On fully wild categories of representations of posets
Assume that I is a finite partially ordered set and k is a field. We prove that if the category prin(kI) of prinjective modules over the incidence k-algebra kI of I is fully k-wild then the category fpr(I,k) of finite dimensional k-representations of I is also fully k-wild. A key argument is a c...
Gespeichert in:
| Veröffentlicht in: | Algebra and Discrete Mathematics |
|---|---|
| Datum: | 2006 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2006
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/157368 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On fully wild categories of representations of posets / S. Kasjan // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 71–91. — Бібліогр.: 29 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-157368 |
|---|---|
| record_format |
dspace |
| spelling |
Kasjan, S. 2019-06-20T03:07:20Z 2019-06-20T03:07:20Z 2006 On fully wild categories of representations of posets / S. Kasjan // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 71–91. — Бібліогр.: 29 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G60, 16G30, 03C60. https://nasplib.isofts.kiev.ua/handle/123456789/157368 Assume that I is a finite partially ordered set and k is a field. We prove that if the category prin(kI) of prinjective modules over the incidence k-algebra kI of I is fully k-wild then the category fpr(I,k) of finite dimensional k-representations of I is also fully k-wild. A key argument is a construction of fully faithful exact endofunctors of the category of finite dimensional khx,yi-modules, with the image contained in certain subcategories. The author thanks Daniel Simson for stimulating remarks and discussions on the subject of this article. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On fully wild categories of representations of posets Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On fully wild categories of representations of posets |
| spellingShingle |
On fully wild categories of representations of posets Kasjan, S. |
| title_short |
On fully wild categories of representations of posets |
| title_full |
On fully wild categories of representations of posets |
| title_fullStr |
On fully wild categories of representations of posets |
| title_full_unstemmed |
On fully wild categories of representations of posets |
| title_sort |
on fully wild categories of representations of posets |
| author |
Kasjan, S. |
| author_facet |
Kasjan, S. |
| publishDate |
2006 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
Assume that I is a finite partially ordered set and
k is a field. We prove that if the category prin(kI) of prinjective
modules over the incidence k-algebra kI of I is fully k-wild then the
category fpr(I,k) of finite dimensional k-representations of I is also
fully k-wild. A key argument is a construction of fully faithful exact
endofunctors of the category of finite dimensional khx,yi-modules,
with the image contained in certain subcategories.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/157368 |
| citation_txt |
On fully wild categories of representations of posets / S. Kasjan // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 71–91. — Бібліогр.: 29 назв. — англ. |
| work_keys_str_mv |
AT kasjans onfullywildcategoriesofrepresentationsofposets |
| first_indexed |
2025-11-24T16:09:58Z |
| last_indexed |
2025-11-24T16:09:58Z |
| _version_ |
1850850947482255360 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2006). pp. 71 – 91
c© Journal “Algebra and Discrete Mathematics”
On fully wild categories of representations of
posets
Stanis law Kasjan
Communicated by D. Simson
Abstract. Assume that I is a finite partially ordered set and
k is a field. We prove that if the category prin(kI) of prinjective
modules over the incidence k-algebra kI of I is fully k-wild then the
category fpr(I, k) of finite dimensional k-representations of I is also
fully k-wild. A key argument is a construction of fully faithful exact
endofunctors of the category of finite dimensional k〈x, y〉-modules,
with the image contained in certain subcategories.
1. Introduction
Throughout k is a field, I stands for a finite partially ordered set (poset).
Representations of posets have been successfully applied to investi-
gate in particular: lattices over orders, Cohen-Macaulay modules, see
[24], and abelian groups [1], see also [2], [3]. The present paper is moti-
vated by the latter family of applications. More precisely, the motivation
comes from the works of Arnold and Simson on realization of algebras
as endomorphism algebras of so called filtered representations of posets
(see [3]). This leads to the concept of k-endo-wildness [3], [28]. The
main result of [3] is that k-endo wildness is equivalent to k-wildness for
the category of k-representations of a poset I having a unique maximal
element (for arbitrary, possibly finite, field). A case of our main result is
used to obtain it in [3].
Supported by Polish KBN Grant 5 P03A 015 21
2000 Mathematics Subject Classification: 16G60, 16G30, 03C60.
Key words and phrases: representations of posets, wild, fully wild representa-
tion type, endofunctors of wild module category.
72 On fully wild categories
Assume for a moment that k is algebraically closed. A combinatorial
criterion for wildness of the matrix problem associated with I was given
by Nazarova in [15], see [24, Theorem 15.3], in case when I has a unique
maximal element. It has been extended partially to a class of posets
with more maximal elements in [12], [13], [14]. Recall that the matrix
problem associated with a poset I has an interpretation in terms of the
category prin(kI) of prinjective modules or the category modsp(kI) of
socle projective modules over the incidence algebra kI of I, see [25]. A
consequence of the Nazarova’s result and its proof is that the category of
prinjective modules over kI is fully wild (in the sense of [26, Definition
2.4]) provided it is wild, when I has a unique maximal element. This
is no longer true when we consider posets with more maximal elements
(see [6, Remark 5.8]), but still remains true for a wide class of posets
considered in [14].
For the concept of tame and wild representation types and a detailed
discussion of their various aspects the reader is referred e.g. to [7], [24],
[26].
The wildness of the category prin(kI) is equivalent to the wildness of
modsp(kI) by the results of [25, Proposition 2.4], [27], [10]. It was not
clear if this is true with respect to fully wildness as well.
The aim of this paper is to confirm that fully wildness of prin(kI)
implies fully wildness of modsp(kI). To be precise, we work with the
concept of fully k-wildness defined in [3], [28, Definition 2.4] over an
arbitrary, not necessarily algebraically closed field k, see 2.5 below. Note
that the results of [14] are valid over an arbitary field.
The paper is organized as follows. We collect basic concepts and for-
mulate one of our main results, Theorem 2.9, in Section 2. For more
information on posets and their representations the reader is referred to
the monograph by Simson [24]. Sections 3 and 4, devoted to full endo-
functors of the category of k〈x, y〉-modules, can be read independently on
the rest of the paper. Section 3 contains our second main result - Theo-
rem 3.3. This is, together with Theorem 4.2, the main tool for the proof
of Theorem 2.9, which is finished in Section 5. Throughout we formulate
our considerations in terms of modules over the incidence algebra of a
poset. The applications to filtered representations of posets are given in
Section 6.
We use the following notation: N is the set of natural numbers
{0, 1, 2, ...}. Given two indices i, j we put δij = 1 provided i = j and
δij = 0 otherwise. We denote by W the free associative k-algebra k〈x, y〉
with two free noncommuting generators x and y. We often refer to the
natural grading of W in which the generators x, y have degree 1. Given
a k-algebra A let mod (A) be the category of right finitely generated
S. Kasjan 73
A-modules. The full subcategory of modules of finite k-dimension is de-
noted by modf(A). Given a ring S let Mm×n(S) be the S-module of
m×n-matrices with coefficients in S. We often deal with block matrices;
if the block partition of a matrix A is fixed then the (i, j)-th block of A
is denoted usually by Aij . We put Mm(S) for Mm×m(S) equipped with
the natural ring structure. The identity matrix of size m×m is denoted
by Im.
2. Posets and their representations
2.1. Let I = (I,�) be a finite partially ordered set (poset) and denote by
max I the set of its maximal elements. Assume that I = {1, ..., n, p1, ..., pr},
where max I = {p1, ..., pr}. Denote I \ max I by I−.
Let v ∈ NI . Following the idea from [16], [25] given a k-algebra S we
define the variety of I-matrices of size v with coefficients in S as follows
MatI,v(S) = {A = (Api) ∈
∏
i∈I,p∈max I
Mv(p)×v(i)(S) : Api = 0 if i � p}.
We have to admit "degenerated" matrices without rows or without co-
lumns, as in Chapter 2 of [24]. It is convenient to think about the elements
of MatI,v(S) as block matrices with the horizontal blocks indexed by
elements of max I and the vertical ones - by elements of I−.
The reader is referred to [25] for the structure of a G-set on MatI,v(k)
for a suitable algebraic group G.
2.2. Let kI denote the incidence algebra of I with coefficients in k,
that is, the algebra formed by all I×I-matrices [λij ]i,j∈I such that λij = 0
provided i � j in I, [25]. For i � j let eij ∈ kI be the elementary matrix
with 1 at the (i, j)-position and zeros elsewhere. Let ei = eii; ei is the
standard idempotent matrix corresponding to i ∈ I. The algebra kI can
be viewed in a triangular matrix form
[
A M
0 B
]
where A = kI−, B is the semisimple algebra k(max I) and M is the A-
B-bimodule
⊕
i≺p∈max I eikIep. According to this presentation the right
kI-modules can be treated as triples
(X ′
A, X
′′
B, φ : X ′
A ⊗A M −→X ′′
B),
where X ′
A is a right A-module, X ′′
B is a right B-module and φ is a B-
homomorphisms. More precisely, one can define the category of such
74 On fully wild categories
triples with morphisms defined in a usual way and observe that this
category is equivalent to the category of right kI-modules. Dually, we
can view kI-modules as triples
(X ′
A, X
′′
B, φ : X ′
A −→HomB(M,X ′′
B)).
If φ is the homomorphism adjoint to φ then the two above triples rep-
resent isomorphic kI-modules. From now on we denote the functor
HomB(M,−) by | − |. See [20] for details.
2.3. We recall from [25] the construction of the prinjective mod-
ule associated to a block matrix A ∈ MatI,v(k). Given v ∈ NI let
PA(v) be the right projective A-module
⊕
i∈I−(eiA)v(i) and let QB(v) =⊕
p∈max I(epB)v(p). If S is a k-algebra then we put PS
A(v) = S ⊗k PA(v)
and QS
B(v) = S ⊗k QB(v).
Observe that
PA(v)ei ∼=
⊕
j�i
kv(j) ⊗k keji
and
|QB(v)|ei ∼=
⊕
i≺p∈max I
kv(p) ⊗k keip
as k-vector spaces for i ∈ I−. We fix the isomorphisms and treat them
as identities.
Denote by ξt the t-th standard basis element of the free S-module Sl,
where t ≤ l.
Now let A = (Api)i∈I−,p∈max I ∈ MatI,v(S). The matrix A defines an
S-A-bimodule homomorphism
φA : PS
A(v)−→HomB(M,QS
B(v))
such that the restriction of φA to PS
A(v)ei is defined by the block matrix
ΦA[i] =
Aq1j1 Aq1j2 ... Aq1,jt
Aq2j1 Aq2j2 ... Aq2,jt
... ...
Aqsj1 Aqsj2 ... Aqs,jt
with respect to the basis
ξ1 ⊗ ej1,i, ..., ξv(j1) ⊗ ej1,i,
ξ1 ⊗ ej2,i, ..., ξv(j2) ⊗ ej2,i,
..............................
ξ1 ⊗ ejt,i, ..., ξv(jt) ⊗ ejt,i
S. Kasjan 75
of PS
A(v)ei and
ξ1 ⊗ ei,q1
, ..., ξv(q1) ⊗ ei,q1
,
ξ1 ⊗ ei,q2
, ..., ξv(q2) ⊗ ei,q2
,
...............................
ξ1 ⊗ ei,qs
, ..., ξv(qs) ⊗ ei,qs
of |QS
B(v)|ei, where {j1, ..., jt} = {j ∈ I : j � i}, {q1, ..., qs} = {q ∈
max I : i ≺ q}.
The homomorphism φA is represented by the block matrix
ΦA[1] 0 ... 0
0 ΦA[2] ... 0
0 0
. . . 0
0 0 ... ΦA[n]
in the suitable bases of PS
A(v) and |QS
B(v)|.
The S-kI-bimodule identified with the triple
(PS
A(v), QS
B(v), φA : PS
A(v)−→HomB(M,QS
B(v))
will be denoted by ÂS .
If A ∈ MatI,v(k) then Âk is a finite dimensional kI-module which is
prinjective, that is, its restriction to kI− is a projective kI−-module.
The category of all (finite dimensional) prinjective right kI-modules is
denoted by prin(kI). We refer to [25], [18] for a detailed discussion of
this category.
Every module X in prin(kI) is isomorphic to Âk for some A ∈
MatI,v(k) and a uniquely determined v ∈ NI by [25, Proposition 2.3].
Such v is called the coordinate vector of X and it is denoted by
cdn(X). See [24], [25], [18] for a definition of cdn(X) expressed in
terms of the module X. Introduce the following notation: cdn(X) =
(cdn′(X), cdn′′(X)), where cdn′(X) and cdn′′(X) are the projections
of cdn(X) onto NI− and Nmax I respectively.
2.4. Lemma. Let X = (X ′
A, X
′′
B, φ : X ′
A −→|X ′′
B|) and Y =
(Y ′
A, Y
′′
B , ψ : Y ′
A −→|Y ′′
B |) be prinjective kI-modules.
(a) Ext1kI(X,Y ) ∼= HomA(X ′, |Y ′′|)/B(X,Y ), where
B(X,Y ) = {|r′′|φ− ψr′ : r′′ ∈ HomB(X ′′, Y ′′), r′ ∈ HomA(X ′, Y ′)}.
(b) There is a k-linear surjection
ΞX,Y : MatI,v(k)−→Ext1kI(X,Y )
76 On fully wild categories
where v = (cdn′(X), cdn′′(Y )).
Proof. The proof of (a) is standard, whereas (b) follows from (a):
we define the homomorphism
MatI,v(k)−→HomA(X ′, |Y ′′|)
in the same way as the map A 7→ φA in 2.3 and observe that it is surjec-
tive.
2.5. Let C be a full exact additive subcategory of mod (kI). Fol-
lowing [26], [24, Section 14.2], [28] we say that the category C is of fully
k-wild representation type if there exists a full, faithful and exact k-
linear functor T : modf(W)−→ mod (kI) with the image contained
in C. The category C is k-wild if there exists an exact k-linear func-
tor T : modf(W)−→ mod (kI) preserving indecomposability, respecting
isomorphism classes and with the image contained in C. It follows from
the Wildness Correction Lemma in [26] that prin(kI) is of fully k-wild
representation type if and only if it is of strictly wild representation type
in the sense of [4] and [17], that is, there exists a W-kI-bimodule WNkI
which is a finitely generated free W-module and induces a fully faithful
exact functor (−)⊗WNkI : modf(W)−→ mod (kI) with the image in C,
[28, Lemma 2.5]. If k is algebraically closed then k-wildness is equivalent
to wildness in the usual sense, see [7].
Our proofs of fully wildness are based on the idea from [19]: a suitable
functor is determined by a pair of orthogonal "bricks" X, Y with at least
3-dimensional extension group Ext1(X,Y ).
2.6. Lemma. Assume that the category prin(kI) is of fully k-wild
prinjective type. Then there exists prinjective modules X,Y satisfying the
conditions:
1. EndkI(X) ∼= EndkI(Y ) ∼= k,
2. HomkI(X,Y ) = HomkI(Y,X) = 0,
3. dimk Ext1kI(X,Y ) ≥ 3.
For the proof see e.g. Lemma 3.6 in [14] and its proof.
We will show that if the category prin(kI) is fully k-wild then there is
a bimodule WNkI defining its fully k-wildness and having a very special
form.
2.7. Lemma. Assume that the cateory prin(kI) is of fully k-wild
prinjective type. Then there exists v ∈ NI and N ∈ MatI,v(W) such that
S. Kasjan 77
the induced functor
(−) ⊗W N̂W : modf(W)−→ prin(kI)
is full, N has only two non-constant entries and they have degree 1.
Proof. Assume that X = Âk and Y = B̂k are prinjective modules
satisfying the conditions of Lemma 2.6. Let v = (cdn′(X), cdn′′(Y )) and
let E1, E2, E3 ∈ MatI,v(k) be elements such that
ΞX,Y (E1),ΞX,Y (E2),ΞX,Y (E3) ∈ Ext1kI(X,Y )
are linearly independent (see 2.4) and if Ei = (Ei
pj)j∈I−,p∈max I then only
one of the matrices Ei
pj is nonzero and it has only one nonzero entry, for
i = 2, 3. Then it follows by Lemmas 1.5 and 8.6 [19], (see also [14]) that
N = (Npj)j∈I−,p∈max I , where each Npj has the form
Npj =
[
Apj E1
pj + xE2
pj + yE3
pj
0 Bpj
]
satisfies the required condition.
2.8. Following [22], [20] a right kI-module is called socle projective
if its right socle is a projective kI-module. A module X identified with a
triple
(X ′
A, X
′′
B, φ : X ′
A −→|X ′′
B|)
is socle projective if and only if the map φ is injective. The category of
socle projective kI-modules is denoted by modsp(kI).
There is a nice adjustment functor introduced in [20], [25], [18]
ΘB : prin(kI)−→modsp(kI)
defined as the restriction of the functor
Θ′
B : mod (kI)−→modsp(kI)
associating to a triple
(X ′
A, X
′′
B, φ : X ′
A −→|X ′′
B|)
the triple (Im(φ), X ′′
B, u : Im(φ)−→|X ′′
B|), where u is the identity embed-
ding. More generally, given a ring S and an S-kI-bimodule X we denote
by ΘB(X) the S-kI-bimodule defined in the same way.
Recall the relevant properties of the functor ΘB.
Theorem [25, Lemma 2.1, Proposition 2.4]. Let k be a field and I a
finite poset.
78 On fully wild categories
(a) The functor ΘB is full, dense and the kernel of ΘB is the ideal in
the category prin(kI) consisting of all homomorphism factorizing through
modules of the form (X ′
A, 0, 0).
(b) The category prin(kI) is of finite representation type (that is, ad-
mits only finitely many isomorphism classes of indecomposable objects) if
and only if modsp(kI) is of finite representation type.
(c) If k is algebraically closed then the category prin(kI) is of tame
(resp. wild) representation type if and only if modsp(kI) is of tame (resp.
wild) representation type.
Proof. The assertions (a), (b) follow from Lemma 2.1(c) in [25]
whereas (c) from Proposition 2.4 in [25], see also [27], [10].
2.9. One of the main results of this paper is the following theorem.
Theorem. Let k be a field and I a finite poset. Then the category
modsp(kI) is of fully k-wild representation type provided the following
equivalent conditions hold.
(a) The category prin(kI) is of fully k-wild representation type.
(b) The integral Tits quadratic form qI : ZI → Z,
qI(x) =
∑
i∈I
x2
i +
∑
i≺j∈I−
xixj −
∑
p∈max I
(
∑
i≺p
xi)xp
is not weakly non-negative, that is, there exists a vector v ∈ NI such
qI(v) < 0.
(c) The poset I contains as a full peak subposet (see [25]) one of the
hypercritical irreducible posets listed in Table 1 in [14], or a poset which
is peak-reducible to any of the above ones (see Section 3 of [14]) in [14].
The equivalence of (a), (b) and (c) is the main result of [14]. In 5.4
we prove that (a) implies fully k-wildness of modsp(kI).
Remark. We believe that the converse implication holds for every
finite poset. It follows easily for classes of posets for which there are
criteria for tameness in terms of weak nonnegativity of the Tits quadratic
form. One-peak posets and thin two-peak posets (see [13]) form such
classes thanks to Nazarova theorem [15], [24, Theorem 15.3] and the
results of [13], [12]. The key argument is that wildness implies fully
wildness of prin(kI) for such posets I, see [14]. We do not know a proof
valid for arbitrary posets.
S. Kasjan 79
3. Full endofunctors of modf(k〈x, y〉)
In this section we prove the second of our main results - Theorem 3.3.
3.1. Given A,B ∈ Mm(W) denote by M [A,B] the W-W-bimodule
isomorphic to Wm as a left W-module and with the right multiplication
by x and y defined in the standard basis by the matrices A and B respec-
tively. If C,D ∈ Mn(W) then A(C,D) denotes the matrix obtained from
A by substituting x by C and y by D. Clearly, a scalar entry λ of A is
replaced by λIn. Note that A(C,D) ∈ Mnm(W) and A(C,D) ∈ Mnm(k)
if C,D ∈ Mn(k). Moreover M [A(C,D),B(C,D)] ∼= M [C,D] ⊗W M [A,B]
as W-W-bimodules.
Definition. A pair (A,B) of square W-matrices of size n is a full
pair if the functor
(−) ⊗W M [A,B] : modf(W)−→modf(W)
is full.
Lemma. Assume that (A,B) is a full pair of matrices of size n. Then
(1) (a11A + a21B + λIn, a12A + a22B + µIn) is a full pair provided
a11, a12, a21, a22, λ, µ ∈ k and a11a22 − a12a21 6= 0.
(2) if (C,D) is another full pair then (A(C,D),B(C,D)) is a full pair.
The proof of (a) is straightforward, whereas (b) follows from the iso-
morphism M [A(C,D),B(C,D)] ∼= M [C,D] ⊗W M [A,B].
3.2. Introduce the following notation. Given m ∈ N, µ ∈ k and
ρ = (ρ0, ρ1, ..., ρm) ∈ km+1 let
Xm,µ,ρ =
µ 0 0 ... 0 0 0
1 µ 0 ... 0 0 0
ρ0x 0 µ ... 0 0 0
...
. . .
...
ρmx 0 0 ... 0 µ 0
y 0 0 ... 0 0 µ
∈ Mm+4(W)
Ym =
0 1 0 ... 0 0 0
0 0 1 ... 0 0 0
...
. . .
...
0 0 0 ... 0 1 0
0 0 0 ... 0 0 1
0 0 0 ... 0 0 0
∈ Mm+4(W)
80 On fully wild categories
Lemma. For every m ∈ N, µ ∈ k and ρ 6= 0 the pair (Xm,µ,ρ,Ym) is
full.
Proof. For simplicity we present the proof in the case m = 0, ρ0 =
1, µ = 0. The general proof does not differ essentially. Consider a
block matrix F = [fij ]i,j=1,...,4 with fij ∈ Ms(k) for some s and assume
that F commutes with X = X0,1,1(X,Y ) and Y = Y0(X,Y ) for some
matrices X,Y ∈ Ms(k). The latter commutativity implies the following
conditions:
fij = 0, i > j,
f11 = f22 = f33 = f44,
f12 = f23 = f34,
f13 = f24.
Now the commutativity of X and F gives f12 = f13 = f14 = 0 and
f11X = Xf11, f11Y = Y f11.
From now on let Fi = αix+ βiy + γi be fixed nonzero polynomials of
degree at most 1, i = 1, ...,m. Without loss of generality we can assume
that Fi = x+βiy+γi for i = 1, ...,m′ and Fi = y+γi for i = m′+1, ...,m
for some m′ ≤ m. If k is infinite we can assume m′ = m thanks to Lemma
3.2.
3.3. Theorem. Under the notation above:
(1) There exists a full pair (X ,Y) of size m′ + 4 and invertible W-
matrices Ci, Di ∈ Mm′+4(W), i = 1, ...,m, such that CiFi(X ,Y)Di ∈
Mm′+4(k) or
CiFi(X ,Y)Di =
[
0 Im′+3
γ′i + β′iy 0
]
for some γ′i, β
′
i ∈ k, i = 1, ...,m. For the matrices X ,Y one can choose
Xm′,µ,ρ and Ym′ for some µ ∈ k, ρ ∈ km′+1.
(2) There exists a full pair (Z, T ) of size 4(m′ + 4) and invertible
W-matrices C ′
i, D
′
i ∈ M4(m′+4)(W), i = 1, ...,m, such that
C ′
iFi(Z, T )D′
i ∈ M4(m′+4)(k)
for i = 1, ...,m.
(3) If k is infinite then there exists a full pair (Z, T ) of size 4(m+ 4)
such that the matrices Fi(Z, T ) are invertible for i = 1, ...,m.
The matrices Z, T in (2) and (3) are: Z = X (X0,0,1,Y0), T =
S. Kasjan 81
Y(X0,0,1,Y0), where X ,Y are as in (1).
We precede the proof of the theorem by a series of lemmas.
3.4. First we observe that for an element of Mm(W) to be invertible
it is enough to have one-sided inverse.
Lemma. Let A,B ∈ Mn(W) and AB = In. Then BA = In.
Proof. For every m ∈ N and a, b ∈ Mm(k) we have A(a, b)B(a, b) =
Imn and therefore B(a, b)A(a, b) = Inm. This means that
n∑
j=1
Bij(a, b)Ajl(a, b) = δilIm
for every m ∈ N, i, l = 1, ..., n, and a, b ∈ Mm(k). Recall that the k-
algebra Mm(k) has no polynomial identity of degree less than 2m, see
e.g. [9, Lemma 6.3.1]. Taking m large enough we conclude that
n∑
j=1
BijAjl = δil
for every i, l, that is, BA = Im.
3.5. Recall that Fi = x+ βiy + γi for i = 1, ...,m′.
Lemma. The determinant of the matrix
Fi(Xm′,µ,ρ,Ym′)
treated as a matrix with coefficients in k[x, y], equals
λm′+4
i − βiλ
m′+2
i +
+[ρ0β
2
i λ
m′+1
i + ...+ (−1)m′
ρm′βm′+2
i λi]x+ (−1)m′+1βm′+3
i y,
where λi = µ+ γi, for i = 1, ...m′.
Proof. Follows by direct calculation.
3.6. Lemma. Given µ ∈ k there exist ρ ∈ km′+1, ρ 6= 0, such that
there are matrices Mi ∈ Mm′+4(W) satisfying
Fi(Xm′,µ,ρ,Ym′)Mi = (λm′+4
i + (−1)m′+1βm′+3
i y)Im′+4,
where λi = µ+ γi, for i = 1, ...,m′.
Proof. Note that
ρ0β
2
i λ
m′+1
i + ...(−1)m′
ρm′βm′+2
i λi = 0, i = 1, ...,m′.
82 On fully wild categories
is a system of m′ linear equations with m′+1 unknowns ρ0, ..., ρm′ , there-
fore it has a nonzero solution ρ = (ρ0, ..., ρm′). When ρ is so then
det(Fi(Xm′,µ,ρ,Ym′) = λm′+4
i + (−1)m′
βm′+3
i y
for i = 1, ...,m′ by 3.5.
Treat Fi(Xm′,µ,ρ,Ym′) as a matrix over k[x, y] and let Mi be the
matrix adjoint to Fi(Xm′,µ,ρ,Ym′). Then the required equality holds if
we view the coefficients as elements of k[x, y]. But observe that every
entry of Mi has degree at most 1 and the first row of Mi contains only
constants. Therefore the equality is true also over k〈x, y〉.
3.7. Proof of Theorem 3.3. We keep the notation introduced in
3.5 and 3.6 above.
(1) First consider i = 1, ...,m′. Take µ ∈ k arbitrary and let ρ be as in
3.6. Let X = Xm′,µ,ρ and Y = Ym′ . Then Fi(X ,Y) is a square W-matrix
with m′ + 4 rows and columns and having nonconstant terms only in the
first column. As before we set λi = µ+ γi.
If λi = βi = 0 then Fi(X ,Y) has only one nonzero column containing
an entry 1 thus it can be reduced by elementary transformations on rows
and columns to a matrix having only one nonzero entry equal 1.
Assume that λi 6= 0 or βi 6= 0. The matrix obtained from Fi(X ,Y)
by deleting the first column has rank m′ + 3. After suitable elementary
operations on rows and columns of Fi(X ,Y) we can reduce it to the form
[
0 Im′+3
G(x, y) 0
]
where G(x, y) is an element of W of degree one. Now
G(x, y) = (−1)m′+5(λm′+4
i + (−1)m′
βm′+3
i y)
by Lemma 3.5.
For i > m′ we note that Fi(X ,Y) = Y + γiIm′+4 has all entries in k.
(2) Let X , Y be as in (1) and
U = X0,0,1 =
0 0 0 0
1 0 0 0
x 0 0 0
y 0 0 0
V = Y0 =
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
.
Set Z = X (U ,V), T = Y(U ,V). Then (Z, T ) is a full pair by by Lemma
3.1 and satisfies the claim by (1).
S. Kasjan 83
(3) Since k is infinite we can choose µ ∈ k such that λi = µ+ γi 6= 0
for i = 1, ...,m′ = m. By 3.6 for any i = 1, ...,m there exists a matrix
Mi ∈ Mm+4(W) such that
Fi(X ,Y)Mi = (λm+4
i + (−1)mβm+3
i y)Im+4
Let U , V, Z, T be as in (2). Then
Fi(Z, T )Mi(Z, T ) = diag(λm+4
i I4 + (−1)mβiV,m+ 4).
(Given a square matrix A we denote by diag(A,m) the block matrix
with m blocks A at the diagonal and zeros outside.) This is an invertible
matrix in M4(m+4)(k), let Li be its inverse. Then
Fi(Z, T )Mi(Z, T )Li = I4(m+4)
and Fi(Z, T ) is invertible by 3.4. As above, (Z, T ) is a full pair by
Lemma 3.1.
Remarks. (a) We expect that Theorem 3.3 can be improved by
skipping the assumption that k is infinite in (3).
(b) Let Σ ⊆ W be a finite set of nonzero elements of degree 1 and
denote by modf(WΣ) the full subcategory of modf(W) formed by all mod-
ules of the form U(a, b) (see 5.1 below for the notation) such that F (a, b)
is an invertible matrix for every F ∈ Σ. Theorem 3.3 proves that the cat-
egory modf(WΣ) is fully k-wild provided k is an infinite field. It would be
interesting to generalize this assertion to arbitrary finite set Σ of nonzero
elements of W. Note that if k is algebraically closed then wildness (not
fully wildness) of modf(WΣ) follows by Theorem 2 in [8], since modf(WΣ)
is an open subcategory of modf(W) in the sense of [8].
Example. We present a full pair (X ,Y) such that the matrix XY −
YX is invertible.
0 0 0 0 0 0 0
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
y 0 0 0 1 0 0
x 0 0 0 0 1 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
0 1 0 0 0 0 0
0 0 0 1 1 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 0
The above matrices were found with a help of a computer, partially by
a random search. Unfortunately our methods do not suggest how to
generalize Theorem 3.3 to arbitrary finite sets Σ of nonzero elements of
k〈x, y〉.
84 On fully wild categories
4. Pure k〈x, y〉-matrices
4.1. Lemma. Let F0, F1 be free left W-modules of finite rank and
assume that f : F0−→F1 is a W-homomorphism. Let A be the matrix of
f with respect to some bases of F0 and F1. Then
(1) The module Im f is free.
(2) The following conditions are equivalent:
(a) Im f is a direct summand of F1 (that is, Im f →֒ F1 is a pure
monomorphism),
(b) there exist invertible square W-matrices B,C such that BAC is
a block matrix of the form [
Ir 0
0 0
]
for some r ∈ N,
(c) the canonical homomorphism
U ⊗W Im f−→Im(U ⊗W f)
is an isomorphism for every right W-module U .
If this is the case then r is the W-rank of Im f .
Proof. (1) follows since W is a free ideal ring [5, §2.4, Proposition
2.1]. An equivalence of (a) and (b) is easy thanks to (1), similarly as the
implication (a) ⇒ (c). In order to prove the converse implication assume
that F1/Im f is not projective. Then it is enough to take U such that
TorW1 (U,F1/Im f) 6= 0.
Definition. A matrix A ∈ Mn×m(W) is pure if it satisfies the
condition (2.b) of the lemma above.
4.2. Theorem. Let Gi be W-matrices (of arbitrary sizes), i =
1, ...,m. Assume that every entry of Gi has degree at most 1 and there
is at most one column containing a non-constant entry for i = 1, ...,m.
Then there exists a full pair (Z, T ) such that the matrices Gi(Z, T ) are
pure for i = 1, ...,m.
Proof. Using elementary transformations on rows and columns we
can reduce each Gi to a block matrix of the shape:
[
Iri
0 0
0 0 Fi
]
where ri ∈ N and Fi is a column whose entries have degree at most 1.
After applying suitable elementary operations on Fi we can assume that
either Fi has at most one nonzero entry or it has exactly two nonzero
S. Kasjan 85
entries of the form x+ a, y + b for some a, b ∈ k. Let 1 ≤ m1 ≤ m2 ≤ m
be such that Fi has unique nonzero entry Φi (Φi ∈ W is an element of
degree at most 1) for i = 1, ...,m1, Fi has two nonzero entries x + ai,
y+ bi for i = m1 +1, ...,m2 and Fi is a zero column for i = m2 +1, ...,m.
Let (Z, T ) be the full pair of matrices of size s = 4(m1 + 4) such that
C ′
iΦi(Z, T )D′
i has all the entries in k for some invertible W-matrices C ′
i,
D′
i for i = 1, ...,m1, see Theorem 3.3 (2). Then the matrices Gi(Z, T ) are
pure for i = 1, ...,m1.
Observe that the remaining matrices Gi(Z, T ) are also pure. Indeed,
analysis of the shapes of Z and T constructed in Theorem 3.3 shows that
the block matrix [
Z + aiIs
T + biIs
]
can be reduced by elementary transformations to
[
0
Is
]
.
5. Proof of Theorem 2.9
5.1 We say that a W-kI-bimodule N = (N ′
A, N
′′
B, φ : N ′
A −→|N ′′
B|), free
as a left W-module, is purely defined if φ satisfies the equivalent con-
ditions in Lemma 4.1 (2), that is, it is defined by a pure matrix with
respect to some/any bases.
Given m ∈ N and a, b ∈ Mm(k) let U(a, b) be the right W-module
isomorphic to km as a k-module, equipped with the right action of x
(resp. y) defined by the matrix a (resp. b) with respect to the standard
basis of km. The following assertion is clear.
Lemma. Let C ∈ MatI,v(W). Then
U(a, b) ⊗W ĈW ∼= Ĉ(a, b)
k
.
5.2. Lemma. Assume that N = (N ′
A, N
′′
B, φ : N ′
A −→|N ′′
B|) is a
W-kI-bimodule which is free as a W-module and purely defined. Then
ΘB(N) is a W-kI-bimodule free as a W-module and
ΘB(U ⊗W N) ∼= U ⊗W ΘB(N)
86 On fully wild categories
as kI-modules for every module U in modf(W).
Proof. First note that, since B is a semisimple k-algebra,
U ⊗W HomB(M,N ′′
B) ∼= HomB(M,U ⊗W N ′′
B)
as right A-modules for every right W-module U . Now the lemma follows
by Lemma 4.1. (c) and the definition of ΘB.
5.3. Lemma. Assume that C ∈ MatI,v(W) has only two non-
constant entries and they have degree 1. Then there exists a full pair
(X ,Y) such that the bimodule ̂C(X ,Y)
W
is purely defined.
Proof. After suitable linear change of variables we can assume that
the non-constant entries are equal either x and y + γ respectively or y
and y+γ for some γ ∈ k, see Lemma 3.1. Replacing C by C(U ,V), where
U =
0 0 0 0
1 0 0 0
x 0 0 0
y 0 0 0
V =
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
we can assume that x and y appear in the same column. By Theorem 4.2
we conclude that there exist a full pair (Z, T ) such that all the matrices
ΦC [i](Z, T ) (see 2.3 for the notation) are pure for i ∈ I−. This means
that the module ̂C(Z, T )
W
is purely defined.
Corollary. Assume that the category prin(kI) is fully k-wild. Then
there exists a purely defined W-kI-bimodule N such that the induced func-
tor
(−) ⊗W NkI : modf(W)−→ mod (kI)
is full and its image is contained in prin(kI).
Proof. Apply the above lemma to the matrix N defined in Lemma
2.7.
5.4. Proof of Theorem 2.9. Assume that the category prin(kI)
is fully k-wild. By Corollary 5.3 there exists a purely defined W-kI-
bimodule N defining fully k-wildness of prin(kI).
We obtain the functor
ΘB((−) ⊗W N) : modf(W)−→ mod (kI)
which is full (see 2.8), faithful and exact by 5.2 and its image is contained
in modsp(kI). Therefore the category modsp(kI) is fully k-wild.
5.5. Example. Let I be a poset with a unique maximal element p
and containing five pairwise incomparable elements i1, ..., i5. Then the
S. Kasjan 87
category prin(kI) is of fully k-wild representation type by Nazarova’s
Theorem (see [24, Theorem 15.3]) and [14]. A functor satisfying the
conditions of the definition of fully k-wildness is also described in [24]: it
is the functor determined by the matrix N ∈ MatI,v(W), where v(p) = 2,
v(ij) = 1, j = 1, ..., 5 and v(j) = 0 for j /∈ {p, i1, ..., i5} and
[Ni1p|Ni2p|Ni3p|Ni4p|Ni5p] =
[
1 0 1 1 1
0 1 1 x y
]
The bimodule N̂ is not necessarily purely defined, it depends on the
position of i1, ..., i5 in I. For example, when I \ {p} is the following:
i1 i2 i3 j
ր տ
i4 i5
(an arrow i−→j indicates the relation i ≺ j), then the matrix
Φj(N ) =
[
1 1
x y
]
is not pure.
In order to guarantee the pure definitness in this case it is enough to
substitute (x, y) by a full pair (X ,Y) turning the polynomial x − y into
a invertible matrix. Theorem 3.3 asserts that there exists such a pair of
matrices of size 20 × 20. As usual, a particular case admits a simpler
solution than the one suggested by a general theory: it is enough to take
X =
0 0 0 0
1 0 0 0
x 1 0 0
0 y 1 0
Y =
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
.
Below we present the matrix inverse to X − Y:
y 1 0 1
−1 0 0 0
y 0 0 1
−1 + xy x −1 x
.
6. On filtered representations of posets
Let S be a ring. Following [3] (see [25]) repfg(I, S) denotes the category
of filtered finitely generated S-representations of I. The objects of this
88 On fully wild categories
category are systems U = (Ui)i∈I of finitely generated left S-modules such
that Ui ⊆
⊕
i�p∈max I Up ⊆
⊕
p∈max I Up for every i ∈ I and πj(Ui) ⊆
Uj provided i � j in I, where πi is the composition of the canonical
homomorphisms
⊕
p∈max I
Up −→
⊕
i�p∈max I
Up −→
⊕
p∈max I
Up
for i ∈ I.
A morphism from U = (Ui)i∈I to V = (Vi)i∈I in repfg(I, S) is an
S-module homomorphism f :
⊕
p∈max I Up −→
⊕
p∈max I Vp such that
f(Ui) ⊆ Vi for every i ∈ I.
Further, fspr(I, S) (resp. fpr(I, S)) denotes the full subcategory of
repfg(I, S) with objects (Ui)i∈I such that
⊕
p∈max I Up is a projective S-
module (resp.
⊕
p∈max I Up and
⊕
p∈max I Up/Ui are free S-modules for
all i ∈ I).
When S = k is a field the categories repfg(I, S), fspr(I, S) and
fpr(I, S) coincide with the category I − spr of peak I-spaces defined
in [25].
Recall that
Θ′
B : mod (kI)−→modsp(kI)
is the functor, whose restriction to prin(kI) is the adjustment functor ΘB
(see 2.8)
Recall from [25] the definition of the functor
ΘI : mod (kI)−→ repfg(I, k).
Given a kI-module X identified with the triple (X ′
A, X
′′
B, φ : X ′
A → |X ′′
B|)
the representation ΘI(X) is defined as (X i)i∈I where Xp = X ′′
Bep for
p ∈ max I and X i = φ(X ′
Aei) for i ∈ I−. This correspondence extends
to a functor in a natural way.
It is proved in [25] that the restriction of this functor to modsp(kI)
yields an equivalence of the categories modsp(kI) and repfg(I, k). The
inverse functor
ρ : repfg(I, k)−→modsp(kI)
sends a representation (Xi)i∈I to a kI-module X isomorphic to
⊕
i∈I Xi
with the right multiplication given by xiejl = πl(xi) when i = j and
xiejl = 0 when i 6= j for j � l and xi ∈ Xi.
Given a ring S we extend the above definitions to S-representations
of I: ΘI sends S-kI-bimodules to objects of repfg(I, S) and ρ acts in a
inverse direction.
S. Kasjan 89
The relevant properties of these correspondences are listed in the fol-
lowing lemma.
6.1. Lemma Let N be a W-kI-bimodule free finitely generated as a
left W-module and purely defined. Then
(a) ΘI(N) is an object of fpr(I,W).
(b) There is a natural isomorphism of k-representations of I:
U ⊗W ΘI(N) ∼= ΘI(U ⊗W N)
for any U ∈ modf(W) (the tensor product at the left hand side is defined
in a natural way, see [3, Sect. 3]).
(c) There is a natural isomorphism of kI-modules:
ρ(U ⊗W ΘI(N)) ∼= U ⊗W ΘB(N)
for any U ∈ modf(W).
Proof. The assertion (a) is a direct consequence of the definition of
pure definitness, whereas (b) follows by 4.1, as in the proof of Lemma
5.2. In order to prove (c) observe that the functors ρ ◦ ΘI and Θ′
B are
naturally equivalent and the proof follows by Lemma 5.2 and (b).
Now we can formulate a version of the main statement of Theorem
2.9 in terms of the category fpr(I, k).
Corollary (cf. [3]). Let I be a poset such that the category prin(kI) is
fully k-wild. There exists an object N of fpr(I,W) such that the functor
(−) ⊗W N : modf(W)−→ fpr(I, k)
is exact, full and faithful. Therefore the category fpr(I, k) is fully k-wild
in the sense of [3].
Proof. Let M be a purely defined W-kI-bimodule defining the fully
k-wildness of prin(kI). Such a bimodule exists by Corollary 5.3. Then
the representation N = ΘI(M) satisfies the conditions of the corollary
thanks to the lemma above, see the proof of Theorem 2.9.
Acknowledgement. The author thanks Daniel Simson for stimulat-
ing remarks and discussions on the subject of this article.
References
[1] D. M. Arnold, "Abelian Groups and Representations of finite Partially Ordered
Sets", CMS Books in Mathematics, Springer-Verlag, New-York, 2000.
[2] D. M. Arnold and D. Simson, Representations of finite partially ordered sets over
commutative uniserial rings, J. Pure Appl. Algebra 205, 2006, pp.640-659.
90 On fully wild categories
[3] D. M. Arnold and D. Simson, Endo-wild representation type and generic repre-
sentations of finite posets, Pacific J. Math. 219, 2005, pp.101-126.
[4] W. Crawley-Boevey, Modules of finite length over their endomorphism rings, in
"Representations of Algebras and Related Topics", London Math. Soc. Lecture
Notes 168, 1992, pp.127 - 184.
[5] P. M. Cohn, Free Rings and Their Relations, Acad. Press, New York 1971.
[6] P. Dowbor and S. Kasjan, Galois covering technique and tame non-simply con-
nected posets of polynomial growth, J. Pure. Appl. Algebra 147, 2000, pp.1-24.
[7] Yu. A. Drozd, Tame and wild matrix problems, in "Representations and
Quadratic Forms", Kiev 1979, pp.39-74 (in Russian).
[8] Y. A. Drozd and G.-M. Greuel, Tame-wild dichotomy for Cohen-Macaulay mod-
ules, Math. Ann. 294, 1992, pp.387-394.
[9] I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs,
No. 15, The Mathematical Association of America, 1968.
[10] S. Kasjan, Adjustment functors and tame representation type, Comm. Algebra,
22, 1994, pp.5587-5597.
[11] S. Kasjan and D. Simson, Varieties of poset representations and minimal posets
of wild prinjective type, in Proceedings of the Sixth International Conference on
Representations of Algebras, Canadian Mathematical Society Conference Pro-
ceedings, Vol. 14, 1993, pp.245-284.
[12] S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets
I, J. Pure Appl. Algebra 106, 1996, pp.307-330.
[13] S. Kasjan i D. Simson, Tame prinjective type and Tits form of two-peak posets
II, J. Algebra 187, 1997, pp.71-96.
[14] S. Kasjan and D. Simson, Fully wild prinjective type of posets and their quadratic
forms, J. Algebra 172, 1995, pp.506-529.
[15] L. A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR
39, 1975, pp.963-991 (in Russian).
[16] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets,
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 28, 1972,
pp.69-92 (in Russian).
[17] J. A. de la Peña, On the dimension of module varieties of tame and wild algebras,
Comm. Algebra 19, 1991, pp.1795-1807.
[18] J. A. de la Peña and D. Simson, Prinjective modules, reflection functors,
quadratic forms and Auslander-Reiten sequences, Trans. Amer. Math. Soc. 329,
1992, pp.733 - 753.
[19] C. M. Ringel, Representations of K-species and bimodules, J. Algebra, 41,
1976, pp.269 - 302.
[20] D. Simson, Moduled categories and adjusted modules over traced rings, Disser-
tationes Math. 269, 1990, pp.1-67.
[21] D. Simson, On the representation type of stratified posets, C. R. Acad. Sci. Paris,
311, 1990, pp.5 - 10.
[22] D. Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138,
1991, pp.113-137.
S. Kasjan 91
[23] D. Simson, Right peak algebras of two-separate stratified posets, their Galois
coverings and socle projective modules, Comm. Algebra 20, 1992, pp.3541 -
3591.
[24] D. Simson, "Linear Representations of Partially Ordered Sets and Vector Space
Categories", Algebra, Logic and Applications, Vol. 4, Gordon & Breach Science
Publishers, 1992.
[25] D. Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl.
Algebra, 90, 1993, pp.77-103.
[26] D. Simson, On representation types of module subcategories and orders, Bull.
Polish Acad. Sci. 41, 1993, pp.77-93.
[27] D. Simson, Prinjective modules, propartite modules, representations of bocses
and lattices over orders, J. Math. Soc. Japan 49, 1997, pp.31-68.
[28] D. Simson, On large indecomposable modules, endo-wild representation type and
right pure semisimple rings, Algebra and Discrete Math. 2, 2003, pp.93-118.
[29] D. Vossieck, Représentations de bifoncteurs et interprétation en termes de mod-
ules, C. R. Acad. Sci. Paris, 307, 1988, pp.713 - 716.
Contact information
S. Kasjan Faculty of Mathematics and Computer
Science, Nicolaus Copernicus University,
Chopina 12/18, 87-100 Toruń, Poland
E-Mail: skasjan@mat.uni.torun.pl
Received by the editors: 01.06.2005
and in final form 22.11.2006.
|