 1 for all indecomposable finitely
presented modules N over R .
n
Lemma2.3. IfK ∈ Mod−R thenHom(F L ,F K)= 0. HenceF F K ∼=
n 1 n 0 1 0
F F K.
0 0
Proof. Suppose(f ,f )isahomomorphismfromF L toF K. Thenf = 0
0 1 1 n 0 0
and hence f ◦1 = 0. So f = 0. (cid:3)
1 Ln 1
Remark 2.4. Suppose M ∈ mod − R , M a finite-dimensional k-vector
n 0
space, Γ :M → Hom(L ,M) is an injective k-vector space homomorphism
0 n
and α is an automorphism of M. There is a basis v ,...,v for M and
1 n 0
orthogonal idempotent endomorphisms e of M suchthat e Γ(v ) = Γ(v )and
i i i i
e M is indecomposable if and only if there is a basis v ,...,v for M and
i 1 n 0
orthogonal idempotent endomorphisms e of M such that e αΓ(v )= αΓ(v )
i i i i
and e M is indecomposable.
i
Lemma 2.5. If A holds then all finitely presented right R -modules are
n n+1
direct sumsof modules of the form T := (k,0,0), (0,M ,0)and (k,M ,Γ)
n+1 1 1
where M is a finitely presented indecomposable right R -module and Γ is
1 n
an injective k-vector space homomorphism.
Proof. Let (M ,M ,Γ) be an R -module. If (M ,M ,Γ) is finitely pre-
0 1 n+1 0 1
sented then M is finitely presented and M is finite-dimensional by 2.2. If
1 0
Γ is not injective then
∼
(M ,M ,Γ) =(kerΓ,0,0)⊕(M /kerΓ,M ,Γ).
0 1 0 1
So, without loss of generality, we may assume Γ is injective. By A there
n
exists a basis v ,...,v for M and orthogonal idempotent endomorphisms
1 n 0
e ,...,e of M such that e Γ(v ) = Γ(v ). For each 1 ≤ i≤ n, let t : M →
1 n i i i i 0
M be a k-linear map such that t (v ) = v and t (v ) = 0 if i 6= j. So
0 i i i i j
(t ,e ),...,(t ,e ) are orthogonal idempotents for (M ,M ,Γ). Thus
1 1 n n 0 1
n n
(M ,M ,Γ) = (0,(1− e )M ,0)⊕ (v k,e M ,Γ| )
0 1 i 1 i i 1 vik
Xi=1 Mi=1
as required. (cid:3)
Lemma 2.6. For all n ∈ N, dimHom(L ,L ) = 1.
n n
Proof. SincedimHom(L ,L ) = 1andF isfullandfaithful,dimHom(L ,L ) =
0 0 1 n n
1 for all n ∈N. (cid:3)
Lemma 2.7. Let M be a k-vector space, v ,...,v a basis for M and
0 1 m 0
Γ : M → Hom(L ,Tl) be an injective k-vector space homomorphism.
0 n n
There exist e ,...e orthogonal idempotent endomorphism of Tl such that
1 m n
e Γ(v )= Γ(v ) and e Tl is indecomposable.
i i i i n
8 LORNAGREGORY
Proof. Let (α ,0) = Γ(v ),...,(α ,0) = Γ(v ) where each α is a k-linear
1 1 m m i
mapfromHom(Ln−1,Ln−1)tokl. SinceΓisinjectiveanddimHom(Ln−1,Ln−1)=
1, it follows that l ≥ m and α (Id ),...,α (Id ) ∈ kl are linearly
1 Ln−1 m Ln−1
independent. Let ǫ ,...,ǫ be the idempotent endomorphisms of kl such
1 m
that ǫ α (Id )= α (Id )and dimimǫ = 1. Let e = (ǫ ,0),...,e =
i i Ln−1 i Ln−1 i 1 1 m
(ǫ ,0). By definition, e ,...,e have the required properties. (cid:3)
m 1 m
Proposition 2.8. For all n ≥ 1, An−1 and Bn imply An.
Proof. Let M be a finitely presented module over R , M a k-vector space
1 n 0
andΓ :M → Hom(L ,M ) injective. By B , M = F N⊕F K⊕Tl where
0 n 1 n 1 1 0 n
K,N ∈ mod−Rn−1 and l ∈ N0.
By 2.3, Hom(F1Ln−1,F0K)= 0, so we may as well assume K = 0.
Let v ,...,v be a basis for M such that
1 n 0
Γ(v ) = (f ,w ),...,Γ(v ) = (f ,w )
1 1 1 m m m
and
Γ(v )= (0,w ),...,Γ(v ) = (0,w )
m+1 m+1 n n
where f ,...,f ∈ Hom(L ,F N) are linearly independent over k and
1 m n 1
w ,...,w ∈ Hom(L ,Tl).
1 n n n
Since F1 is full, there exist f1∗,...,fm∗ ∈ Hom(Ln−1,N) such that F1f1∗ =
f ,...,F f∗ = f .
1 1 m m
Let w∗,...,w∗ be such that w = (t ,0) and w∗ = t (Id ). Note that
1 n i i i i Ln−1
since Hom(Ln−1,Ln−1) is 1-dimensional, ti and hence wi is determined by
w∗ = t (Id ).
i i Ln−1
Here is a diagram for Γ(v ):
i
Hom(Ln−1,Ln−1) Id // Hom(Ln−1,Ln−1)
g g
7− 7−
→ →
(fi∗◦g,ti(g)) fi∗◦g
(cid:15)(cid:15) (Id,0) (cid:15)(cid:15)
Hom(Ln−1,N)⊕kl // Hom(Ln−1,N)
Let α = (α ,α ) : F N ⊕ Tl → F N ⊕ Tl be such that α ◦ (f∗,0) =
0 1 1 n 1 n 0 i
(f∗,−w∗) for i = 1,...,m, α ◦(0,w∗) = (0,w∗) for i = m+1...,n and
i i 0 i i
α = Id .
1 N
Soα◦Γ(v ) = (f ,0),...α◦Γ(v ) = (f ,0)andα◦Γ(v ) = (0,w ),...,α◦
1 1 m m m+1 m+1
Γ(v )= (0,w ). By 2.4, we may replace Γ by α◦Γ.
n n
Let M′ be the span of v ,...,v . By definition of M′, if u ∈ M′ then
0 1 m 0 0
Γ(u) = (Γ (u),0) ∈ Hom(L ,F N)⊕Hom(L ,Tl) = Hom(L ,F N ⊕Tl).
0 n 1 n n n 1 n
Let ∆ : M0′ → Hom(Ln−1,N) be defined by setting F1∆(u) = Γ0(u). By
An−1, there exists e1,...,em orthogonal idempotent endomorphisms of N
such that e ∆(v ) = ∆(v ) and e N is indecomposable for 1 ≤ i ≤ m. Thus
i i i i
F (e )Γ(v ) = Γ(v ) and F (e )F (N)= F (e N) which is indecomposable.
1 i i i 1 i 1 1 i
ZIEGLER CLOSURES OF SOME UNSTABLE TUBES 9
Let M′′ be the span of v ,...,v . By 2.7, there exist e ,...e
0 m+1 n m+1 n
orthogonal idempotent endomorphism of Tl such that e Γ(v ) = Γ(v ) and
n i i i
e Tl is indecomposable.
i n
So σ = (e ,0),...,σ = (e ,0) and σ = (0,e ),...,σ = (0,e )
1 1 m m m+1 m+1 n n
are orthogonal idempotent endomorphismsof F N⊕Tl such that σ Γ(v ) =
1 n i i
Γ(v ) and σ (F N ⊕Tl) is indecomposable. (cid:3)
i i 1 n
Proposition 2.9. For all n ∈N, B and A imply B .
n n n+1
Proof. By 2.5, A implies that each finitely presented module over R is
n n+1
a direct sum of modules of the form T ,(k,M ,Γ) and F M = (0,M ,0)
n+1 1 1 1 1
where Γ is injective and M is an indecomposable R -module. So in order
1 n
to show that the first clause of B is true, we need now consider modules
n+1
of the form (k,M ,Γ). By B , M is either F N, F K or T where N,K
1 n 1 1 0 n
are indecomposable Rn−1-modules.
Since Hom(L ,F K) = 0, if M = F K then Γ is not injective.
n 0 1 0
IfM = T then(k,T ,Γ)isisomorphictoF T sincedim Hom(L ,T )=
1 n n 1 n k n n
1 i.e. (Γ,Id ): (k,T ,Γ) → F T is an isomorphism.
Tn n 1 n
Now suppose that M1 = F1N for N an indecomposable Rn−1-module.
Since Γ : k → Hom(L ,F N) is injective, dimHom(L ,F N) 6= 0. By
n 1 n 1
Bn−1, dimHom(Ln−1,N) ≤ 1. So dimHom(Ln,F1N) = 1. Thus (Γ,IdN) :
(k,N,Γ) → F N is an isomorphism.
1
It now remains to show that dimHom(L ,M) ≤ 1 for all indecom-
n+1
posable M ∈ mod − R . If M = F K then dimHom(L ,M) = 0.
n+1 0 n+1
If M = F N then dimHom(L ,M) = dimHom(L ,N) ≤ 1 by B . If
1 n+1 n n
M = T then dimHom(L ,M) ≤ 1 follows from 2.6 and the definition
n+1 n+1
of T .
n+1
(cid:3)
We now consider the base cases, A and B .
0 1
Lemma 2.10. Let V be a discrete valuation domain with maximal ideal m,
M a finitely presented V-module and M ⊆ Hom(V/m,M) a V/m-vector
0
subspace. There exists v ,...,v a basis for M as a V/m-vector space and
1 n 0
orthogonal idempotent endomorphisms of M such that e M is indecompos-
i
able and e v = v for 1 ≤ i≤ n.
i i i
Proof. Since V is a principal ideal domain, M = M′ ⊕Vm where M′ is a
torsion module. Since Hom(V/m,V) = 0, we may replace M by M′. Let
W = {f(a) ∈ M |a ∈ V/m and f ∈M }. NotethatW isasubmoduleofM.
0
Let p ∈ V generate m and let L = {m ∈ M | 0 6= mpl ∈ W for some l ∈ N}.
Since V is an RD-ring (see [Pre09, section 2.4.2]), L is pure in M. Since
M and hence L is finite-length, L is pure-injective. Therefore L is a direct
summand of M.
The lemma now follows from the structure theorem for finitely generated
module over principle ideal domains.
(cid:3)
10 LORNAGREGORY
Lemma 2.11. B holds.
1
Proof. We need to show that all finitely presented modules over R are
1
direct sums of modules of the form F M := (0,M,0), F M and T :=
0 1 1
(k,0,0) whereM is a finitely presented indecomposable moduleover V. Let
(M ,M,Γ) be an arbitrary finitely presented module over R . As usual, we
0 1
may assume Γ is injective. Letting e ,...,e be as in 2.10, we have that
1 n
n n
(M ,M,Γ) ∼= (0,(1− e )M,0)⊕ (v k,e M,Γ| ).
0 i i i vik
Xi=1 Mi=1
Themodule(0,(1− n e )M,0) is equaltoF (1− n e )M andforeach
i=1 i 0 i=1 i
1≤ i ≤ n, (vik,eiMP,Γ|vik) is isomorphic to F1eiM.P
That Hom(L ,N) is 1-dimensional is proved exactly as in the proof of
1
2.9. (cid:3)
Corollary 2.12. For all n ≥ 0 and m ≥ 1, A and B hold.
n m
Corollary 2.13. All indecomposable finitely presented modules over R
n
have local endomorphism rings.
3. Indecomposable pure-injectives and the Ziegler spectrum
Throughout this section, let V be a valuation domain with maximal idea
m.
Lemma 3.1. All finitely presented indecomposable right R -modules are
n+1
pure-injective except for Fn+1V. The module Fn+1V is pure-injective if and
0 0
only if V is pure-injective as a right module over itself.
Proof. This follows directly from 2.1, the fact that the functors F and F
0 1
preserve pure-injectivity and that (k,0,0) is finite-length and hence pure-
injective. (cid:3)
Proposition 3.2. Every indecomposable pure-injective module over R
n+1
is of the form F N, F N or T := (k,0,0) for some indecomposable pure-
0 1 n
injective R -module N.
n
Proof. Since all finitely presented indecomposable modules over R are
n+1
pure-injective except Fn+1V, the set of finitely presented pure-injective
0
modulestogether with Fn+1V isdensein Zg . SinceF andF commute
0 Rn+1 0 1
with direct limits and products, the images of F and F are definable sub-
0 1
b
categories of Zg after closing under direct summands (see [Pre12, 3.8]).
Rn+1
Thus Add(imF )∩Zg =: C and Add(imF )∩Zg =: C are closed
0 Rn+1 0 1 Rn+1 1
subsets of Zg . The point (k,0,0) is a closed point of Zg . Since
Rn+1 Rn+1
C ∪ C ∪ {(k,0,0)} is a dense closed subset of Zg it is all of Zg .
1 2 Rn+1 Rn+1
Thus all indecomposable pure-injecitve R -modules are of the required
n+1
form. (cid:3)
Theorem 3.3. The indecomposable pure-injective modules over R are
n