Projective resolution of irreducible modules over tiled order
We indicate the method for computing the kernels of projective resolution of irreducible module over tiled order. On the base of this method we construct projective resolution of irreducible module and calculate the global dimension of tiled order. The evident view of kernels of projective resolutio...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Projective resolution of irreducible modules over tiled order / V. Zhuravlev, D. Zhuravlyov // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 323–336. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1522472019-06-10T01:25:38Z Projective resolution of irreducible modules over tiled order Zhuravlev, V. Zhuravlyov, D. We indicate the method for computing the kernels of projective resolution of irreducible module over tiled order. On the base of this method we construct projective resolution of irreducible module and calculate the global dimension of tiled order. The evident view of kernels of projective resolution allows to check easily the regularity of tiled order. 2012 Article Projective resolution of irreducible modules over tiled order / V. Zhuravlev, D. Zhuravlyov // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 323–336. — Бібліогр.: 10 назв. — англ. 1726-3255 2010 MSC:16E05, 16G20, 16G10, 16D40. http://dspace.nbuv.gov.ua/handle/123456789/152247 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We indicate the method for computing the kernels of projective resolution of irreducible module over tiled order. On the base of this method we construct projective resolution of irreducible module and calculate the global dimension of tiled order. The evident view of kernels of projective resolution allows to check easily the regularity of tiled order. |
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Zhuravlev, V. Zhuravlyov, D. |
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Zhuravlev, V. Zhuravlyov, D. Projective resolution of irreducible modules over tiled order Algebra and Discrete Mathematics |
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Zhuravlev, V. Zhuravlyov, D. |
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Zhuravlev, V. |
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Projective resolution of irreducible modules over tiled order |
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Projective resolution of irreducible modules over tiled order |
| title_full |
Projective resolution of irreducible modules over tiled order |
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Projective resolution of irreducible modules over tiled order |
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Projective resolution of irreducible modules over tiled order |
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projective resolution of irreducible modules over tiled order |
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Інститут прикладної математики і механіки НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/152247 |
| citation_txt |
Projective resolution of irreducible modules over tiled order / V. Zhuravlev, D. Zhuravlyov // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 323–336. — Бібліогр.: 10 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT zhuravlevv projectiveresolutionofirreduciblemodulesovertiledorder AT zhuravlyovd projectiveresolutionofirreduciblemodulesovertiledorder |
| first_indexed |
2025-07-13T02:38:53Z |
| last_indexed |
2025-07-13T02:38:53Z |
| _version_ |
1837497678767849472 |
| fulltext |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 323 – 336
c© Journal “Algebra and Discrete Mathematics”
Projective resolution
of irreducible modules over tiled order
Viktor Zhuravlev, Dmytro Zhuravlyov
Communicated by V. V. Kirichenko
Abstract. We indicate the method for computing the kernels
of projective resolution of irreducible module over tiled order. On the
base of this method we construct projective resolution of irreducible
module and calculate the global dimension of tiled order. The evident
view of kernels of projective resolution allows to check easily the
regularity of tiled order.
1. Tiled orders over discrete valuation rings
Recall [2] that a semimaximal ring is a semiperfect semiprime right
Noetherian ring A such that for each primitive idempotent e ∈ A the ring
eAe is a discrete valuation ring (not necessarily commutative).
Denote by Mn(B) the ring of all n× n matrices over a ring B.
Theorem 1 (see [2]). Each semimaximal ring is isomorphic to a finite
direct product of prime rings of the following form:
Λ =
O πα12O . . . πα1nO
πα21O O . . . πα2nO
. . . . . . . . . . . . . . . . . . . . . . . . . . .
παn1O παn2O . . . O
, (1)
2010 MSC: 16E05, 16G20, 16G10, 16D40.
Key words and phrases: tiled order, projective resolution, distributive module,
kernel of epimorphism.
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324 Projective resolution of irreducible modules
where n ≥ 1, O is a discrete valuation ring with a prime element π, and
αij are integers such that
αij + αjk ≥ αik, αii = 0
for all i, j, k.
The ring O is embedded into its classical division ring of fractions D,
and (1) is the set of all matrices (aij) ∈ Mn(D) such that
aij ∈ παij O = eiiΛejj ,
where e11, . . . , enn are the matrix units of Mn(D). It is clear that Q =
Mn(D) is the classical ring of fractions of Λ.
Obviously, the ring A is right and left Noetherian.
Definition 1. A module M is distributive if its lattice of submodules is
distributive, i.e.,
K ∩ (L+N) = K ∩ L+K ∩N
for all submodules K, L, and N .
Clearly, any submodule and any factormodule of a distributive module
are distributive modules.
A semidistributive module is a direct sum of distributive modules. A
ring A is right (left) semidistributive if it is semidistributive as the right
(left) module over itself. A ring A is semidistributive if it is both left and
right semidistributive (see [9]).
Theorem 2 (see [8]). The following conditions for a semiperfect semi-
prime right Noetherian ring A are equivalent:
• A is semidistributive;
• A is a direct product of a semisimple artinian ring and a semimaxi-
mal ring.
By a tiled order over a discrete valuation ring, we mean a Noetherian
prime semiperfect semidistributive ring Λ with nonzero Jacobson radical.
In this case, O = eΛe is a discrete valuation ring with a primitive
idempotent e ∈ Λ.
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V. Zhuravlev, D. Zhuravlyov 325
Definition 2. An integer matrix E = (αij) ∈ Mn(Z) is called
• an exponent matrix if αij + αjk ≥ αik and αii = 0 for all i, j, k;
• a reduced exponent matrix if αij + αji > 0 for all i, j, i 6= j.
We use the following notation: Λ = {O, E(Λ)}, where E(Λ) = (αij) is
the exponent matrix of the ring Λ, i.e.
Λ =
n∑
i,j=1
eijπ
αij O,
in which eij are the matrix units. If a tiled order is reduced, i.e., Λ/R(Λ)
is the direct product of division rings, then αij + αji > 0 if i 6= j, i.e.,
E(Λ) is reduced.
We denote by M(Λ) the poset (ordered by inclusion) of all projective
right Λ-modules that are contained in a fixed simple Q-module U . All
simple Q-modules are isomorphic, so we can choose one of them. Note
that the partially ordered sets Ml(Λ) and Mr(Λ) corresponding to the
left and the right modules are anti-isomorphic.
The set M(Λ) is completely determined by the exponent matrix
E(Λ) = (αij). Namely, if Λ is reduced, then
M(Λ) = {pz
i | i = 1, . . . n, and z ∈ Z},
where
pz
i ≤ pz′
j ⇐⇒
{
z − z′ ≥ αij if M(Λ) = Ml(Λ),
z − z′ ≥ αji if M(Λ) = Mr(Λ).
Obviously, M(Λ) is an infinite periodic set.
Let P be an arbitrary poset. A subset of P is called a chain if any
two of its elements are related. A subset of P is called a antichain if no
two distinct elements of the subset are related.
Definition 3. A right (resp. left) Λ-module M (resp. N) is called a right
(resp. left) Λ-lattice if M (resp. N) is a finitely generated free O-module.
Given a tiled order Λ we denote Latr(Λ) (resp. Latl(Λ)) the category
of right (resp. left) Λ-lattices. We denote by Sr(Λ) (resp. Sl(Λ)) the
partially ordered by inclusion set, formed by all Λ-lattices contained in a
fixed simple Mn(D)-module W (resp. in a left simple Mn(D)-module V ).
Such Λ-lattices are called irreducible.
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326 Projective resolution of irreducible modules
Let Λ = {O, E(Λ)} be a tiled order, W (resp. V ) is a simple right
(resp. left) Mn(D)-module with D-basis e1, . . . , en such that eiejk = δijek
(eijek = δjkei).
Then any right (resp. left) irreducible Λ-lattice M (resp. N), lying in
W (resp. in V ) is a Λ-module with O-basis (πα1e1, . . . , π
αnen), while
{
αi + αij ≥ αj , for the right case;
αij + αj ≥ αi, for the left case.
(2)
Thus, irreducible Λ-lattices M can be identified with integer-valued
vector (α1, . . . , αn) satisfying (2). We shall write E(M) = (α1, . . . , αn)
or M = (α1, . . . , αn).
The order relation on the set of such vectors and the operations on
them corresponding to sum and intersection of irreducible lattices are
obvious.
Remark 1. Obviously, irreducible Λ-lattices M1 = (α1, . . . , αn) and
M2 = (β1, . . . , βn) are isomorphic if and only if αi = βi + z for i =
1, . . . , n and z ∈ Z.
2. Kernel of epimorphism from direct sum of modules to
their sum
Let Λ be the reduced tiled order with the exponent matrix E(Λ) = (αij),
M is irreducible right Λ-module and P (M) is its projective cover.
The following statement holds.
Proposition 1 ([10]). Let X1, . . . , Xs be the set of all maximal sub-
modules of irreducible and non-projective Λ-module M with E(M) =
(α1, . . . , αn) and E(Xi) = E(M) + eji
, where ek = (0, . . . , 0
︸ ︷︷ ︸
k−1
, 1, 0, . . . , 0).
Then
P (M) =
s
⊕
i=1
παjiPji
and M =
s∑
i=1
παjiPji
.
This statement allows to find easily the projective cover of irreducible
module over tiled order.
Theorem 3 ([10]). Let M1, . . . ,Mn be submodules of distributive module
M =
n∑
i=1
Mi and epimorphism ϕ :
n
⊕
i=1
Mi 7→ M operates by the rule
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V. Zhuravlev, D. Zhuravlyov 327
ϕ(m1, . . . ,mn) = m1 + . . . + mn. Then ker ϕ = {(y1, . . . , yn) | yi =
∑
j 6=i
mij , mij = −mji ∈ Mi ∩Mj}.
Since the tiled order is a semidistributive ring we have the following
corollary.
Corollary 1 ([10]). Let M be irreducible Λ-module and P (M) =
s
⊕
i=1
παjiPji
, M =
s∑
i=1
παjiPji
. Then the kernel of epimorphism ϕ : P (M) 7→
M equals to ker ϕ = {(y1, . . . , yn) | yi =
∑
k 6=i
mik, mik = −mki ∈
παjiPji
∩ παjkPjk
}.
The kernel K as the submodule in
n
⊕
i=1
Mi can be formally written as
K =
∑
i<j
(Mi ∩Mj)(ei − ej),where ek = (0, . . . , 0
︸ ︷︷ ︸
k−1
, 1, 0, . . . , 0).
3. Distributive equations and the system of distributive
equations
Let O be a discrete valuation ring with unique maximal ideal m = πO,
where π is a prime element of a ring. O, F = O/πO — is a skew field,
Λ = {O, E(Λ) = (αij)} — is a reduced tiled order over a discrete valuation
ring O with exponent matrix E(Λ) = (αij) ∈ Mn(Z).
The equation of the form
s∑
i=1
aimi = 0, where ai ∈ F , mi ∈ Mi,
M1, . . . ,Ms are the submodules of the distributive module M , we will
call distributive.
Let M1, . . . ,Ms be the submodules of distributive module M =
s∑
i=1
Mi
and epimorphism ϕ :
s
⊕
i=1
Mi → M operates by the rule ϕ(m1, . . . ,ms) =
m1 + · · · + ms. Then by the theorem 3 ker ϕ = {(y1, . . . , ys) ∈
s
⊕
i=1
Mi | yi = −m1i − · · · − mi−1i + mii+1 + · · · + mis, where mij ∈
Mi ∩Mj , i < j}.
In other words, distributive equation
m1 + · · · +ms = 0, (3)
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328 Projective resolution of irreducible modules
where mi ∈ Mi i = 1, . . . , s, M1, . . . ,Ms are the submodules of the
distributive module M , has solution
m1 = m12 + · · · +m1s,
m2 = −m12 +m23 + · · · +m2s,
· · · · · · · · · · · · · · · · · · · · ·
mi = −m1i − · · · −mi−1i +mii+1 + · · · +mis,
· · · · · · · · · · · · · · · · · · · · ·
ms = −m1s − · · · −ms−1s.
(4)
Distributive equation
a1m1 + · · · + asms = 0 (5)
where ai ∈ F , mi ∈ Mi, ai 6= 0 for all i = 1, . . . s by replacement
aimi = zi ∈ Mi reduces to the equation (3) and then
a1m1 = m12 + · · · +m1s,
a2m2 = −m12 +m23 + · · · +m2s,
· · · · · · · · · · · · · · · · · · · · ·
aimi = −m1i − · · · −mi−1i +mii+1 + · · · +mis,
· · · · · · · · · · · · · · · · · · · · ·
asms = −m1s − · · · −ms−1s.
(6)
where mij ∈ Mi ∩Mj for all i, j i 6= j. Hence
m1 = a−1
1 (m12 + · · · +m1s),
m2 = a−1
2 (−m12 +m23 + · · · +m2s),
· · · · · · · · · · · · · · · · · · · · ·
mi = a−1
i (−m1i − · · · −mi−1i +mii+1 + · · · +mis),
· · · · · · · · · · · · · · · · · · · · ·
ms = a−1
s (−m1s − · · · −ms−1s).
(7)
Let now consider the system of distributive equations.
a11m1 + · · · + a1sms = 0,
a21m1 + · · · + a2sms = 0,
· · · · · · · · · · · · · · · · · · · · ·
at1m1 + · · · + atsms = 0,
(8)
where mi ∈ Mi for all i,M1, . . . ,Ms are the submodules of the distributive
module M , aij ∈ F for all i, j.
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V. Zhuravlev, D. Zhuravlyov 329
Denote A = (aij) ∈ Mt×s(F ). We can consider that vectors which are
the rows of the matrix A are linearly independent. Indeed, if
(at1, . . . , ats) = a1(a11, . . . , a1s) + · · · + at−1(at−11, . . . , at−1s),
then
at1m1 + · · · + atsms =
= a1(a11m1 + · · · a1sms) + · · · + at−1(at−11m1 + · · · + at−1sms).
Thus the last equation is a corollary of previous equations. Therefore
t ≤ s. When t = s are the vectors which are the rows of the matrix A
and construct the basis of space F s. Then
(1, 0, . . . , 0) = c11(a11, . . . , a1s) + · · · + c1s(as1, . . . , ass),
(0, 1, . . . , 0) = c21(a11, . . . , a1s) + · · · + c2s(as1, . . . , ass),
· · · · · · · · · · · · · · · · · · · · ·
(0, 0, . . . , 1) = cs1(a11, . . . , a1s) + · · · + css(as1, . . . , ass).
(9)
where cij ∈ F for all i, j. Hence
m1 = c11(a11m1 + . . .+ a1sms) + · · · + c1s(as1m1 + . . .+ assms),
m2 = c21(a11m1 + . . .+ a1sms) + · · · + c2s(as1m1 + . . .+ assms),
· · · · · · · · · · · · · · · · · · · · ·
ms = cs1(a11m1 + . . .+ a1sms) + · · · + css(as1m1 + . . .+ assms).
(10)
Thus, for existence of nonzero solution of the system (8) it is needed
rank A < s, that is t < s.
Obviously, the solution of the system (8) depends on the set
M1, . . . ,Ms. Denote M = (M1, . . . ,Ms). Thus the system (8) uniquely
is determined by the matrix A and the set of the irreducible modules
M1, . . . ,Ms and we will denote it by (A,M).
Consider the method of solving the system of distributive equa-
tions (A,M). Determine the sequence of systems (A1,M1), (A2,M2),
. . ., (At,M t) inductively. Notably, let A1 = A, M1 = M . Let we have
the system (Ak,Mk), where Ak = (a
(k)
ij ) — is rectangular matrix of size
(t+ 1 − k) × vk, Mk = (M
(k)
1 , . . . ,M
(k)
vk ).
Let in the first row of the matrix Ak the elements a
(k)
1j1
, a
(k)
1j2
, . . . , a
(k)
1jlk
are not equal to 0, and all other elements are equal to 0. That is the first
equality of the system is of the form
a
(k)
1j1
m
(k)
j1
+ · · · + a
(k)
1jlk
m
(k)
jlk
= 0.
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330 Projective resolution of irreducible modules
Then by (7)
m
(k)
j1
= (a
(k)
1j1
)−1(m
(k)
j1j2
+ · · · +m
(k)
j1jlk
),
m
(k)
j2
= (a
(k)
1j2
)−1(−m
(k)
j1j2
+ · · · +m
(k)
j2jlk
),
· · · · · · · · · · · · · · · · · · · · ·
m
(k)
jlk
= (a
(k)
1jlk
)−1(−m
(k)
j1j2
− · · · −m
(k)
jlk−1jlk
).
(11)
Substituting these expressions for m
(k)
jl
when l = 1, . . . , lk in other
equalities of the system (Ak,Mk), we will have the system of t−k equalities
lk∑
r=1
a
(k)
ijr
(a
(k)
1jr
)−1(−m
(k)
j1jr
− · · · −m
(k)
jr−1jr
+m
(k)
jrjr+1
+ · · · +m
(k)
jrjlk
)+
+
∑
j 6=j1,...,jl
a
(k)
ij m
(k)
j = 0
i = 2, . . . , t+1−k. This is the system of C2
lk
+(vk − lk) = vk+1 unknowns:
m
(k)
jxjy
(x 6= y, x, y = 1, . . . , lk) and m
(k)
j (j 6= jr, r = 1, . . . , lk), where
m
(k)
jxjy
∈ M
(k)
jx
∩M
(k)
jy
.
Let
Mk+1 = {M
(k)
jx
∩M
(k)
jy
, x 6= y, x, y = 1, . . . , lk}∪
∪ {M
(k)
j , 1 ≤ j ≤ Vk, j 6= j1, . . . , jlk}.
Obtained rectangular matrix of the size (t− k) × vk+1 from unknowns
m
(k)
jxjy
, x 6= y, x, y = 1, . . . , lk, m
(k)
j and m
(k)
j , j 6= jr, r = 1, . . . , lk
we denote by Ak+1. Thus, we have obtained the system of distributive
equations (Ak+1,Mk+1).
Solving only one equation from each system (A1,M1), (A2,M2), . . .,
(At−1,M t−1), we will obtain the system (At,M t) of one distributive
equation, which we will obtain also by the following formulas (7). We will
have the (t − 1)-th union of relations (11) for k = 1, . . . , t − 1 between
old unknowns of the system (Ak,Mk) and innovated unknowns of the
system (Ak+1,Mk+1). Using these relations in inverse order, firstly when
k = t− 1, then when k = t− 2 and so forth, we will obtain the solution
of the system (A1,M1) = (A,M).
Note that since module Mk+1 either coincides with the module Mk
or is the intersection of two modules from Mk, and the solution of the
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system of distributive equations (8) is of the form
m1 = b11x1 + b12x2 + · · · + b1rxr,
m2 = b21x1 + b22x2 + · · · + b2rxr,
· · · · · · · · · · · · · · · · · · · · ·
ms = bs1x1 + bs2x2 + · · · + bsrxr.
(12)
where bij ∈ F , xi ∈ Xi, Xi — is the intersection of not more than the
(t+ 1)’s module from the set M1,M2, . . . ,Ms.
Proposition 2. The solution (12) of the system of distributive equations
(8), which is obtained by mentioned method, is the general solution of this
system.
Proof. The proof will be made with the help of induction by the number
of equations of the system(8). The base of the induction is t = 1. The
solution of distributive equation (5) is of the form (7) and is the general
solution of the equation by the theorem 3.
Let we have the system (8) of t equations and let the solution of
the system of (t − 1) distributive equations, which is obtained by the
mentioned method, be the general solution of the system of distributive
equations.
Let a11, . . . , a1l in the system (8) be the elements of the first row,
which are not equal to 0, a1l+1 = · · · = a1s = 0. The the system (8)
subject to the solution of the first equation from the system is equivalent
to the system
m1 = a−1
11 (m12 + · · · +m1l),
m2 = a−1
12 (−m12 +m23 + · · · +m2l),
· · · · · · · · · · · · · · · · · · · · ·
mi = a−1
1i (−m1i − · · · −mi−1i +mii+1 + · · · +mil),
· · · · · · · · · · · · · · · · · · · · ·
ml = a−1
1l (−m1l − · · · −ml−1l).
(13)
a21a
−1
11
l∑
k=2
m1k + · · · + a2la
−1
1l
l−1∑
k=1
mkl +
s∑
k=l+1
a2kmk = 0,
· · · · · · · · · · · · · · · · · · · · ·
at1a
−1
11
l∑
k=2
m1k + · · · + atla
−1
1l
l−1∑
k=1
mkl +
s∑
k=l+1
atkmk = 0.
(14)
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332 Projective resolution of irreducible modules
From (13) it follows that m1 + · · ·+ml = 0, and it is the first equation
of the system (8). By assumption of induction the solution of the system
(14) is a general
m12 = b11x1 + b12x2 + · · · + b1rxr,
· · · · · · · · · · · · · · · · · · · · ·
ml−1l = bu1x1 + bu2x2 + · · · + burxr,
ml+1 = bu+11x1 + bu+12x2 + · · · + bu+1rxr,
· · · · · · · · · · · · · · · · · · · · ·
ms = bu+s−l1x1 + bu+s−l2x2 + · · · + bu+s−lrxr.
(15)
where u = C2
l , bij ∈ F , xk ∈ Xk, k = 1, . . . , r, Xk — is the intersection
of not more than t modules from the set M1, . . . ,Ms.
Then the expression for m1, . . . ,ml we obtain from (13) and (15).
Moreover, from the generality of the solution (15) and the unambiguity
of expressions for m1, . . . ,ml in (13) we will obtain the generality of
the system’s solution (13) – (14). From the equivalence of the systems
(13) – (14) and (8) we obtain that the solution of the system (8), which
is obtained by the mentioned method, is the general. The proposition is
proved.
The solution (12) of the system of distributive equations (8) we will
write as
X1
b11
...
bs1
+Xr
b1r
...
bsr
.
Remark 2. The form of the solution (12) depends on the set of equations
which are solved from each system (Ak,Mk).
If Y1
c11
...
cs1
+Yp
c1p
...
csp
is the another solution of the system (12),
then from the generality of the solutions we have
X1
b11
...
bs1
+Xr
b1r
...
bsr
= Y1
c11
...
cs1
+ Yp
c1p
...
csp
.
The set of irreducible modules {M1, . . . ,Mv} constructs the partially
ordered set by inclusion.
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V. Zhuravlev, D. Zhuravlyov 333
Let us consider the cases when the expression for the set of system’s
solutions can be simplified.
• Let the module M1 be the submodule of modules M i1
, . . . ,M iz and
vector b̄1 is linearly expressed over F through vectors b̄i1
, . . . , b̄iz :
b̄1 = α1b̄i1
+ · · · + αz b̄iz , where αj ∈ F . Then
b̄1M1 + b̄i1
M i1
+ · · · + b̄izM iz =
= (α1b̄i1
+ · · · + αz b̄iz )M1 + b̄i1
M i1
+ · · · + b̄izM iz =
= b̄i1
(α1M1 +M i1
) + b̄i2
(α2M1 +M i2
) + · · · + b̄iz (αzM1 +M iz ) =
= b̄i1
M i1
+ b̄i2
M i2
+ · · · + b̄izM iz .
• Let the vectors b̄i and b̄j be collinear and not equal to zero. Then
b̄iM i + b̄jM j = b̄iM i + λb̄iM j = b̄i(M i + λM j) = b̄i(M i +M j).
4. The construction of projective resolution of irreducible
modules over tiled order
Let X1, . . . , Xs be the set of all maximal submodules of irreducible
and non-projective Λ-module M with E(M) = (α1, . . . , αn) and E(Xi) =
E(M) + eji
, where ek = (0, . . . , 0
︸ ︷︷ ︸
k−1
, 1, 0, . . . , 0). Then by the proposition 1
P (M) =
s
⊕
i=1
παjiPji
and M =
s∑
i=1
παjiPji
.
By the corollary 1 the kernel of epimorphism ϕ : P (M) 7→ M equals
to ker ϕ = {(y1, . . . , yn) | yi =
∑
k 6=i
mik, mik = −mki ∈ Pji
∩ Pjk
}.
The kernel K1 as submodule in
n
⊕
i=1
Mi can be formally written as
K1 =
∑
i<j
(Mi ∩Mj)(ei − ej),where ek = (0, . . . , 0
︸ ︷︷ ︸
k−1
, 1, 0, . . . , 0).
Let the n-th kernel of projective resolution of irreducible module be
of the form Kn =
s∑
i=1
Mif i, where Mi — are irreducible modules and
P (Mi) =
li
⊕
k=1
P
(i)
jk
.
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334 Projective resolution of irreducible modules
Obviously, there is the epimorphism ψ :
li
⊕
k=1
P (Mi) → Kn, which
operates by the formula
ψ(m1
1, . . . ,m
1
l1
,m2
1, . . . ,m
2
l2
, . . . ,ms
1, . . . ,m
s
ls
) =
= (m1
1 + . . .+m1
l1
)f1 + . . .+ (ms
1 + . . .+ms
ls
)fs.
Proposition 3. Let M1, . . . ,Ms be Λ-modules, P (Mi) =
li
⊕
k=1
P
(i)
jk
be the
projective cover of module Mi and the epimorphism ψ :
li
⊕
k=1
P (Mi) → Kn
operates by the rule
ψ(m1
1, . . . ,m
1
l1
,m2
1, . . . ,m
2
l2
, . . . ,ms
1, . . . ,m
s
ls
) =
= (m1
1 + . . .+m1
l1
)f1 + . . .+ (ms
1 + . . .+ms
ls
)fs.
If projective module P
(1)
1 is the submodule of projective modules
P
(2)
1 , P
(3)
1 , . . . , P
(t)
1 and f1 = α2f2 + α3f3 + · · · + αtf t, where αi ∈ F ,
then there exists the epimorphism ψ :
(
l1
⊕
k=2
P
(1)
k
)
⊕
(
s
⊕
i=2
P (Mi)
)
→ Kn,
which operates by the rule
ψ(m1
2, . . . ,m
1
l1
,m2
1, . . . ,m
2
l2
, . . . ,ms
1, . . . ,m
s
ls
) =
= (m1
2 + . . .+m1
l1
)f1 + . . .+ (ms
1 + . . .+ms
ls
)fs.
Proof. Let write down the operation of the epimorphism ψ in the form
ψ(m1
1, . . . ,m
s
ls
) =
= m1
1f1+(m1
2+. . .+m1
l1
)f1+(m2
1+. . .+ms
l2
)f2 . . .+(ms
1+. . .+ms
ls
)fs =
= m1
1(α2f2 + . . .+αtf t)+(m1
2 + . . .+m1
l1
)f1 + . . .+(ms
1 + . . .+ms
ls
)fs =
= (m1
2 + . . .+m1
l1
)f1 +
(
(m2
1 + α2m
1
1) +m2
2 + . . .+m2
l2
)
f2 + . . .
+
(
(mt
1 + αtm
1
1) +mt
2 + . . .+mt
lt
)
f t + (mt+1
1 + . . .+mt+1
lt+1
)f t+1 + . . .
+ (ms
1 + . . .+ms
ls
)fs.
Since P
(1)
1 ⊆ P
(k)
1 for all k = 2, . . . , t, then mk
1 + αkm
1
1 = mk
1 for all
k = 2, . . . , t, where mk
1 ∈ P
(k)
1 .
Thus,
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V. Zhuravlev, D. Zhuravlyov 335
ψ(m1
1, . . . ,m
s
ls
) = (m1
2 + · · · +m1
l1
)f1+
+ (m2
1 +m2
2 + · · · +m2
l2
)f2 + · · · + (mt
1 +mt
2 + · · · +mt
lt
)f t+
+ (mt+1
1 + · · · +mt+1
lt+1
)f t+1 + · · · + (ms
1 + · · · +ms
ls
)fs =
= ψ
(
m1
2, . . . ,m
1
l1
,m2
1,m
2
2, . . . ,m
2
l2
,m3
1, . . . ,m
3
l3
, . . . ,
. . . ,mt
1,m
t
2,m
t
lt
,mt+1
1 , . . . ,mt+1
lt+1
, . . . ,ms
1,m
s
ls
)
.
Corollary 2. In conditions of proposition 3 projective module P
(1)
1 isn’t
included in projective cover of module P (Kn).
By the kernel Kn we construct the epimorphism ψ :
li
⊕
k=1
P (Mi) → Kn,
which operates by the rule
ψ(m1
1, . . . ,m
1
l1
,m2
1, . . . ,m
2
l2
, . . . ,ms
1, . . . ,m
s
ls
) =
= (m1
1 + . . .+m1
l1
)f1 + . . .+ (ms
1 + . . .+ms
ls
)fs.
Using proposition 3 by the epimorphism ψ we construct the epimorphism
ψ with the minimal number of direct summand. For the kernel Kn+1
we obtain the system of distributive equations. This system is solved by
the method given in the section 3. Hereby, we construct the projective
resolution of irreducible module, indicating not only projective modules
but also all intermediate kernels.
Conclusion
The results obtained in sections 3 and 4 allow to construct projec-
tive resolutions of irreducible modules over tiled order and calculate the
global dimension of order. There is the program, which is written in Java
programming language, on the basis of performed researches. It allows to
calculate projective resolutions of irreducible modules over tiled order of
any finite length.
Specifying the kernels of resolution in explicit form allows to determine
easily whether the global dimension of tiled order depends on characteristic
of skew field, i.e. whether the order is regular.
References
[1] Yu.A. Drozd, V.V. Kirichenko, Finite Dimensional Algebras. Springer-Verlag,
Berlin-Heidelberg-New York 1994.
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336 Projective resolution of irreducible modules
[2] A.G. Zavadskij and V.V. Kirichenko, Torsion-free Modules over Prime Rings, Zap.
Nauch. Seminar. Leningrad. Otdel. Mat. Steklov. Inst. (LOMI) - 1976. - v. 57. - p.
100-116 (in Russian). English translation in J. of Soviet Math., v. 11, N 4, April
1979, p. 598-612.
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Miroshnichenko, and V. N. Zhuravlev, Tiled orders over discrete valuation rings,
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[4] Zh. T. Chernousova, M. A. Dokuchaev, M. A. Khibina, V. V. Kirichenko, S. G.
Miroshnichenko, and V. N. Zhuravlev, Tiled orders over discrete valuation rings,
finite Markov chains and partially ordered sets. II. Algebra and Discrete Math. 2
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[5] M. Hazewinkel, N. Gubareni and V. V. Kirichenko, Algebras, Rings and Modules.
Vol. 1, Series: Mathematics and Its Applications, 575, Kluwer Acad. Publish.,
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Vol. 2, Series: Mathematics and Its Applications, 586. Springer, Dordrecht, 2007.
xii+400pp.
[7] V. V. Kirichenko, A. V. Zelensky, V. N. Zhuravlev, Exponent matrices and tiled
orders over discrete valuation rings, Algebra and Computation Vol. 15 No 5 & 6
(2005) 997–1012.
[8] V. V. Kirichenko and M. A. Khibina, Semi-perfect semi-distributive rings, In:
Infinite Groups and Related Algebraic Topics, Institute of Mathematics NAS
Ukraine, 1993, pp. 457–480 (in Russian).
[9] A. A. Tuganbaev, Semidistributive modules and rings, Kluwer Acad. Publ., Dor-
drecht, 1998.
[10] V. Zhuravlev, D. Zhuravlyov, Tiled orders of width 3, Algebra and Discrete Math. 1
(2009) 111–123.
Contact information
V. Zhuravlev Department of Mechanics and Mathematics,
Taras Shevchenko National University of Kyiv,
Volodymyrska, 64, Kyiv 01033, Ukraine
E-Mail: vshur@univ.kiev.ua
D. Zhuravlyov JSCB “Industrialbank”, Processing Center,
Kutuzova street, 18/7, 01133 Kyiv, Ukraine
E-Mail: dzhuravlev@ukr.net
Received by the editors: 12.08.2012
and in final form 12.08.2012.
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