AMERICAN MATHEMATICAL SOCIETY
Volume 364, Number 3, March 2012, Pages 1395–1411 S 00029947(2011)054103
Article electronically published on November 7, 2011
APPROACH TO ARTINIAN ALGEBRAS VIA NATURAL QUIVERS
FANG LI AND ZONGZHU LIN
Abstract. Given an Artinian algebraAover a ﬁeldk, there are several com binatorial objects associated toA. They are the diagramDA as deﬁned by Drozd and Kirichenko, the natural quiver ΔA deﬁned by Li (cf. Section 2), and a generalized version ofkspecies (A/r, r/r^{2}) withrbeing the Jacobson radical ofA. WhenAis splitting over the ﬁeldk, the diagramDA and the wellknown Extquiver ΓA are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebraA.
1. Introduction
1.1. Given an Artinian algebra A over a ﬁeld k, there are several combinatorial objects associated toA. They are the diagramDA as deﬁned in [DK], the natural quiver ΔA deﬁned in [Li] (cf. Section 2), and a generalized version of kspecies (A/r, r/r^{2}) withr being the Jacobson radical ofA. WhenA is splitting over the ﬁeldk, the diagramDAand the wellknown Extquiver ΓAare the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra A.
For a given ArtiniankalgebraA, let{S1, S2,· · ·, Ss}be the complete set of non isomorphic irreducible Amodules. Set Di = EndA(Si), which is a division ring, and Ext^{1}_{A}(Si, Sj), which is a DiDjbimodule. A is said to split over the ground ﬁeldk, or say, tobeksplitting, if dimkEndA(Si) = 1 (i.e.,Di=kfor all irreducible Amodules Si). Recall that a quiver is a ﬁnite directed graph Γ = (Γ0,Γ1) with vertex set Γ0 and arrow set Γ1. For a ksplitting algebra A, one can deﬁne a ﬁnite quiver ΓA called the Extquiverof Aby setting Γ0 ={1,2,· · · , s} and with mij = dimk(Ext^{1}_{A}(Si, Sj)) being the number of arrows from itoj.
There is another way to characterize the Extquiver for a ksplitting algebra Artinian algebra. By [ARS] and [Liu], whenAis a ﬁnite dimensional algebra over a ﬁeldkand 1 =ε1+· · ·+εsis a decomposition of 1 into a sum of primitive orthogonal idempotents, then we can reindex{S1, S2,· · ·, Ss} such thatSi ∼=Aεi/rεi where ris the radical ofA, and moreover, dimkExtA(Si, Sj) = dimk(εjr/r^{2}εi).
Received by the editors August 14, 2009 and, in revised form, May 18, 2010.
2010Mathematics Subject Classiﬁcation. Primary 16G10, 16G20; Secondary 16P20, 13E10.
This project was supported by the National Natural Science Foundation of China (No.
10871170) and the Natural Science Foundation of Zhejiang Province of China (No. D7080064).
The second author was supported in part by an NSA grant and the NSF I/RD program.
c2011 American Mathematical Society Reverts to public domain 28 years from publication 1395
If A is basic and ksplitting, then A is a quotient of the path algebra of ΓA. The properties of this quiver ΓA, in particular, its relation to the representations of A, has been extensively studied in the ﬁeld of representations of algebras. ΓA
is invariant under Morita equivalence, i.e., ifA and B are two Morita equivalent kalgebras, then ΓA is isomorphic to ΓB.
If A is not basic, it is no longer isomorphic to a quotient of the path algebra kΓA. It is discussed in this paper and in [Li] as to how to get the analogue of the Gabriel theorem in this case.
1.2. In recent years, geometric methods have been heavily used in representation theory of algebras. To each ﬁnite dimensional algebraAover an algebraically closed ﬁeldk, one can associate a sequence of algebraic varieties Mod^{d}k(A) (d= 1,2,3. . .) as closed subvarieties of the aﬃne spaces k^{d}^{×}^{d}. The association of the varieties depends on the presentation of the algebra A using ﬁnitely many generators and ﬁnitely many relations. In [B], it is proved that two algebrasAandBare isomorphic if and only if the associated varieties Mod^{d}_{k}(A) and Mod^{d}_{k}(B) are isomorphic as GLd(k)varieties. Thus having a more standard presentation of the algebraA will help with studying these varieties. The purpose of this paper is to explore relations of the Extquiver of A and the natural quiver which will be deﬁned in Section 2.
The natural quiver will have fewer arrows than the Extquiver when the algebraA is not basic. Natural quivers are not invariant under the Morita equivalence and are much closer to reﬂect the structure of the algebra rather than just its module category. There are numerous cases, even in representation theory, where one needs the structure of the algebras. For example, the character values of ﬁnite groups in a block cannot be preserved through Morita equivalence.
1.3. The paper is organized as follows. In Section 2 we recall the deﬁnition of the natural quiver ΔA of an Artiniankalgebra Aand provide a precise relation with the Extquiver ΓAwhen the algebraAis splitting over the ground ﬁeldk. In Section 3 we prove in Theorem 3.4 that any Artinian algebra, which is splitting over its radical, is a quotient of the generalized path algebra of its natural quiver associated to A/r. This gives a presentation of the algebra A. Although there is always a surjective algebra homomorphism from the path algebra of the natural quiver ΔAto the tensor algebraTA/r(r/r^{2}), the above surjective map toAdoes not always factor through TA/r(r/r^{2}) (Example 3.5). There have been numerous generalizations of WedderburnMalcev theorems to characterize an Artinian algebra that is splitting over its radical. By using the generalized path algebra of the natural quiver, we give another characterization of an Artinian algebraAwhich is splitting over its radical;
see Corollary 3.6. Moreover, we discuss the relations among the natural quiver and the Extquiver of an Artinian algebra and the associated generalized path algebra (see the ﬁgure at the end of Section 4) and, furthermore, their relationship with the diagram of an Artinian algebra as deﬁned in [DK]. The main results are the formulae in Theorems 2.2 and 5.3. As an application, in Section 5, we discuss the relationship between the diagram and natural quiver of an Artinian algebra that is not splitting over the ground ﬁeld. In Section 6 we prove that a (not necessarily basic) hereditary Artinian algebra which is splitting over its radical is isomorphic to the generalized path algebras of its natural quiver, provided the deﬁning ideal as described in Theorem 3.4 does not intersect with the arrow space (see Theorem 6.5).
2. The relation between natural quiver and Extquiver
Suppose thatAis a left Artiniankalgebra andr=r(A) is its Jacobson radical.
Write A/r = s
i=1Ai, where Ai are twosided simple ideals of A/r. Such a decomposition ofA/ris also called a block decomposition of the algebraA/r. Then, r/r^{2} is anA/rbimodule. Let iMj =Ai·r/r^{2}·Aj, which is ﬁnitely generated as anAiAjbimodule for each pair (i, j).
For two rings A and B, and a ﬁnitely generated ABbimodule M, deﬁne rkA,B(M) to be the minimal number of generators ofM as anABbimodule among all generating sets. As a convention, we always denote rkA,B(0) = 0.
The isomorphism classes of irreducible Amodules is indexed by the set Δ0 = {1,· · ·, s} corresponding to the set of blocks of A/r. We now deﬁne the natural quiver ΔA = (Δ0,Δ1) with Δ0 being the vertex set and, for i, j ∈ Δ0, tij = rkA_{j},A_{i}(jMi) being the number of arrows from i to j in ΔA. Obviously, there is no arrow fromi to j ifjMi = 0. This quiver ΔA = (Δ0,Δ1) is called the natural quiver ofA.
The notion of a natural quiver was ﬁrst introduced in [Li], where the aim was to use the generalized path algebra from the natural quiver of an Artinian algebra Ato characterize Athrough the generalized Gabriel theorem. The advantage of a generalized path algebra is that valued quiver information is already encoded in the generalized path algebras. In the language of Kontsevich and Soilbelman [KY], the Gabrieltype algebra cannot be stated as an aﬃne noncommutative scheme which can be embedded into a thin scheme in a sense that they are “inﬁnitesimally”
isomorphic. The aim of this paper will be to ﬁnd a “smallest” embedding.
For a quiverQ= (Q0, Q1), a subquiver Q^{} of Qis called denseif (Q^{})0 =Q0, and for any verticesi, j, there exists an arrow fromitoj inQ^{} if and only if there exists an arrow fromito j inQ.
WhenAis splitting over the ground ﬁeldk, thenAi∼=Mn_{i}(k) and the irreducible moduleSi haskdimension ni. In this case the Extquiver ΓA ofA is deﬁned (cf.
Section 1.1). It is proved in [Liu, Prop. 7.4.3] that tij ≤mij≤ninjtij.
In addition, ifAis basic, then ΔA= ΓAas discussed in Section 1.1. In general these two constructions will give two diﬀerent quivers if A is not basic. The following results were proved in [LC].
Proposition 2.1. Let A be a ksplitting Artiniankalgebra over a ﬁeldk and let B be the corresponding basic algebra of A. Then:
(i) ΓA= ΓB; (ii) ΓB = ΔB;
(iii) ΔA is a dense subquiver of ΓA, and thus a dense subquiver of ΓB and ΔB.
Now, we will just give the exact relation betweentij and mij when the algebra Ais splitting overk. First, we recall that for any real numbera, the ceiling ofais deﬁned to be
a= min{n∈Zn≥a}.
Theorem 2.2. LetAbe an ArtiniankalgebraAwhich is splitting overk. Assume ΓA is the Extquiver ofAandΔA is the natural quiver ofA. Thentij =_{n}^{m}_{i}_{n}^{ij}_{j}for i, j∈(ΓA)0= (ΔA)0. Heremij is the number of arrows from itoj in ΓA.
Proof. The proof involves computing a minimal generating set ofiMj =Ai·(r/r^{2})· Aj as AiAjbimodules. We ﬁrst note that bothAi and Aj are simple kalgebras and are split overk. HenceAi⊗kA^{op}_{j} is a also a simple centralkalgebra, isomorphic toMninj(k). Sincer/r^{2}is a semisimple leftAimodule and a semisimple rightAj module,iMjis a semisimpleAi⊗kA^{op}_{j} module with simple components isomorphic toSi⊗kS^{op}_{j} . HereS_{j}^{op} is the right irreducibleAjmodule. Lete^{i}_{kl} be the standard matrix basis elements ofAi=Mn_{i}(k). Then dimkExt^{1}A(Si, Sj) = dimke^{i}11r/r^{2}e^{j}_{11}. We claim thatiMjhas exactly dimke^{i}_{11}r/r^{2}e^{j}_{11}manyAi⊗kA^{op}_{j} composition factors.
Indeede^{i}_{11}⊗e^{j}_{11}is a primitive idempotent ofAi⊗kA^{op}_{j} . Now the theorem follows from the following lemma for simple Artinian rings.
Lemma 2.3. Let R=Mn(D) be the ring of all n×nmatrices with entries in a division ring D and letM be an Rmodule. LetL=D^{n} be the natural irreducible Rmodule of column vectors. Then:
(i) M is a semisimpleRmodule isomorphic toL^{⊕}^{m}=L⊕L⊕ · · · ⊕L, where m= dimD(e11M);
(ii) M can be generated by^{m}_{n}many elements overRbut cannot be generated by a fewer number of elements.
Proof. SinceRis simple, all ﬁnitely generatedRmodules are semisimple. Note that every irreducibleRmodule is isomorphic toL=D^{n}. Then (i) follows from the fact that dimDe11L= 1. To show (ii), we note thatR=n
s=1Ress∼=L^{⊕}^{n} as a leftR module. IfM is generated bytmany elements, thenM is a quotient ofR^{⊕}^{t}as a left Rmodule. Hence,M has at mosttncomposition factors counting multiplicity, i.e., m≤tnand^{m}_{n} ≤t. ThusM cannot be generated by less than^{m}_{n}many elements.
On the other hand, letNbe anyRmodule of lengthp≤n. ThenNis isomorphic to L^{⊕}^{p}. There is a surjective homomorphismφ:R→N of leftRmodules. By writing m=tn+s, with 0≤s < n, we have^{m}_{n}=tifs= 0 and^{m}_{n}=t+1 ifs >0. Hence we can construct a surjective homomorphism R^{}^{m}^{n}^{} → L^{⊕}^{tn+s} = (L^{⊕}^{n})^{⊕}^{t}⊕L^{⊕}^{s}.
HenceM is generated by^{m}_{n}many elements.
As we note in Proposition 2.1 (ii), the Extquiver and the natural quiver of a ﬁnite dimensional basic algebra coincide with each other. As an application of Theorem 2.2, we give an example which means that the coincidence is also possible to happen for some nonbasic algebras.
Example 2.4. Letkbe a ﬁeld of characteristic diﬀerent from 2 and letQbe the quiver:
•e1

•
e2^{}
•e3^{}.
•e2
β α α^{} β^{}
• e3
Let Λ =kQ be the path algebra ofQand G=σbe the automorphism group ofQof order 2. Thenσdeﬁnes akalgebra automorphism ofkQ. Now, we consider the Extquiver and the natural quiver of the skew group algebra ΛG(see [ARS]).
Letrbe the Jacobson radical of Λ. By Proposition 4.11 in [ARS], (rΛG) is the Jacobson radical of ΛG. It is easy to see that (ΛG)/(rΛG)∼= (Λ/r)G. On Page 84 of [ARS], it was given that (Λ/r)G∼=A1×A2×A3×A4=k×k×
k k k k
× k k
k k as algebras and that the associated basic algebraB is obtained in the reduced form
from ΛG, which is Mortia equivalent to ΛG. Moreover, it was proved in [ARS] that B is isomorphic to the path algebra of the following quiver
e(1)• e(2)
? 
• ν e(3)
 e(4)• .
• λ μ
This quiver is just the Extquiver ΓΛG of ΛG. Therefore, allmij = 0 or 1. For i = 1,2,3,4, dimkAi =n^{2}_{i}, where n1 =n2 = 1,n3 =n4 = 2. By Theorem 2.2, tij =_{n}^{m}_{i}_{n}^{ij}_{j}. Then, for each pair (i, j), we have tij =mij = 0 or 1. Therefore, the natural quiver ΔΛG is equal to the Extquiver ΓΛG.
3. Algebras splitting over radicals
The concept of a generalized path algebra was introduced early in [CL]. Here we review a diﬀerent but equivalent deﬁnition.
Given a quiverQ= (Q0, Q1) and a collection of kalgebrasA={Ai i∈Q0}, let ei ∈ Ai be the identity. Let A0 =
i∈Q0Ai be the direct product kalgebra.
Note thatei are orthogonal central idempotents ofA0.
For i, j ∈ Q0, let Ω(i, j) be the subset of arrows in Q1 from i to j. Deﬁne AiΩ(i, j)Ajto be the freeAiAjbimodule (in the category ofkvector spaces) with basis Ω(i, j). This is the freeAi⊗kA^{op}_{j} module over the set Ω(i, j).
ThenM =
i,jAiΩ(i, j)Ajis anA0A0bimodule. The generalized path algebra is deﬁned to be the tensor algebra
(1) TA0(M) =
∞ n=0
M^{⊗}^{A}^{0}^{n}.
Here M^{⊗}^{A}^{0}^{n} = M ⊗A0 M ⊗A0 · · · ⊗A0 M and M^{⊗}^{A}^{0}^{0} = A0. We denote the generalized path algebra byk(Q,A). Elements inM^{⊗}^{A}^{0}^{n}are calledvirtualApaths of length n. In cases of path algebras, virtualApaths are linear combinations of Apaths of equal length. We denoteJ =+∞
n=1M^{⊗}^{A}^{0}^{n}. Thenk(Q,A)/J∼=A0. The generalized path algebra has the following universal mapping property. For anykalgebraB with anykalgebra homomorphismφ0:A0→B (thus makingB anA0A0bimodule) and anyA0A0bimodule homomorphismφ1 :M →B, there is a uniquekalgebra homomorphismφ:k(Q,A)→B extendingφ0 andφ1.
This deﬁnition is equivalent to the original deﬁnition in [CL] and has the ad vantage of the abovementioned universal mapping property. As a matter of fact, the classical path algebras are the special cases by taking Ai =k. We are more interested in the case whenAi are simplekalgebras, in particular when allAi are central simple algebras. The generalized path algebrak(Q,A) is callednormal if allAi are simplekalgebras.
A kalgebra A is said to be splitting over its radical r if there is a kalgebra homomorphism ρ : A/r → A such that π◦ρ = IdA/r. For example, if A/r is separable and A is Artinian, then A is always splitting over r as a result of the WedderburnMalcev theorem (and many generalizations in the literature [P]). Note that a normal generalized path algebrak(Q,A) with an acyclic (no oriented cycles) quiver Qis always splitting over its radical rad(k(Q,A)) = J. This equality fails in general for normal generalized path algebras.
Proposition 3.1. Let Q be a ﬁnite quiver and k(Q,A) be a normal generalized path algebra with eachAi being ﬁnite dimensional. IfI is an ideal ofk(Q,A)with J^{s} ⊆ I ⊂ J for some positive integer s, then k(Q,A)/I is a ﬁnitedimensional algebra and rad(k(Q,A)/I) =J/I.
Proof. SinceQis a ﬁnite quiver andAiare ﬁnite dimensional, we have thatM^{⊗}^{A}^{0}^{n} in (1) is ﬁnite dimensional overk for alln. Hence k(Q,A)/J^{s} is also ﬁnite dimen sional. It follows thatk(Q,A)/Iis ﬁnite dimensional. On the other hand, J/I is a nilpotent ideal ofk(Q,A)/Iwith (J/I)^{s}= 0 and (k(Q,A)/I)/(J/I)∼=k(Q,A)/J∼=
i∈Q_{0}Ai is a semisimple algebra. This implies rad(k(Q,A)/I) =J/I. We remark that the ﬁnite dimensionality ofAi overk cannot be removed. For example, take the Dynkin quiver Q of type A2 with two vertices {1,2} and one arrow αfrom 1 to 2. Let A1 and A2 be two inﬁnitedimensional ﬁeld extensions ofk. AlthoughA1 andA2 are two simplekalgebras, the path algebrak(Q,A) is neither left nor right Artinian since
A2αV ={
i
aiαbi:ai∈A2, bi∈V} is a left ideal ofk(Q,A) for any vector subspaceV ofA1.
In this section we will show that every Artinian algebraAwhich is splitting over its radical will be a quotient of a generalized path algebra of its natural quiver.
For an Artinian algebra A with radical r, let ΔA be its natural quiver and A/r=
i∈(ΔA)0Ai, where allAi are simple algebras. DenoteA={Ai : i∈Δ0}.
Then, the generalized path algebra k(ΔA,A) is called the associated generalized path algebraofA.
In [LC], we introduce the following notion of a Gabrieltype algebra.
Deﬁnition 3.2. Let A be an Artinian kalgebra and k(ΔA,A) be its associated generalized path algebra. A is said to be of Gabriel type for a generalized path algebraifA∼=k(ΔA,A)/I for some ideal I ofk(ΔA,A) contained inJ. We callI the deﬁning ideal ofA.
In [Li] and [LC], the conditions for certain Artinian algebras to be of Gabriel type were discussed under diﬀerent assumptions. Here we give a more general description.
Theorem 3.3. Let A be a Gabrieltype Artinian algebra for a generalized path algebra over a ﬁeld k such that A ∼= k(ΔA,A)/I with the ideal I of k(ΔA,A) satisfying I⊂J. Then,Ais splitting over its radical r.
Proof. The assumptionI⊆J implies that
(k(ΔA,A)/I)/(J/I)∼=k(ΔA,A)/J ∼=A1⊕ · · · ⊕Ap∼=A/r
is semisimple; then radA ∼= rad(k(ΔA,A)/I) = J/I. We have A1⊕ · · · ⊕Ap ⊂ k(ΔA,A) and (A1⊕ · · · ⊕Ap)∩I ⊂ (A1⊕ · · · ⊕Ap)∩J = 0. Hence, we get A1⊕· · ·⊕Ap
→ε k(ΔA,A)/I ^{η} (k(ΔA,A)/I)/(J/I)∼=k(ΔA,A)/J∼=A1⊕· · ·⊕Ap, which implies ηε= 1. Therefore,Ais splitting overr.
In fact, in Theorem 8.5.4 of [DK], it was proven that, for a ﬁnitedimensional algebra A with radical r, if the quotient algebra A/r is separable, then A is iso morphic to a quotient algebra of the tensor algebra TA/r(r/r^{2}) by an idealI such
thatJ^{s}⊂I⊂J^{2}for some positive integers. The separability condition plays two roles in the proof. First, it guarantees the WedderburnMalcev theorem to get
(a) Ais splitting over its radical.
Secondly, the seperability also implies that
(b) r/r^{2}is isomorphic to a direct summand ofr as anA0A0bimodule.
It turns out that the properties (a) and (b) together are equivalent to the existence of the surjective mapφin the theorem.
It is proved in [Li] that there always exists a surjective homomorphism of algebras π : k(ΔA,A) → TA/r(r/r^{2}). This can be seen from the universal property of k(ΔA,A). Hence any Artinian algebraAwith separable quotientA/ris isomorphic to a quotient algebra ofk(ΔA,A) by an ideal I as in Theorem 3.3. The diﬀerence between the points given in this paper and those in [Li] is that the algebra A is splitting over its radical and the given idealI is included inJ but not inJ^{2}. The following theorem is an improvement of the result in [Li].
Theorem 3.4. Let A be an Artinian kalgebra such that A is splitting over its radical. Then there is a surjective algebra homomorphism φ:k(ΔA,A)→A with J^{s}⊆ker(φ)⊆J for some positive integers.
Proof. LetA/r=A0=
i∈Δ_{0}Ai. ThenA0is semisimple andr/r^{2}is a semisimple A0module. Since A splits over r, we can regard A0 as a subalgebra of A, and henceris anA0A0bimodule under the multiplication inA and the quotient map
¯:r→r/r^{2} is a homomorphism ofA0A0bimodules. We will use this property in the following argument.
For each pair (i, j), let iMj =Ai(r/r^{2})Aj. LetXij ={fl^{ij} ∈ AirAj ⊆r l = 1, . . . , tij} such that the image ¯Xij = {f¯_{l}^{ij}  l = 1, . . . , tij} in iMj is a minimal generating set as an AiAjbimodule. Let X =
i,j∈Δ_{0}Xij. Then, its image X¯ =
i,j∈Δ0
X¯ij generatesr/r^{2}= _{i,j}_{∈}_{Δ}_{0}Ai(r/r^{2})Aj as anA0A0bimodule.
We now show that the subalgebra S of A generated by A0 and the set X is actually A. Consider the associated graded algebra gr(A) =
s≥0r^{s}/r^{s+1}, with r^{0}=A. Then gr(S) =
s≥0(S∩r^{s})/(S∩r^{s+1}) is the graded subalgebra of gr(A) generated byA0 and ¯X ={f¯_{l}^{ij}1≤l≤tij, i, j∈Δ0}.
We claim that gr(S) = gr(A). We will show that Sp = r^{p}/r^{p+1}, where Sp = (S∩r^{p})/(S∩r^{p+1}) are the homogeneous components of the graded algebras grS.
Note that bothSpandr^{p}/r^{p+1}areA0A0bimodules. It follows from the deﬁnition that S1 = (A0XA0)/(r^{2}∩S) ⊆A0XA¯ 0 = r/r^{2}. By the choice of the set ¯X, we have thatA0XA0 maps ontor/r^{2} (under the quotient mapr→r/r^{2}). Hence, we haveS1=r/r^{2}. The multiplication
r/r^{2}⊗A0r/r^{2}⊗A0· · · ⊗A0r/r^{2}
ptimes
→r^{p}/r^{p+1}
in gr(A) is surjective following the deﬁnition ofr^{p}. Thus r^{p}/r^{p+1}=S 1· · ·S1
ptimes
⊆Sp.
However, we haveSp⊆r^{p}/r^{p+1}. HenceSp=r^{p}/r^{p+1} and gr(A) = gr(S).
We now prove thatS=A. Otherwise, we must haverS sinceAis generated byA0andr. Letpbe maximal such thatr^{p}⊆S is not true. Such apexists since
r^{m}= 0 for m0 (Ais Artinian) and p≥1. Takea∈r^{p}\S and let ¯a∈r^{p}/r^{p+1} be the image of a. Since Sp = r^{p}/r^{p+1}, there is b ∈ S∩r^{p} such that ¯b = ¯a in r^{p}/r^{p+1}. Thus a−b∈r^{p+1} ⊆S (pis maximal). Hence, a=b+ (a−b)∈S∩r^{p}. This is a contradiction. Hence we haveS=A.
The fact that A splits over r and A0 is a subalgebra of A such that the quo tient map A → A/r restricts to A0 is the identity map. Then A0XA0 ⊆ A is an A0A0subbimodule of A. Since Ω(i, j) = tij = rk(iMj), there is a surjec tive homomorphism of A0A0bimodules: A0Ω(i, j)A0 → A0XijA0. Now by the universal mapping property of the generalized path algebra, there is a surjective algebra homomorphismφ:k(ΔA,A)→S, sinceS is generated byA0and the set X as a subalgebra ofA. Hence φ(k(Δ,A)) =S =A, as we have just proved, and φA_{0}= idA_{0}.
LetI= ker(φ) andJ be the ideal ofk(ΔA,A) generated by all virtualApaths of length 1. Thenφ(J) =rand thus inducesk(Δ,A)/J∼=A/r. Hence ker(φ)⊆J. Since φ(J^{p}) = r^{p} for all p ≥ 0, we have φ(J^{s}) = r^{s} = 0 for some s > 0. This
completes the proof of the theorem.
Theorem 3.4 requires that the Artinian algebraAis splitting over its radical, i.e., the property (a) only without (b) as mentioned above, although the tensor algebra has to be replaced by the generalized path algebra. In the case r_{A}^{2} = 0, condition (b) is automatic provided (a) holds.
For an Artinian algebraA, setA0=A/rA, which is a semisimple algebra with an A0A0bimodule structure on M =rA/r_{A}^{2}. Then we can call the ordered pair (A0, rA/r_{A}^{2}) a generalized kspecies in the language of kspecies discussed in [R].
The tensor algebraTA_{0}(M) =_{∞}
n=0M^{⊗}^{A}^{0}^{n}is an associate algebra (not necessarily Artinian). The generalized path algebra of a quiver is naturally the tensor algebra of a generalizedkspecies. The properties of this path algebra are controlled by the bimodule structure ofM. This algebra plays an important role in noncommutative geometry, which will be studied in the next paper.
The following example shows the diﬀerence between the generalized path algebra and the tensor algebra of the associated generalizedkspecies in the fact thatφdoes not exist with condition (a) only.
One also notes that the surjective algebra homomorphism π : k(ΔA,A) → TA0(rA/r^{2}_{A}) is an isomorphism if and only if Ai(rA/r_{A}^{2})Aj is a free Ai ⊗k Aj bimodule for all i, j. One natural question is, for a general π, whether there is a map ψ : TA0(rA/r_{A}^{2})→ A such that ψ◦π = φ. The following example gives an answer to this question. It also shows that one cannot expect to generalize [DK, Th. 8.5.2] to nonseparable cases.
Example 3.5. LetD=k[α] withk=Fp(t) being the transcendental extension of the ﬁnite ﬁeld withpelements andα= √^{p}
t. Then D is a purely inseparable ﬁeld extension ofk. NowB=D⊗kD is not semisimple, and its radicalrB is nilpotent and generated byx=α⊗k1−1⊗kαbe aDDsubbimodule ofB. LetA=D⊕rB
be akvector space with multiplication deﬁned by (d, z)(d^{}, z) = (dd^{}, dz^{}+zd^{}+zz^{}) withd, d^{} ∈D andz, z^{} ∈rB . Heredd^{} andzz^{} are the multiplications inD and rB respectively by noting the DDbimodule structure on rB. This makes A an associative algebra andA/rA=Dis not separable. However,Ais splitting over its radical. Ais a not a commutative algebra (unlessp= 2) andrA/r_{A}^{2} =rB/r^{2}_{B}∼=D are DDbimodules whose actions are coinciding with the left and right actions
of D. Note that the left and right actions of D on rB are not the same unless p= 2. Hence the tensor algebraTD(rA/r_{A}^{2}) is a commutative algebra, butAis not commutative. Hence there is no surjectivekalgebra mapψ:TD(rA/r_{A}^{2})→A.
This example shows that under the assumption thatAis splitting over its radical, the surjective mapφ:k(ΔA,A)→A in Theorem 3.4 cannot be factored through TA_{0}(rA/r^{2}_{A}).
As mentioned earlier the separability condition onA0implies thatAis splitting over its radical, by the WedderburnMalcev theorem. There have been numerous generalizations of the WedderburnMalcev theorem in the literature. Combining Theorem 3.3 and Theorem 3.4, the following gives a characterization of Artinian algebras that is splitting over its radical in terms of natural quivers and the asso ciated generalized path algebras.
Corollary 3.6. An Artiniankalgebra Ais splitting over its radical if and only if A is of Gabriel type for generalized path algebras (cf. Deﬁnition 3.2); i.e., there is a surjective algebra homomorphismπ:k(ΔA,A)→A such that ker(π)⊆J.
We will see in Section 6 that it will be important if the condition ker(π)⊂J in Corollary 3.6 is replaced by the stronger one, that is, ker(π)⊂J^{2}. In this case, the idealI= ker(π) is said to beadmissible.
4. The relations among quivers arisen from an Artinian algebra to its generalized path algebra
In this section, we use the relation between the Extquiver and the natural quiver in Section 2 to study the associated normal generalized path algebras of Artinian algebras. Note that the deﬁnition of the Extquiver always requires that an Artinian algebraA is splitting over the ﬁeldk, which we will assume in this section.
The natural quiver ΔA of an Artinian algebra Ais always ﬁnite, i.e., including ﬁnitely many vertices and ﬁnitely many arrows. In the sequel, we always assume that ΔA is acyclic (i.e., ΔA does not have oriented cycles of length at least 1).
Trivially, this is the suﬃcient and necessary condition under which the associated k(ΔA,A) is Artinian.
By deﬁnition, acyclicity of ΔA implies that the natural quiver of k(ΔA,A) is just that of A, that is, Δk(Δ_{A},A) = ΔA. Denote by mij and gij, respectively, the arrow multiplicities in the Extquivers ΓA and Γk(Δ_{A},A), respectively. Note that they should not be confused with each other. In general, mij and gij are quite diﬀerent since the representation theories ofAandk(ΔA,A) are quite diﬀerent (cf.
Section 5).
For the radicalJ of k(ΔA,A), k(ΔA,A)/J ∼=A/rA=
i∈Δ0Ai. By Theorem 2.2, the natural quiver Δk(Δ_{A},A) is a subquiver of the Extquiver Γk(Δ_{A},A), and exactly,
(2) tij = gij
ninj.
DenoteI= (ΔA)0. Acomplete set of nonisomorphic primitive orthogonal idem potents of A is a set of primitive orthogonal idempotents {ei : i ∈ I} such that Aei ∼=Aej as leftAmodules for anyi=j in I and for each primitive idempotent ethe moduleAe is isomorphic to one of the modulesAei (i∈I).
Letei =ei+rA. Then, by [LC],{ei:i∈I} is a complete set of nonisomorphic primitive orthogonal idempotents ofA/r. Let ˜ei ∈k(ΔA,A) be the lift of ¯ei (we
have assumed that k(ΔA,A) is Artinian). Then, Si ∼= Aei/rAei = (A/rA)ei ∼= (k(ΔA,A)˜ei)/(Je˜i), as i∈I, gives a list of all nonisomorphic irreducible modules for bothAandk(ΔA,A).
For i ∈ I, the identity 1Ai of Ai can be decomposed into a sum of primitive idempotents, i.e., 1Ai=e^{i}_{11}+· · ·+e^{i}_{n}_{i}_{n}_{i}, and we can assumee^{i}_{11}=ei.
Note thatA isksplitting. By [ARS], Proposition III.1.14, we have gij= dimkExt^{1}_{k(Δ}_{A}_{,}_{A}_{)}(Si, Sj) = dimk(˜e^{j}_{kk}J/J^{2}e˜^{i}_{ll}) for allk, l. Therefore,
dimkAjJ/J^{2}Ai = dimk1A_{j}J/J^{2}1A_{i} =
n_{j}
k=1 n_{i}
l=1
dimke^{j}_{kk}J/J^{2}e^{i}_{ll}
= ninjdimkExt^{1}_{k(Δ}_{A}_{,}_{A}_{)}(Si, Sj) =ninjgij.
Recall that the number of arrows from ito j in Δk(ΔA,A) = ΔA istij. These tij
arrows generate freely theAjAibimoduleAjJ/J^{2}Ai. Thus,
dimkAjJ/J^{2}Ai= dimkAjdimk(J/J^{2}) dimkAi =n^{2}_{j}tijn^{2}_{i}. Hence,gij=ninjtij, that is, we obtain the following:
Proposition 4.1. Let A be a ksplitting ﬁnitedimensional algebra with radicalr, whose natural quiverΔAis acyclic. WriteA/r=
i∈(ΔA)0Ai, whereAiare simple algebra for all i with ni=√
dimkAi. Then, the natural quiver of k(ΔA,A)is the same as that ofA and
gij=ninjtij,
wheregij is the number of arrows fromi toj inΓk(Δ_{A},A) andtij is the number of arrows fromitoj inΔA.
For the number mij of arrows fromi toj in ΓA, by Theorem 2.2,tij =_{n}^{m}_{i}_{n}^{ij}_{j}. Then,_{n}^{m}_{i}_{n}^{ij}_{j}= _{n}^{g}^{ij}
in_{j}. Equivalently, we have Corollary 4.2. mij ≤gij< mij+ninj.
The set{Pi=Aeii∈I}is a complete set of representatives of the isoclass of indecomposable projectiveAmodules. Then the basic algebraB ofAis given by
B= EndA(
i∈I
Pi)∼=
i,j∈I
HomA(Pi, Pj)∼=
i,j∈I
eiAej.
Similarly the basic algebra ofk(ΔA,A) is C= Endk(ΔA,A)(
i∈I
k(ΔA,A)ei)∼=
i, j∈I
eik(ΔA,A)ej.
The relationship between the two basic algebrasB andC is given in [LC] whenA is of Gabriel type.
As we have said,k(ΔA,A) is not Artinian when ΔAhas an oriented cycle. Hence, we cannot aﬃrm whetherC is Morita equivalent tok(ΔA,A) in general. However, C is still decided uniquely by k(ΔA,A). Therefore, we call C the basic algebra associated tok(ΔA,A).
As we have assumed ΔA is acyclic in this section, C is Morita equivalent to k(ΔA,A) sincek(ΔA,A) is Artinian. Hence, their Extquivers are the same, that is, Γk(Δ_{A},A)= ΓC. Also, sinceC is basic, ΓC= ΔC.
To sum up, assuming that ΔA is acyclic, we get the following diagram:
ΓA
t_{ij}=_{ninj}^{mij}
⊃ ΔA = Δk(Δ_{A},A)
t_{ij}= ^{gij}
⊂ninj Γk(Δ_{A},A)
∩ ∩
ΓB = ΔB
m_{ij}≤g_{ij}<m_{ij}+n_{i}n_{j}
⊂ ΔC = ΓC,
where⊂,⊃and∩mean the embeddings of the dense subquivers.
5. Diagram for nonsplitting algebras
In Sections 2 and 4, the Artinian algebra Ais required to be splitting over the ground ﬁeld k due to the deﬁnition of the Extquiver. If A is not splitting over k, usually the Extquiver and its representations have to be respectively replaced by the socalled valued quiveror kspecies. On the other hand, the notion of the diagram of an Artinian algebra is introduced in [DK] in the case when A is not necessarily splitting over k. Now, we recall the deﬁnition of the diagram of an Artinian algebraA.
LetP1, P2,· · · , Psbe pairwise nonisomorphic principal indecomposable projec tive modules over an Artinian algebraA, corresponding to the simple components A1, A2,· · · , As of the semisimple algebra A =A/r = s
i=1Ai. Write Ri =rPi; then Si = Pi/Ri is the corresponding irreducible Amodule. At the same time, Vi=Ri/rRi is a semisimple left Amodule which has a direct sum decomposition Vi ∼=s
j=1S_{j}^{⊕}^{h}^{ij} for some unique integers hij. Deﬁne the quiverDA = (D0, D1) by setting D0 = {1,· · · , s} and the arrow set D1 such that there are exactly hij
many arrows fromitoj fori, j∈D0. This quiverDA is called the diagram of the algebraA.
Observe that: (i) the projective cover of Vi is P(Ri)∼=s
j=1P_{j}^{⊕}^{h}^{ij} for each i;
(ii) two Morita equivalent algebras have the same diagrams; (iii) the diagrams ofA andA/r^{2} coincide.
Proposition 5.1. Let S1, S2,· · ·, Ss be pairwise nonisomorphic irreducible mod ules over an Artinian algebra A. Then, in the diagram DA of A, the number of arrows from the vertex ito the vertex j is
hij= dimD_{j}ExtA(Si, Sj), whereDj= EndA(Sj)is a division algebra.
Proof. Applying the functor HomA(−, Sj) to the short exact sequence 0 →Ri → Pi→Si→0, one gets the long exact sequence, which gives the isomorphisms
Ext^{p}_{A}(Ri, Sj)∼= Ext^{p+1}_{A} (Si, Sj) (p≥0).
SinceRi/rRi =s
l=1S_{l}^{⊕}^{h}^{il}, we have HomA(Ri/rRi, Sj)∼=D^{h}_{j}^{ij}. Now the propo sition follows from HomA(Ri/rRi, Sj)∼= HomA(Ri, Sj).
When A is ksplitting, each Dj = k. Then hij = dimkExt^{1}_{A}(Si, Sj) = mij. Hence, in this case, the Extquiver ΓAis equal to the diagramDA, whose relation ship with ΔA is given in Theorem 2.2. When Ais not ksplitting, in place of the Extquiver ΓAis a valued quiver with modulation. The following gives a relation of the diagramDA with the valued quiver and associated modulation as well as with the natural quiver ΔA for a general Artinian algebraA.