Irreducible components of varieties of
representations II
B. Huisgen-Zimmermann∗ and I. Shipman†
September 26, 2015
Abstract
This article is part of a program to evolve the generic representation theory of basic
finite dimensional algebras Λ over an algebraically closed field K; in other words, the
goal is to determine the irreducible components of the varieties Repd (Λ) parametrizing the finite dimensional representations with dimension vector d, and to generically
describe the representations encoded by the components. Here we primarily target
truncated path algebras, i.e., algebras of the form Λ = KQ/I for a quiver Q, where I
is generated by all paths of some fixed length in the path algebra KQ. The main result
characterizes the irreducible components of the affine (resp. projective) parametrizing variety Repd (Λ) (resp. GRASSd (Λ)) in case Q is acyclic. Our classification is
in representation-theoretic terms, permitting to list the components from the quiver
and Loewy length of Λ. Combined with existing theory, it moreover yields an array of
generic features of the modules parametrized by the irreducible components, such as
generic minimal projective presentations, generic skeleta (“path bases” recruited from
a finite set of eligible paths), generic dimensions of endomorphism rings, generic socles,
etc.
The information on truncated path algebras with acyclic quiver supplements the
comparatively well-developed theory available in the special case where Λ is hereditary,
i.e., for I = 0: On one hand, we add to the classical generic results regarding the ddimensional KQ-modules; they address only the modules of maximal Loewy length.
On the other hand, the more general theory for I 6= 0 developed here fills in generic
data on the d-dimensional KQ-modules of any fixed Loewy length.
1
Introduction and main result
The overarching objective is to generically organize the representation theory of basic finite
dimensional algebras Λ over an algebraically closed field K. This approach to representation
theory was pioneered by Kac and further developed by Schofield in the hereditary case, that
is, when Λ = KQ is a path algebra; see [Kac80, Kac82] and [Sch92]. In this situation, the
parametrizing varieties of the modules with any fixed dimension vector d are irreducible
(in fact, they are full affine spaces), and the task amounts to relating generic features of
∗ Department
† Department
of Mathematics, University of California, Santa Barbara, CA 93106. birge@math.ucsb.edu
of Mathematics, University of Utah, Salt Lake City, UT 84112. ian.shipman@gmail.com
1
the d-dimensional modules to the quiver Q. Properties which are generically constant on
Repd (Λ) include decomposition properties, tops, socles, submodule lattices, homological
dimensions, etc.
On departing from the hereditary scenario, one is confronted with the initial roadblock
(major) that, typically, the parametrizing varieties Repd (Λ) are reducible and sport a
highly complex geometry. So pinning down the components in a representation-theoretically
useful format inevitably becomes the first step in any non-hereditary generic representation
theory. For prior work on the subject we refer to [BCHZ15, CW13, CIFR13, CBS02, DF77,
ES89, Ger61, Gur92, Kra82, Mor80, RRS11, Sch04], and the overview in the introduction to
[HZ14]. We point out that, until recently, it was only in tame cases that complete solutions
to the non-hereditary component problem were available.
Two types of parametrizing varieties are under consideration, the classical affine varieties
Repd (Λ) and their projective counterparts GRASSd (Λ); see Section 2.B. The problem of
identifying the components is equivalent in the two settings; indeed, any information gained
in one can readily be transferred to the other. But the two scenarios draw on different
caches of geometric techniques, making it advantageous to switch back and forth.
Our main result, stated below, addresses truncated path algebras, that is, algebras of the
form Λ = KQ/hall paths of length L + 1i for a quiver Q and a fixed positive integer L. This
class includes the hereditary finite dimensional algebras and those with vanishing radical
square. To indicate the prominent place held by truncated path algebras, we note that every
basic finite dimensional algebra Λ0 is a factor algebra of a unique truncated path algebra
Λ which has the same quiver and Loewy length as Λ0 ; in this pairing, each of the varieties
Repd (Λ0 ) arises as a subvariety of the corresponding Repd (Λ), which makes the truncated
case a natural first target in the quest for generic information.
In case Λ is truncated, one always has a supply of comparatively large irreducible subvarieties
of Repd (Λ) at one’s disposal (see [BHZT09]), namely, the varieties labeled Rep S, to be
introduced next. They partition Repd (Λ) into finitely many strata. Since any irreducible
component of Repd (Λ) intersects precisely one of them in a nontrivial open set, all of the
components are among the closures Rep S.
To introduce Rep S, let J be the Jacobson radical of Λ. Then J L+1 = 0. The radical layering
of a module M is the sequence S(M ) = J l M/J l+1 M 0≤l≤L , and the socle layering, S∗ (M ),
is defined dually. Both of the isomorphism invariants S(M ) and S∗ (M ) are semisimple
sequences having the same dimension vector as M , i.e., they are sequences ofPthe form S =
(S0 , S1 , . . . , SL ) whose entries Sl are semisimple modules such that dim S := 0≤l≤L dim Sl
equals dim M . Identifying isomorphic semisimple modules, we define Rep S to be the locally
closed subvariety of Repd (Λ) that consists of the points corresponding to modules with
radical layering S; thus Rep S is pinned down by an (L + 1) × |Q0 |-matrix of nonnegative
integers.
In the truncated situation, our intitial task is reduced to characterizing those semisimple sequences which arise as the generic radical layerings of the irreducible components of
Repd (Λ). For the purpose of sifting them out of the set Seq(d) of all semisimple sequences
with dimension vector d, our chief tool is the following upper semicontinuous map:
Θ : Repd (Λ) → Seq(d) × Seq(d), x 7→ S(Mx ), S∗ (Mx ) ,
where Seq(d) is partially ordered by the dominance order – see under Conventions below –
2
and Mx is the Λ-module corresponding to x. Due to semicontinuity of Θ, the small pairs in
the image are those of interest with regard to the components. As closer scrutiny reveals,
the number of minimal elements in the image of Θ, i.e., in the finite set
rad-soc(d) := { S(M ), S∗ (M ) | M ∈ Λ-mod, dim M = d},
increases steeply as a function of |d| in general. Our point of departure is the following
Observation 1.1. [HZ14, Theorem 3.1] Let Λ be truncated. If (S, S∗ ) is a minimal element
of rad-soc(d), then the closure Rep S is an irreducible component of Repd (Λ).
The converse fails, in general; see [HZ14, Example 4.5]. But, as was proved in [HZ14], it
does hold in case Λ is local. Here we show that the same is true for truncated path algebras
whose quivers are located at the opposite end of the spectrum, that is, do not contain any
oriented cycles.
Main Theorem. Let Λ be a truncated path algebra based on an acyclic quiver Q, and let
d be a dimension vector. Then the following conditions are equivalent for any semisimple
sequence S with dim S = d:
(1) The closure Rep S of Rep S is an irreducible component of Repd (Λ).
(1’) The closure GRASS(S) is an irreducible component of the projective variety GRASSd (Λ).
(See Section 2.B for notation.)
(2) S occurs as the first entry of a minimal pair (S, S∗ ) in rad-soc(d).
The situation is symmetric in S and S∗ : Whenever (S, S∗ ) is a minimal element of rad-soc(d),
then S∗ is the generic socle layering of Rep S; conversely, S∗ determines the generic radical
layering S of the modules with socle layering S∗ .
Given that each irreducible component of Repd (Λ) is of the form Rep S, the Main Theorem resolves the problem of identifying the components (combine it with Theorem 3.3 and
Remark 3.8). It is proved in Section 6.
In the special case of a hereditary algebra Λ = KQ, we first determine the (unique) generic
radical layering S of the Λ-modules with dimension vector d. Then we exploit this information towards a more detailed generic analysis of these modules, on the basis of our general
investigation of Rep S in Section 3. By its nature, Kac’s and Schofield’s seminal work on
path algebras limits its focus to the d-dimensional representations of Q that have maximal
Loewy length L(d). Our results on truncations KQ/I provide an extension of these results
towards a generic understanding of the KQ-modules with dimension vector d and arbitrary
Loewy length < L(d). Indeed, excising the open subvariety of Repd (KQ) which encodes the
modules of Loewy length L(d) leaves us with a copy of the variety Repd KQ/hL(d) − 1i ,
where hmi denotes the ideal of KQ generated by all paths of length m. Iterating, we remove
the points that represent modules
with Loewy length L(d) − 2, thereby shifting the generic
focus to Repd KQ/hL(d) − 2i , etc. As is immediate from the Main Theorem, the locally
closed subvariety of Repd (Λ) representing the KQ-modules of any Loewy length m < L(d)
splits into multiple irreducible components
in general. In exploring the components of the
truncated path algebras Repd KQ/hmi for decreasing values of m, we are thus, in effect,
3
targeting the generic behavior of successive classes of d-dimensional representations of Q,
each step moving us to an irreducible subvariety of Repd (Λ) that had been blended out in
the previous steps (cf. Example 5.1).
As we mentioned ahead of the Main Theorem, for a truncated path algebra Λ based on an
arbitrary quiver, the map Θ may fail to detect all irreducible components of the varieties
Repd (Λ). So we are left with the problem of compensating for the blind spot of Θ in the general case. To formulate our suggestion, let p1 , . . . , pm ∈ KQ \ I be a finite collection of paths
with shared starting and endpoints; for a Λ-module
X, denote by rankX (p1 , . . . , pm ) the
P
rank of the map X m → X, (x1 , . . . , xm ) 7→ 1≤i≤m pi xi . The dual, rankD(X) (p1 , . . . , pm )
refers to the analogous map D(X)m → D(X) based on the right
Λ-structure of D(X).
Moreover, we denote by f (X) the family − rankX (p1 , . . . , pm ) (p1 ,...,pm ) for m ≤ dim X,
and by f ∗ (X) the family of negative ranks of the dual D(X).
Problem 1.2. Let Λ be an arbitrary truncated path algebra and d a dimension vector.
Decide whether all irreducible components of Repd (Λ) are of the form Rep S where S =
S(M ) is the first entry of a quadruple S(M ), S∗ (M ), f (M ), f ∗ (M ) which is minimal in the
set
{ S(X), S∗ (X), f (X), f ∗ (X) | dim X = d}.
(Here we impose the componentwise partial order on tuples of integers.)
In Section 2, we assemble foundational material; this section addresses arbitrary basic finite
dimensional algebras. It is only in Section 3 that we specialize to truncated path algebras,
in order to (a) prepare specific tools for the proof of the Main Theorem, and (b) provide
the theoretical means for an effective application of this theorem. In particular, it is shown
in 3.A how, in the truncated case, the generic radical layering of an irreducible component
provides access to generic modules, generic skeleta, generic minimal projective presentations (Theorem 3.3), and generic endomorphism rings; moreover, the generic behavior of
submodules of generic modules is explored (Corollary 3.5). In Section 4, we narrow the
focus from the truncated to the hereditary scenario. In Section 5, we illustrate the theory
with non-hereditary examples. Section 6, finally, contains a proof of the Main Theorem.
Conventions: For our technique of graphing Λ-modules, we refer to [BHZT09, Definition
3.9 and subsequent examples]. Throughout, Λ is a basic finite dimensional algebra over an
algebraically closed field K, and Λ-mod (resp. mod-Λ) the category of finitely generated
left (resp. right) Λ-modules. By J we denote the Jacobson radical of Λ; say J L+1 = 0.
Without loss of generality, we assume that Λ = KQ/I for some quiver Q and admissible
ideal I ⊆ KQ. By our convention, the product of two paths p, q in KQ is the concatenation
“p after q” if the starting vertex of p equals the terminal vertex of q, and is 0 otherwise.
The set Q0 = {e1 , . . . , en } of vertices of Q will be identified with a full set of primitive
idempotents of Λ. Hence, the simple left Λ-modules are Si = Λei /Jei , 1 ≤ i ≤ n, up to
isomorphism. Unless we want to distinguish among different embeddings, we systematically identify isomorphic semisimple modules; in other words, we identify finitely generated
semisimples with their dimension vectors. This provides us with a partial order on the set
of finite dimensional semisimple modules: Namely, U ⊆ V if and only dim U ≤ dim V under
the componentwise order on Nn0 .
Let S be a semisimple sequence, that is, a sequence of the
P form S = (S0 , S1 , . . . , SL ) such
that each Sl is a semisimple module, and define dim S = 0≤l≤L dim Sl . When Sl = 0 for
4
all l ≥ m + 1, we will also write S in the clipped form (S0 , . . . , Sm ). The set Seq(d) of all
semisimple sequences with fixed dimension vector d is endowed with the following partial
order, dubbed the dominance order :
M
M
S ≤ S0
⇐⇒
Sj ⊆
S0j for l ∈ {0, 1, . . . , L}.
0≤j≤l
0≤j≤l
An element x of a Λ-module M is said to be normed if x = ei x for some i. A top element
of M is a normed element in M \ JM , and a full sequence of top elements of M is any
generating set of M consisting of top elements which are K-linearly independent modulo
JM .
For a brief introduction of the parametrizing varieties, we refer to Section 2.B. Given any
subset U of Repd (Λ) or GRASSd (Λ), the modules corresponding to the points in U are
called the modules “in” U. When U is irreducible, the modules in U are said to generically
have property (∗) in case all modules in some dense open subset of U satisfy (∗). Due to
upper semicontinuity of the map Θ, the radical layerings and socle layerings of the modules
are generically constant on any irreducible subvariety of Repd (Λ) (we re-emphasize our
convention of identifying isomorphic semisimple modules). Hence it is meaningful to speak
of the generic radical and socle layerings of the irreducible components of Repd (Λ).
Acknowledgment. The authors wish to thank Eric Babson for numerous stimulating conversations on the subject of components at MSRI. The first author was partially supported
by an NSF grant while carrying out this work. While in residence at MSRI, Berkeley, the
authors were supported by NSF grant 0932078 000. The second author was also partially
supported by NSF award DMS-1204733.
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2.A
Facts for arbitrary basic algebras Λ
Radical and socle layerings
Recall that the radical layering S(M ) of M ∈ Λ-mod is the sequence S0 (M ), . . . , SL (M )
with Sl (M ) = J l M/J l+1 M ; according to the above conventions, it will be identified with
the shortened sequence S0 (M ), . . . , Sm (M ) whenever J m+1 M = 0 for some m < L.
Dually, the socle layering of M is the sequence S∗ (M ) = S∗l (M ) 0≤l≤L with S∗l (M ) =
socl M/ socl−1 M . Here soc−1 M = 0, and soc0 M ⊆ soc1 M ⊆ · · · ⊆ socL M is the
standard socle series of M , i.e., soc0 M = soc M , and
socl+1 M/ socl (M ) = soc(M/ socl M ).
Clearly, dim S(M ) = dim S∗ (M ) = dim M .
Our interest in semisimple sequences is restricted to those which arise as radical or socle
layerings of Λ-modules. Due to the duality addressed in Lemma 2.2, we typically have a
choice of prioritizing either radical or socle filtrations. We will usually focus on radical
layerings, whence the bias in the following definition.
Definition 2.1. A semisimple sequence S is said to be realizable if there exists a left Λmodule M with S(M ) = S.
5
There is an algorithm for deciding realizability over general path algebras modulo relations.
However, in the truncated case, this condition may be checked by mere inspection of the
quiver; see Section 3.A. We will freely use the elementary properties of radical and socle
layerings assembled in the following lemma.
Lemma 2.2. Let M, N ∈ Λ-mod with dim M = dim N .
• Duality: The radical and socle layerings are dual to each other, in the sense that
S(D(M )) = D(S∗0 (M )), · · · , D(S∗L (M ) and S∗ (D(M )) = D(S0 (M )), · · · , D(SL (M ) ,
where D denotes the duality HomK (−, K) : Λ-mod → mod-Λ.
L
• Radical layering: dim J l M = dim l≤j≤L Sj (M ); in particular
S(M ) ≤ S(N )
⇐⇒
dim J l M ≥ dim J l N for all l ∈ {0, . . . , L}.
• Socle layering: socl M = annM J l+1 and dim socl M = dim
ticular,
S∗ (M ) ≤ S∗ (N )
⇐⇒
0≤j≤l
S∗j (M ); in par-
dim socl M ≤ dim socl N for all l ∈ {0, . . . , L}.
• Connection: J l M ⊆ socL−l M , and hence
2.B
L
L
l≤j≤L
Sj (M ) ⊆
L
0≤j≤L−l
S∗j (M ).
The parametrizing varieties
We recall the definitions of the relevant varieties parametrizing the isomorphism classes of
(left) Λ-modules with dimension vector d = (d1 , . . . , dn ). Recall that Λ = KQ/I.
The affine setting: The classical affine variety is Repd (Λ) =
Y
(xα )α∈Q1 ∈
HomK K dstart(α) , K dend(α) | the xα satisfy all relations in I ,
α∈Q1
where Q1 is the set of arrows of the quiver Q. This variety carries an obvious conjugation
action by GL(d) := GLd1 (K)×· · ·×GLdn (K), the orbits of which are in bijective correspondence with the isomorphism classes of d-dimensional modules. Throughout, we
L denote by
Mx ∈ Λ-mod the module that corresponds to a point x ∈ Repd (Λ), i.e., Mx = 1≤i≤n K di
such that multiplication by an arrow α : i → j is given by xα : K di → K dj . Given any
semisimple sequence S with dimension vector d, we set
Rep S = {x ∈ Repd (Λ) | S(Mx ) = S}.
The set Rep S is always a locally closed subset of Repd (Λ); it is nonempty if and only if S
is realizable in the sense of Definition 2.1.
P
L
The projective setting: Let d = |d| = i di . We fix a projective module P = 1≤r≤d Λzr
z1 , . . . , zd is a full sequence of top
whose top has dimension vector dim (P/JP) = d; here L
di
elements of P. In other words, P is a projective cover of
1≤i≤n Si , and thus is the smallest projective Λ-module with the property that every module with dimension vector d is
6
isomorphic to a quotient P/C for some submodule C ⊆ P. The variety Gr (dim P − d), P
is the vector space Grassmannian of all (dim P − d)-dimensional K-subspaces of P, and
GRASSd (Λ) is the closed subset consisting of the Λ-submodules C of P with the property that dim P/C = d. Under the canonical action of the automorphism group AutΛ (P)
on GRASSd (Λ), the orbits are again in one-to-one correspondence with the isomorphism
classes of modules with dimension vector d. In parallel with the affine setting: For any
semisimple sequence S with dimension vector d, we denote by GRASS(S) the locally closed
subvariety of GRASSd (Λ) picking out the points C with S(P/C) = S. (Caveat: The variety
GRASS(S) introduced here is not to be confused with the much smaller one, Grass S, used
in [BHZT09], for instance; it is in this smaller variety that information on GRASS(S) is
preferably gleaned.)
Connection between the two settings: The horizontal double arrows in the diagram below
point to the transfer of geometric information spelled out in the upcoming proposition. It
was proved in [BHZ01], modulo the unirationality statement which was added in [HZG12].
GRASS(S)
O
Rep S
⊆
⊆
GRASSd (Λ)
O
Repd (Λ)
Diagram 2.3
Proposition 2.3. Consider the one-to-one correspondence between the orbits of GRASSd (Λ)
and Repd (Λ) assigning to any orbit AutΛ (P).C ⊆ GRASSd (Λ) the orbit GL(d).x ⊆
Repd (Λ) that represents the same isomorphism class of Λ-modules. This correspondence
extends to an inclusion-preserving bijection
Φ : {AutΛ (P)-stable subsets of GRASSd (Λ)} → {GL(d)-stable subsets of Repd (Λ)}
which preserves and reflects openness, closures, irreducibility, smoothness and unirationality.
The following observation is an immediate consequence of the fact that the radical layerings
of the modules in an irreducible component of Repd (Λ) are generically constant.
Observation 2.4. All the irreducible components of Repd (Λ) are among the closures of
the irreducible components of the Rep S, where S traces the semisimple sequences with
dimension vector d.
By Proposition 2.3, the preceding observation has an analogue relating the irreducible components of GRASSd (Λ) to those of the subvarieties GRASS(S).
For more detail regarding the parametrizing varieties, we refer to [HZ09].
2.C
Skeleta and generic modules
We recall the following concepts from [BHZT09] and [HZ09]. Let S = (S0 , . . . , SL ) be a
semisimple sequence, and P a projective cover of S0 ; in particular, dim P/JP = dim S0 .
7
L
Fix P = 1≤r≤t Λzr , together with a full sequence z1 , . . . , zt of top elements of P , where
t = dim S0 ; say zr = e(r)zr with e(r) ∈ {e1 , . . . , en }. When the ideal I factored out of the
path-length-graded algebra KQ is not homogeneous, we lack a convenient notion of “path
length” of an element of the form pzr ∈ P , where p = p e(r) is a path in KQ\I. To meet this
L
difficulty, we define skeleta as subsets of the projective KQ-module Pb = 1≤r≤t KQ zr . We
do not make any formal distinction between the Λ- and the induced KQ-module structure
of a left Λ-module, but rely on the context.
Definition 2.5. and remarks. Let S be a semisimple sequence, and P , Pb as above. A
path of length l in Pb is any element of the form pzr , where p is a path of length l in KQ \ I
with start(p) = e(r). In particular, the canonical image in P of any path in Pb is nonzero.
1. An (abstract) skeleton with layering S is a subset σ of Pb consisting of paths in Pb which
satisfies the following two conditions:
• It is closed under initial subpaths, i.e., whenever pzr ∈ σ, and q is an initial
subpath of p (meaning p = q 0 q for some path q 0 ), the path qzr again belongs to
σ.
• For each l ∈ {0, . . . , L}, the number of those paths of length l in σ which end
in a particular vertex ei coincides with the multiplicity of Si in the semisimple
module Sl .
In particular, a skeleton σ with layering S includes the paths z1 , . . . , zt of length zero
in Pb. Note that the set of abstract skeleta with any fixed layering S is finite.
2. Suppose M ∈ Λ-mod. We call an abstract skeleton σ a skeleton of M in case M has
a full sequence m1 , . . . , mt of top elements with mr = e(r)mr such that
• {pmr | pzr ∈ σ} is a K-basis for M , and
• the layering of σ coincides with the radical layering S(M ) of M .
Observe: For every M ∈ Λ-mod, the set of skeleta of M is non-empty. On the other hand,
the set of all skeleta of modules with a fixed dimension vector is finite.
In particular, a skeleton of a module M yields a basis for M which consists of (generalized)
paths with the following property: the length of any of these paths indicates the depth of
the corresponding basis element in the radical layering of M (for x ∈ M \ {0}, the depth of
x in M is the largest integer l with x ∈ J l M ). Closedness under initial subpaths is a strong
additional condition. The motivation for this extra requirement is geometric: The points in
Rep S that correspond to modules with a fixed skeleton form an open affine subvariety of
Rep S.
The following is a rudimentary version of a result proved in [BHZT09, Theorem 4.3]; it is
adapted to our present needs. Let K0 be the smallest subfield of K such that Λ = KQ/I
is defined over K0 , meaning that K0 Q contains generators for I. Moreover, let K0 be
the algebraic closure of K0 in K. Clearly, K0 then has finite transcendence degree over
the prime field
P of K. Any
P automorphism φ ∈ Gal(K:K0 ) induces a ring automorphism
KQ → KQ, i ki pi 7→ i φ(ki )pi , which maps I to I and thus lifts to a ring automorphism
of Λ; the latter, in turn, gives rise to a Morita self-equivalence of Λ-mod, sending a module
8
M to the module whose Λ-structure is that of M twisted by φ. We refer to such a Morita
equivalence as induced by Gal(K:K0 ). Further, we call an attribute of a module Gal(K:K0 )stable if it is preserved by all Gal(K:K0 )-induced self-equivalences of Λ-mod. Note that
dimension vectors are Gal(K:K0 )-stable, for instance; obviously, so are all properties that
are invariant under arbitrary Morita equivalences.
Theorem-Definition 2.6 (Generic skeleta, existence and uniqueness of generic
modules). Assume that the field K has infinite transcendence degree over its prime field.
Whenever C is an irreducible component of Repd (Λ) with generic radical layering S and σ is
a skeleton of some module in C ∩Rep S, all modules in a dense open subset of C have skeleton
σ. In particular, it makes sense to speak of the generic set of skeleta of the modules in C; it
is the union of the sets of skeleta of the modules in C ∩ Rep S and is Gal(K:K0 )-stable.
There exists a generic Λ-module G for C, meaning that
• G belongs to C and
• G has all Gal(K:K0 )-stable generic properties of the modules in C.
Generic modules are unique in the following sense: Whenever G and G0 are generic for C,
there is a Gal(K:K0 )-induced Morita equivalence Λ-mod → Λ-mod which takes the isomorphism class of G to that of G0 .
For concrete illustrations of generic modules and generic skeleta see Section 4.B and Example
5.1.
Clearly, all Morita-invariant generic properties of the modules in C can be traced in any
generic module G. Beyond those: Given a decomposition of G into indecomposable direct
summands, the collection of dimension vectors of the summands of G is generic for decompositions of the modules in C. The same is true for the dimension vectors of the radical
and socle layers of G. Moreover, the dimension of the GL(d)-orbit corresponding to G is
the maximal orbit dimension attained on C; in particular, G is unique up to isomorphism
in case C contains a dense orbit.
In tackling the component problem, the last two of the following comments will allow us to
assume without loss of generality that our base field K has infinite transcendence degree
over its prime field. We will make this assumption whenever it is convenient to have generic
objects G ∈ Λ-mod for the components at our disposal.
Comments 2.7.
1. The set of (points in Repd (Λ) corresponding to the) generic modules for an irreducible
component C of Repd (Λ) is dense in C; see [BHZT09, Corollary 4.5]. However, this
set fails to be open in general.
b be
2. Passage to a base field of infinite transcendence degree over its prime field. Let K
the algebraic closure of a purely transcendental extension field K(Xα | α ∈ A) of K.
b := K
b ⊗K Λ is an algebra which has the same quiver (and hence the same
Then Λ
b∼
b I,
b where Ib is the ideal of KQ
b generated by
dimension vectors) as Λ; indeed, Λ
= KQ/
b is truncated whenever Λ is. The irreducible components of Repd (Λ)
I; in particular Λ
9
b
are in natural inclusion-preserving one-to-one correspondence with those of Repd (Λ).
b be the coordinate ring of Repd (Λ), resp. of Repd (Λ).
b
To see this, let Γ, resp. Γ,
b
b
b
The map Spec Γ → Spec Γ, P 7→ P = K ⊗K P is a well-defined inclusion-preserving
injection; indeed, the tensor product R1 ⊗K R2 of any two zerodivisor-free commutative
algebras R1 , R2 over an algebraically closed field K is in turn a domain (see, e.g.,
[ZS75, Ch.III, Corollary 1 to Theorem 40]). This map restricts to a bijection on the
set of minimal primes: Namely, if P1 , . . . , Pm are the minimal primes in Spec Γ, then
b1 )r1 · · · (P
bm )rm = 0 for suitable ri ≥ 0, whence every minimal prime Q ∈ Spec Γ
b
(P
b
b
contains one of the Pi ; equality Q = Pi follows.
b ∈ Λ-mod
b
3. Now suppose that C is an irreducible component of Repd (Λ) and let G
be
b
b
a generic module for the corresponding irreducible component C of Repd (Λ). Generb which are reflected by the
ically, the modules in C then have all those properties of G
exact and faithful functor
b ⊗K − : Λ-mod → Λ-mod.
b
F =K
In particular, this pertains to dimension vectors, as well as skeleta and direct sum
decompositions. Crucially in the present context: The dimension vectors of the generic
radical and socle layers of C and Cb coincide.
2.D
The role of upper semicontinuous maps in the component
problem
Λ = KQ/I still stands for an arbitrary basic finite dimensional K-algebra. We will rely
on upper semicontinuous maps, defined on the variety Repd (Λ), to detect and separate
irreducible components. The core problem consists of identifying “discerning” and, at the
same time, accessible maps serving this purpose. As is readily illustrated, the complexity
of an upper semicontinuous map Φ would have to be daunting if one aimed at a one-to
one correspondence between the minimal values of Φ and the components of Repd (Λ) for
arbitrary Λ. Useful diagnostic maps need to be adapted to specific classes of algebras. In
the context of truncated path algebras, we will re-encounter the map Θ displayed in the
introduction.
Definition 2.8. Suppose that X is a topological space and (A, ≤) a poset. For a ∈ A, we
denote by [a, ∞) the set {b ∈ A | b ≥ a}; the sets (a, ∞), (−∞, a] and (−∞, a) are defined
analogously.
A map f : X −→ A is called upper semicontinuous if, for every element a ∈ A, the pre-image
of [a, ∞) under f is closed in X.
The map Θ : Repd (Λ) → Seq(d) × Seq(d) of the introduction is upper semicontinuous
(see [HZ14, Observation 2.4]). Moreover, the following numerical module invariants are
well-known to yield upper semicontinuous maps on Repd (Λ): For any fixed N ∈ Λ-mod,
the maps x 7→ dim HomΛ (Mx , N ), x 7→ dim HomΛ (N, Mx ), x 7→ dim Ext1Λ (Mx , N ), and
x 7→ dim Ext1Λ (N, Mx ) are examples; for Ext1 , see [CBS02]. Moreover, for any path p in
KQ \ I, the map x 7→ nullityp Mx is upper semicontinuous; here nullityp Mx is the nullity
of the K-linear map Mx → Mx , m 7→ p m.
The next observation is pivotal in the present context.
10
Lemma-Definition 2.9. Let A be a poset and f : Repd (Λ) → A an upper semicontinuous
map such that Im(f ) is well partially ordered, i.e., Im(f ) has no infinite descending chains
and every nonempty subset has only finitely many minimal elements. Whenever a is a
minimal element in Im(f ), the closure f −1 (a) in Repd (Λ) of the fiber f −1 (a) is a union of
irreducible components of Repd (Λ). We say that f detects an irreducible component C of
X in case the generic value of f on C is minimal in Im(f ); equivalently, C ∩ f −1 (a) 6= ∅
for some minimal element a ∈ Im(f ). Further, f is said to separate irreducible components
in case f −1 (a) is irreducible for every minimal element a ∈ Im(f ).
We illustrate the resulting strategy of detecting and separating.
Example 2.10. Let Λ be an algebra with J 2 = 0 and d any dimension vector. Moreover,
let A be the set of pairs of semisimple modules with dimension vectors ≤ d. As was shown
in [BCHZ15], the function
f : Repd (Λ) → A, x 7→ Mx /JMx , soc Mx
detects all irreducible components of Repd (Λ) and separates them. This allows to list them
from Q in terms of the generic tops of their modules; in this case, it is easy to describe
the full set of isomorphism classes of modules belonging to each component. On the other
hand, it is readily seen that neither of the upper semicontinuous maps x 7→ Mx /JMx or
x 7→ soc Mx by itself suffices for a classification of the components.
Example 2.11. Now let Λ be the local truncated path algebra with Loewy length 3, based
on the quiver with a single vertex and two loops. We denote by S the unique simple object in
Λ-mod. Then [HZ14, Theorem A] guarantees that the irreducible components of Repd (Λ)
for d ∈ N are the closures Rep S corresponding to the minimal pairs (S, S∗ ) in the image of
Θ. In other words, Θ detects all irreducible components of Repd (Λ) and separates them.
If d = 7, for instance, one derives that Repd (Λ) has precisely five irreducible components,
namely the closures Rep S for the following semisimple sequences S:
(S, S 2 , S 4 ), (S 2 , S 2 , S 3 ), (S 2 , S 3 , S 2 ), (S 3 , S 2 , S 2 ) and (S 4 , S 2 , S).
Note that the last of these semisimple sequences is strictly larger than the first, but the
generic socle layerings of the corresponding varieties Rep S are in reverse relation. (In this
local example, the generic socle layerings of any Repd (Λ) are described in [HZ14, Theorem
A].)
Example 2.12. Let Λ = KQ/I, where Q is the quiver
α1
1
*4 2
α2
β1
*4 3
β2
and I is generated by the two paths βj αi for i 6= j. Moreover, take d = (1, 1, 1). Then
Repd (Λ) has two irreducible components with generic modules
1
1
α1
α2
2
2
and
β1
β2
3
3
11
respectively. The map Θ detects, but does not separate them. On the other
hand, the pair
of path nullities, Repd (Λ) → (N0 )2 , x 7→ nullityβ1 α1 Mx , nullityβ2 α2 Mx , obviously again
upper semicontinuous, does detect and separate the components.
Example 2.13. There are truncated path algebras Λ and dimension vectors d for which Θ
does not detect all irreducible components of Repd (Λ); see [HZ14, Example 4.5].
3
Specializing to truncated path algebras
From now on, Λ stands for a truncated path algebra: Λ = KQ/I, where I is the ideal
generated by the paths of length L + 1 for some L ≥ 1.
In Section 3.A, where the first radical layer of projective modules, resp., the first socle layer of
injectives, will play essential roles, we will rely on the following
abbreviations: If ei is a vertex
of Q, we denote by in(ei ) the dimension vector inj (ei ) 1≤j≤n , where inj (ei ) is the number
of arrows in Q1 that start in ej and terminate in ei . Similarly, out(ei ) := outj (ei ) 1≤j≤n ,
where outj (ei ) is the number of arrows that start in ei and end in ej . In other words, in(ei )
is the dimension vector of the socle layer S∗1 E(Si ) of an injective envelope E(Si ) of Si ,
and out(ei ) is the dimension vector of the radical layer S1 (Λei ) of a projective cover of Si .
3.A
Generic radical layerings determine the irreducible components of Repd (Λ)
Theorem 3.1. (see [BHZT09, Theorem 5.3]) Each of the subvarieties Rep S of Repd (Λ) is
irreducible, unirational and smooth, and all irreducible components of Repd (Λ) are among
the closures Rep S, where S traces the semisimple sequences with dimension vector d.
Naturally, in the process of sifting out the semisimple sequences S that give rise to the
irreducible components of Repd (Λ), only the realizable ones are in the running. These
are singled out by the following criterion which may be checked by mere inspection of the
quiver. To conveniently formulate it, we set P1 (T ) = JP (T )/J 2 P (T ) for any semisimple
T ∈ Λ-mod, where P (T ) is a projective cover of T . In other words, P1 (T ) is the first
radical
layer of a projective cover of T and, as such, is determined by its dimension vector
P
1≤i≤n ti · out(ei ), where (t1 , . . . , tn ) is the dimension vector of T .
Theorem 3.2 (Realizability Criterion). For a semisimple sequence S = (S0 , S1 , . . . , SL )
the following conditions are equivalent:
• S is realizable.
• For each l ∈ {0, . . . , L − 1}, the two-term sequence (Sl , Sl+1 ) is realizable.
P
• dim Sl+1 ≤ dim P1 (Sl ) for l ∈ {1, . . . , L}, i.e., dim Sl+1 ≤ 1≤i≤n tli · out(ei ), where
dim Sl = (tl1 , . . . , tln ).
We leave the easy argument to the reader.
12
3.B
Generic minimal projective presentations over truncated path
algebras
For truncated path algebras, Theorem 2.6 has a useful supplement: It permits to pin down
generic minimal projective presentations of the modules in Rep S.
First, we observe that the smallest subfield K0 of K over which Λ is defined is the prime
field of K. Moreover, in the truncated setting, involvement of the projective KQ-module
Pb in the definition of a skeleton becomes superfluous; in fact, Pb may be replaced by P .
The bonus responsible for this simplification is the path length grading of Λ, which
L leads to
an unambiguous notion of length of nonzero elements of the form pzr ∈ P = 1≤r≤t Λzr ,
where p is a path in Q and P is projective with sequence z1 , . . . , zt of top elements.
Our upgraded version of Theorem 2.6 will be used in the proof of the main result. For
convenience, we will typically assume that the field K has infinite transcendence degree over
K0 ; this guarantees that we can locate generic modules for the varieties Rep S within the
category Λ-mod. In light of Comment 2.7(2), this hypothesis will not limit applicability of
the conclusion towards identifying the irreducible components of the parametrizing varieties
Repd (Λ); nor will any of the considered generic module properties of the components be
affected by this assumption.
As in Subsection
L 2.C, we fix a realizable semisimple sequence S and a distinguished projective
cover P = 1≤r≤t Λzr of S0 . As explained above, we may assume skeleta with layering S
to live in P . Given a skeleton σ with layering S, a path qzs ∈ P is called σ-critical if it fails
to belong to σ, while every proper initial subpath q 0 zs of qzs belongs to σ. Moreover, for
any σ-critical path qzs , let σ(qzs ) be the collection of all those paths in σ which are at least
as long as qzs and terminate in the same vertex as qzs .
Theorem 3.3 (Generic modules for Rep S [BHZT09]). Let Λ be a truncated path algebra,
and suppose that the base field K has infinite transcendence degree over its prime field K0 .
Moreover, let S be a realizable semisimple sequence.
If σ is any skeleton with layering S, then the modules in Rep S generically have skeleton
σ, and the generic modules for Rep S are (up to Gal(K:K0 )-induced self-equivalences of
Λ-mod) determined
by minimal projective presentations of the following format: P/R(σ),
L
where P = 1≤r≤t Λzr is the distinguished projective cover of S0 , and
X
X
R(σ) =
Λ qzs −
xqzs , pzr pzr
pzr ∈σ(qzs )
qzs σ-critical
for some family xqzs , pzr of scalars algebraically independent over K0 .
Replacing xv, u in this presentation by an arbitrary family of scalars in K results in a
module in Rep S with skeleton σ, and conversely, every module with skeleton σ is obtained
in this way.
We record an offshoot of Theorem 3.3, which does not require any hypothesis on trdeg(K:K0 ).
Given a skeleton σ with layering S, consider the affine space Mat(σ) consisting of those matrices
C = c v, u v σ-critical
u∈σ
over K with index set {σ-critical paths} × σ for which c v, u = 0 whenever u ∈
/ σ(v).
13
Corollary 3.4. Let Λ be a truncated path algebra, S a realizable semisimple sequence, and σ
any skeleton with layering S. Then, generically, the modules in Rep S have skeleton σ, and
the modules with this skeleton are precisely those having a minimal projective presentation
of the following form:
X
X
P/R(C), with R(C) =
Λ v−
c v, u u
v σ-critical
u∈σ
for some matrix C = c v, u ∈ Mat(σ).
Moreover, generically, the socle layers of the modules in Rep S have the same dimension
b∈Λ
b -mod (see Comment 2.7 (3)).
vectors as those of any generic module G
The next corollary to Theorem 3.3 sets the stage for inductive arguments. Given any module
N ∈ Λ-mod, we call a submodule M layer-stably embedded in N in case M ∩ J l N = J l M
for all l ≥ 0; the latter condition means that, canonically, Sl (M ) ⊆ Sl (N ).
Corollary 3.5 (Submodules and quotients of generic modules). Let Λ be a truncated
path algebra, S = (S0 , . . . , SL ) a realizable semisimple sequence, and G ∈ Λ-mod a generic
module for Rep S.
(a) If U is a submodule of G which is layer-stably embedded in some J m G, then U is generic
for Rep S(U ). In particular: J m G is generic for Rep (Sm , . . . , SL , 0, . . . , 0) whenever
0 ≤ m ≤ L.
(b) Whenever 0 < m ≤ L, the subfactor G/J m G of G is generic for
Rep S0 , . . . , Sm−1 , 0, . . . , 0 .
Proof. (a) In a preliminary step, we verify the special case where U = J m G. In this case,
S(U ) = Sm , . . . , SL , 0, . . . , 0 . We set S0 = S(U ).
L
L
Fix a projective cover PF= 1≤r≤t Λzr of S0 and a projective cover P 0 = 1≤r≤u Λzr0 of
Sm . Moreover, let σ = 0≤l≤L σl ⊆ P be a skeleton of G such that G has a presentation
as specified in
F Theorem 3.3 relative to σ; here σl denotes the set of all paths of length l in
σ. Set σ 0 = m≤l≤L σl , and identify the paths in σm with the distinguished top elements
z10 , . . . , zu0 of P 0 . Under this identification, we find σ 0 to be a skeleton of J m G. Since Λ
is a truncated path algebra, the σ 0 -critical paths in P 0 are then in an obvious one-to-one
correspondence with those σ-critical paths that have length ≥ m + 1: Indeed, given any
σ-critical path qzs of length > m, replace its initial subpath of length m by the appropriate
zj0 (any such initial subpath belongs to σm by the definition of criticality) to arrive at a
σ 0 -critical path q 0 zj0 ; it is routine to check that this yields a bijection as stated. Hence the
description of G in Theorem 3.3 shows J m G to be generic for S0 ; the role played by σ in the
considered presentation of G is taken over by the skeleton σ 0 with layering S0 . This proves
our claim in case U = J m G.
To complete the proof of (a), it thus suffices to show the following: Any layer-stably embedded submodule U of G is generic for S(U ). Again, let z1 , . . . , zt be the fixed full sequence
of top elements of the projective cover P of S0 which provides the coordinate system for
skeleta with layering S. We may assume that dim U/JU = u ≥ 1, and that the distinguished projective cover Q of S0 (U ), on which we base the skeleta with layering S(U ), is of
14
L
the form Q = 1≤r≤u Λzr ; this assumption
F is justified by the inclusion S0 (U ) ⊆ S0 . Pick
any skeleton σ(U ) ⊆ Q of U ; say σ(U ) = 0≤l≤L σl (U ), where σl (U ) is the set of paths of
length l in σ(U ). We embed σ(U ) into a skeleton σ of G as follows: In light of S1 (U ) ⊆ S1 ,
we may supplement σ1 (U ) to a K-basis consisting of paths of length 1 in P . Moreover,
since U is a submodule of G such that JU/J 2 U canonically embeds into JG/J 2 G = S1 , we
may arrange for the paths σ1 \ σ1 (U ) to all start in one of the top elements zu+1 , . . . , zt of
P . Invoking the facts that S2 (U ) ⊆ S2 and U is closed under multiplication by paths, we
may supplement σ2 (U ) to a basis σ2 for S2 such that each path in σ2 \ σ2 (U ) extends one
of the paths in σ1 \ σ1 (U ); in particular each path in σ2 \ σ2 (U ) starts in one of the vertices
zu+1 , . . . , zt . Proceeding
L recursively, we thus arrive at a skeleton σ of G such that σ \ σ(U )
consists of paths in u+1≤r≤t Λzr . In this situation, the σ(U )-critical paths are precisely
those σ-critical paths in P which start in one of the top elements z1 , . . . , zu .
By the uniqueness statement of Theorem 2.6, G has a projective presentation P/R(σ), as
described in Theorem 3.3, based on the skeleton σ we just constructed. Since the residue
classes pzr of the pzr ∈ σ form a basis for G, and the classes represented by the pzr in σ(U )
generate U , we conclude that, for any σ(U )-critical path qzs , the set σ(qzs ) is contained in
σ(U ). Thus Theorem 3.3 exhibits U as generic for Rep S(U ) in the present situation.
Part (b) is proved analogously.
Part (b) of Corollary 3.5 cannot be upgraded to a level matching part (a): If G is as in the
corollary and U ⊆ J m G is layer-stably embedded in J m G, then G/U need not be generic
for the radical layering of G/U . For instance:
Example 3.6. Let Λ = KQ be the Kronecker algebra, i.e., Q is the quiver with 2 vertices,
e1 and e2 say, and two arrows α1 , α2 from e1 to e2 . Then G = Λe1 is generic for S =
(S1 , S22 ), and U = Λα2 is layer-stably embedded in JG. However, G/U fails to be generic
for S0 = S(G/U ) = (S1 , S2 ); indeed, whenever G0 is generic for S0 , we have α2 G0 6= 0.
Using the generic projective resolutions for the modules in Rep S exhibited in Theorem 3.3
and Corollary 3.4, one may compute the generic format of endomorphisms of the modules
in Rep S. In particular, this yields the generic dimension of endomorphism rings.
Corollary 3.7. Let Λ = KQ/I be a truncated path algebra and C = Rep S ⊆ Repd (Λ),
where S is a realizable semisimple sequence with dimension vector d. The generic dimension
of EndΛ (M ) for M in C may be determined from S, Q, and the Loewy length of Λ through
a system of homogeneous linear equations.
Proof. We refer to the notation of Theorem 3.3. In particular, we let σ be a skeleton with layering S, and use the generic form of the minimal projective presentations provided by Corollary 3.4. Clearly any endomorphism φ of P
P/U (σ) is completely determined by the (unique)
scalars arising in the equations φ(zj ) = u∈σ k j, u u (†). Suppose moreover
that, for any
P
path pe ∈ KQ \ I and u ∈ σ, the product pe u expands in the form pe u = v∈σ c(e
p, u, v) v (‡)
with c(e
p, u, v) ∈ K. That φ be an endomorphism of P/U (σ) is equivalent to the following
equalities:
X
q φ(zs ) =
xqzs , pzr p φ(zr )
for all σ-critical paths qzs .
pzr ∈σ(qzs )
15
Expanding both sides of these equalities by first inserting (†), then following with (‡), one
obtains K-linear combinations of the paths in σ on either side. Comparing coefficients of
these basis expansions results in a system of linear equations for the decisive scalars k j, u .
Remark 3.8. Clearly, the minimal elements of rad-soc(d) occur among the pairs (S, S∗ ),
where S traces the realizable semisimple sequences with dimension vector d and S∗ is the
generic socle layering of Rep S. Hence, it is desirable to determine S∗ from S. Theorem
3.3 provides the means to do so. Indeed, let G ∈ Λ-mod be generic for Rep S. Then the
annihilators annG J l are available, these being the terms of the ascending socle series of
G. However, in general it is computationally cumbersome to access S∗ (G) from a minimal
projective presentation of G. In a subsequent article ([HZS]), the authors explore generic
socle series over truncated path algebras (based on arbitrary quivers) from a more conceptual
viewpoint, opening up a low-effort road from S to S∗ .
4
Application to the hereditary case
In this section, we indicate how the information for truncated path algebras which we have
assembled plays out in the special case where Λ is hereditary. Thus we assume that Λ = KQ,
where Q is a quiver without oriented cycles. Moreover, we let L be the maximum of the
path lengths; in particular, this entails J L+1 = 0. Recall that the varieties Repd (Λ) are
full affine spaces in the present situation.
First we show how to find the generic radical layering S of the modules in Repd (Λ), without
resorting to a comparison of pairs in rad-soc(d). Once S is available, the results of Section
3 provide us with the set of generic skeleta for Repd (Λ), as well as with a generic minimal
projective resolution of the modules in Repd (Λ). This information, in turn, serves as a
vehicle for accessing further generic data on the d-dimensional modules; some of them may
alternatively be obtained by the methods of Kac and Schofield ([Kac82] and [Sch92]).
4.A
The generic radical layering S of the modules in Repd (Λ)
It is possible to compute the generic radical layering directly from the dimension vector d.
By Corollary 3.5, if G is a generic module for Repd (Λ) then the radical JG is generic for its
dimension vector. Hence the problem is reduced to computing the generic tops of modules.
Recall that inj (ei ) in the statement of Theorem 4.1 below denotes the number of arrows in
Q which start in vertex j and terminate in vertex i.
Theorem 4.1. The generic top of a Λ-module with dimension vector d has dimension
vector t = (t1 , . . . , tn ) where
X
ti = max{0, di −
dj · inj (ei )}.
j∈Q0
Proof. Let L be the function field of Repd (Λ). Then there is a tautological LQ-module GL
which turns out to have the generic radical and socle layering KQ-modules of dimension
vector d. We will only need to know that the dimension vector of the top of this tautological
module is the generic dimension vector of the tops of KQ-modules in Repd (Λ).
16
Recall that
Repd (Λ) =
Y
HomK (K dstart(α) , K dend(α) ),
α∈Q1
and let (wα,i,j : α ∈ Q1 , 1 ≤ i ≤ dend(α) , 1 ≤ j ≤ dstart(α) ) be the natural coordinates
on Repd (Λ).
LWrite O for the structure sheaf on Repd (Λ) and consider the trivial vector
bundle V = i∈Q0 Odi . Then there is a tautological K-algebra homomorphism Λ → End(V)
which sends ei to the projector onto the summand Odi of V. It maps α ∈ Q1 to the
map Odstart(α) → Odend(α) defined by the matrix (wα,i,j )i,j . This equips V with an action
µ : Λ⊗K V → V of Λ. By construction, µ has the following property: if x ∈ Repd (Λ), then
the action µ|x : Λ⊗K V|x → V|x defines the Λ-module parameterized by the point x.
We consider the map µJ : J⊗K V → V obtained by restricting µ to J⊗K V and let T denote
its cokernel. For each x ∈ U the sequence
J⊗K V|x → V|x → T |x → 0
is exact. However, the image of µJ |x is JV|x and hence T |x is the top of V|x . For all x
in an open, dense set, dimK (ei Tx ) = rank(ei T ). So the vector (rank(ei T )) is the generic
dimension vector of the top of a module in Repd (Λ).
Let η : Spec(L) → Repd (Λ) be the natural inclusion. We define GL = η ∗ V with the induced
Λ-structure. There is an induced exact sequence
J⊗K GL = JL ⊗L GL → GL → η ∗ T → 0
and thus η ∗ T is the top of GL . Now, the dimension vector of η ∗ T over L is simply
(rank(ei T )) and so the dimension vector of the top of GL is the same as the dimension
vector of the top of a general module in Repd (Λ).
Now, we will compute the top of GL . Indeed, the exact sequence
ei J⊗K GL → ei GL → ei TL → 0
implies that
ei T = coker(
M
Ldj → Ldi ).
α:j→i
This map is represented by the matrix Φi = (φα1 φα2 · · · φαr ) where α1 , . . . , αr is an enumeration of the set {α : end(α) = i}. Since the entries of the various φα• are algebraically
independent, all of the maximal minors of Φi are nonzero. Hence the Φi has full rank,
and the formula for the dimension vector of the top follows by examining the sizes of the
matrices Φi .
Remark 4.2. Theorem 4.1 implies the following dual statement: The generic socle of a
Λ-module with dimension vector d is the semisimple module with dimension vector c =
(c1 , . . . , cn ), where
X
ci = max{0, di −
dj outj (ei )}.
j∈Q0
17
4.B
An example illustrating the theory in the hereditary case
Let Λ = CQ, where Q is the quiver
β1
42
α2
β6
/4
α4
1
,3
α1
β5
*4 5
*4 8
46
α6
α5
α3
*
87
/9 9
α7
α8
γ5
β3
and let d = (0, 1, 1, 0, 3, 2, 3, 5, 10) ∈ (N0 )9 . Theorem 4.1 implies that S0 = S2 ⊕S3 ⊕S52 ⊕S8 .
Going over the same sequence of steps with d(1) = (0, 0, 0, 0, 1, 2, 3, 4, 10), we obtain S1 , and
so forth. The resulting generic sequence S may be read off any of the generic skeleta of the
modules in Repd (Λ). We present two of them for further discussion.
z1
z2
z3
z4
z5
2
•
3
5
5
8
•
σ:
7
5
9
6
7
8
8
9
9
9
9
6
9
7
8
8
9
9
9
9
and
σ
e:
z1
z2
ze3
ze4
z5
2
•
3
5
5
8
•
7
5
9
6
7
8
8
9
9
9
9
6
7
9
8
8
9
9
9
9
Consequences: (We note that the information under (c), (d) below, as well as parts of
(e) and (f ), can also be obtained via [Kac82] and [Sch92].)
L
(a) Let P = Pb =
1≤j≤5 Λzr be the distinguished projective cover of S0 . We apply
Theorem 3.3 to the skeleton σ to construct a generic minimal projective presentation of
the d-dimensional Λ-modules. The σ-critical paths in P are α2 z1 , α5 z3 , β5 z4 and α8 z5 .
Choosing elements x1 , . . . , x7 ∈ C which are algebraically independent over Q, we thus
18
obtain the following generic format of a minimal projective presentation: G = P/U (σ),
where U (σ) is generated by the following four elements in P :
α2 z1 , α5 z3 − x1 α5 α3 z2 + x2 β3 z2 + x3 α5 z4 ,
β5 z4 − x4 β5 z3 + x5 β5 α3 z2
and α8 z5 − x6 α7 β3 z2 + x7 γ5 z4 .
(b) From this generic presentation of the d-dimensional Λ-modules, we glean the generic
socle layering S∗ of Repd (Λ): Namely,
S∗ = (S2 ⊕ S910 , S73 ⊕ S85 , S5 ⊕ S62 , S52 , S3 ),
Moreover, we find S2 to be the only semisimple module occurring as a direct summand
of the modules in Rep S.
(c) Generically, the modules with dimension vector d decompose into two indecomposable summands, one isomorphic to S2 , the other with dimension vector d0 = d −
G0 is a generic module for Rep S0 , where
(0, 1, 0, . . . , 0). Indeed, G ∼
= S2 ⊕ G0 , whereL
0
2
0
S = (S3 ⊕S5 ⊕S8 , S1 , . . . , S4 ). Letting P = 2≤j≤5 Λzj be the distinguished projective
cover of S00 , we obtain a generic skeleton σ 0 = σ \ {z1 } ⊆ P 0 for the modules with dimension vector d0 . A generic minimal projective resolution based on σ 0 is G0 = P 0 /U (σ).
Using Corollary 3.7, we find that, generically, the modules with dimension vector d0 have
endomorphism rings isomorphic to C. Since this guarantees generic indecomposability
of the d0 -dimensional modules, our claim is justified.
(d) Generically, the modules with dimension vector d (resp., with dimension vector d0 )
contain a submodule isomorphic to Λe3 ⊕ Λe5 . Consult the generic skeleton σ
e to see
this.
(e) The submodule of G which is generated by (the U (σ)-residue classes of) z2 , z3 , z4 is
layer-stably embedded in G. By Corollary 3.5, the module Λz2 + Λz3 + Λz4 is therefore
generic for its dimension vector. Again applying Corollary 3.7, one obtains generic
indecomposability of the modules with this dimension vector.
(f) The modules with dimension vector (0, 0, 0, 0, 2, 1, 1, 2, 5) have generic skeleta as shown
under z3 , z4 of σ and under ze3 , ze4 in σ
e. Generically, they decompose into two local
modules which are unique up to isomorphism. The local summands are determined by
their graphs, which coincide with the trees displayed under ze3 and ze4 of the skeleton σ
e.
5
Nonhereditary examples
In the examples, Q will be an acyclic graph and d a dimension vector of Q. Our primary
purpose in Example 5.1 is to illustrate the information that results from exploring the generic
behavior of the d-dimensional representations of Q as we vary the allowable Loewy length
L + 1. In the extreme cases, where L is either the maximal path length, i.e. L = 6, on one
hand, or L = 1 on the other, generic direct sum decompositions are already well understood;
see [Kac82], [Sch92], and [BCHZ15]. As is to be expected, the picture is more complex in
the mid-range between these extremes.
19
Example 5.1. Let ΛL = CQ/hthe paths of length L + 1i, where Q is the quiver
1
/2
/& 4
/8 3
/& 6
/8 5
/8 7
and d = (1, 1, . . . , 1) ∈ N7 .
(a) Clearly, if L = 6, i.e., ΛL = KQ, the modules with dimension vector d are generically
uniserial with radical layering (S1 , . . . , S7 ).
(b) For L = 5, the variety Repd (ΛL ) has precisely 6 irreducible components, all of them
representing generically indecomposable modules. The are listed in terms of their generic
modules which, by Theorem 3.3, are available from the generic radical layerings. We
communicate these modules via their graphs, in Diagram 5.1.1; here the solid edge
paths starting at the top represent the chosen skeleton σ in each case, and the edge
paths starting at the top and terminating in a broken edge are the σ-critical paths, tied
in by the relations given in Theorem 3.3.
2
1
1
1
1
3
2
2
2
2
4
3
3
3
5
4
4
6
5
1
7
6
5
7
4
6
7
3
5
1
2
4
3
4
5
5
6
6
6
7
7
7
Diagram 5.1.1. Generic modules for Example 5.1(b), L = 5
For each of the semisimple sequences S which are generic for components of Repd (Λ),
the modules in Rep S have a fine moduli space; see, e.g., [HZ07, Theorem 4.4, Corollary
4.5]. All of these moduli spaces are 4-dimensional.
(c) The case L = 3 is more interesting. Using the Main Theorem, one finds that the
variety Repd (Λ) has precisely 28 irreducible components, 12 of which encode generically
indecomposable modules; the remaining 16 encode modules which generically split into
two indecomposable summands. The dimensions of the moduli spaces representing the
modules with generic radical layering (existent by [HZ07, Theorem 4.4]) vary among
1, 2, 3 for the different components. In particular, none of the components contains a
dense orbit. We list 9 of these components in Diagram 5.1.2, again in terms of graphs
20
of their generic modules.
1
(A)
(B)
2
3
4
5
6
7
1
2
3
4
7
3
1
2
5
2
4
6
6
7
7
2
4
3
5
7
5
3
L
4
5
2
4
L
3
5
6
7
5
2
7
1
2
6
3
4
6
L
6
1
1
3
3
4
1
1
2
5
6
7
•
1
(C)
4
6
5
7
Diagram 5.1.2. Some generic modules for Example 5.1(c), L = 3
Note that the generic radical layering of the component labeled (A) in the diagram is
strictly smaller than that of the component labeled (B), while the socle layerings are in
reverse relation. The generic socle layering of (A) is strictly smaller than that of (C),
but the generic radical layerings of (A) and (C) are not comparable.
If J 2 = 0, the subvarieties Rep S of Repd (Λ) constitute a stratification of Repd (Λ) in the
strict sense, in that all closures of strata are unions of strata. In fact, in Loewy length 2
the strata are organized by the equivalence “Rep S ⊆ Rep b
S ⇐⇒ (b
S, b
S∗ ) ≤ (S, S∗ )”; see
[BCHZ15, Theorem 3.6]. The nontrivial implication fails badly already for J 3 = 0, even
when the underlying quiver is acyclic, as the upcoming example demonstrates.
Example 5.2. Let Λ = KQ/hthe paths of length 3i, where Q is the quiver
1
α1
/2
α2
/3
α5
/6
β
4
α4
/5
Again take d = (1, . . . , 1). Then Repd (Λ) has precisely 3 irreducible components, one of
which is the closure of Rep b
S, where b
S = (S1 ⊕ S3 , S2 ⊕ S5 , S3 ⊕ S6 ). All modules in this
component are annihilated by β. If S = (S1 ⊕ S2 ⊕ S4 ⊕ S6 , S3 ⊕ S5 , 0), and if the the generic
socle layerings of the modules in Rep b
S and Rep S are denoted by b
S∗ and S∗ , respectively,
we find (b
S, b
S∗ ) < (S, S∗ ). On the other hand, Rep S 6⊆ Rep b
S, since, generically, the modules
21
in Rep S are not annihilated by β. However, observe
that Rep S ∩ Rep b
S 6= ∅; indeed, the
direct sum S1 ⊕ Λe2 /(Λα2 + Λβ) ⊕ Λe4 /Λα4 ⊕ S6 belongs to the intersection.
The component containing the irreducible variety Rep S is that with generic radical layering
e
S = (S1 ⊕ S2 ⊕ S4 , S3 ⊕ S5 , S6 ).
6
Proof of the Main Theorem
The equivalence of (1) and (1’) holds without any hypotheses on Λ (see Proposition 2.3);
the implication “(2) =⇒ (1)” only requires that Λ be a truncated path algebra (see [HZ14,
Theorem 3.1]).
Now suppose Λ = KQ/hthe paths of length L + 1i, where Q is an acyclic quiver.
“(1) =⇒ (2)”: Let S be a realizable semisimple sequence with dimension vector d, and let
S∗ be the generic socle layering of the modules in Rep S. Suppose that the pair (S, S∗ )
fails to be minimal in rad-soc(d). We will show that then the closure of Rep S is not
an irreducible component of Repd (Λ). By Comment 2.7(2), we do not lose generality in
passing to a suitable extension field of K which has infinite transcendence degree over its
prime field. Hence we may assume that for any realizable semisimple seqence S0 , there is
a generic Λ-module with radical layering S0 ; see Section 2.C, and 3.B for more detailed
information in the truncated case. Let G be generic for Rep S. In particular, this implies
that S∗ is the socle layering of G.
Our strategy is as follows: We will first pin down a suitable short exact sequence
0→A→G→B→0
representing a class η ∈ Ext1Λ (B, A) say. Then we will construct another class ξ ∈ Ext1Λ (B, A)
and consider the one-parameter family of extensions
0 → A → Gt → B → 0
∼ G, while
corresponding to η + tξ, t ∈ K. Our construction will be to the effect that G0 =
e
S(Gt ) < S(G) for general t. This will yield Rep S $ Rep(S) for the general radical layering
e
S of the family S(Gt ) , thus attesting to failure of maximality of Rep S as an irreducible
subvariety of Repd (Λ).
Our assumption on S provides us with a pair (b
S, b
S∗ ) ∈ rad-soc(d) which is strictly smaller
∗
b S∗ (G))
b for
than (S, S ) under the componentwise dominance order. Say (b
S, b
S∗ ) = (S(G),
b
b
some Λ-module G. It is harmless to assume that G is a generic module for the radical
b = min{S∗ (N ) | N in Rep(b
layering b
S. Clearly, b
S < S, since S∗ (G)
S)} is determined by b
S.
b
b
This means that ⊕l≤j Sl ⊂ ⊕l≤j Sl . In light of the equality ⊕0≤l≤L Sl = ⊕0≤l≤L Sl , this
implies that there exists an index v with the property that b
Sv 6⊂ Sv .
bτ 6⊆ Sτ . Then τ ≥ 1 because S
b0 ⊆ S0 in
We choose τ ∈ {0, . . . , L} minimal with respect to S
b
b
view of S < S. Pick k ∈ {1, . . . , n} such that dimK (ek Sτ ) < dimK (ek Sτ ).
Claim 1. There is an element a ∈ ek G such that a ∈
/ J τ G but J L−τ +1 a = 0.
22
Proof. Note that the annihilator of J L−τ +1 in G coincides with socL−τ (G). If the claim
were false, we would thus obtain ek socL−τ (G) ⊆ ek J τ G. Since always J τ G ⊆ socL−τ (G)
b this would amount to ek socL−τ (G) = ek J τ G. On the other hand, by
(analogously for G),
the construction of τ and k,
dimK (ek Sτ ) = dimK (ek J τ G) − dimK (ek J τ +1 G) <
b − dimK (ek J τ +1 G)
b = dimK (ek b
dimK (ek J τ G)
Sτ ), (1)
and, combining with our assumption, we derive
dimK (ek socL−τ G)) ≤
b − dimK (ek J τ +1 G)] <
dimK (ek J τ G) + [dimK (ek J τ +1 G)
b
dimK (ek socL−τ (G));
(2)
b ≤ S. However, S
b∗ (G)
b due to S
b ≤
keep in mind that dimK (ek J τ +1 G) ≤ dimK (ek J τ +1 G)
∗
b
S (G) implies
b ≤ dimK (ek socL−τ (G)),
dimK (ek socL−τ (G))
a contradiction.
Claim 2. Let A = Λa be the submodule ofG generated by a, and set B = G/A. Then the
semisimple sequence Sτ −1 (B), Sτ (B) ⊕ Sk is realizable.
Proof. We repeatedly use the realizability criterion 3.2. Clearly, it suffices to prove that,
for some submodule S0 of the semisimple module Sτ −1 (B), the sequence S0 , ek Sτ (B) ⊕ Sk
is realizable. Recall that we identify the vertices of Q with the corresponding primitive
idempotents of Λ. Hence it makes sense to consider the sum e of the starting vertices of
the paths of positive length ending in ek ; clearly, e is an idempotent
in Λ. We will show
realizability of the semisimple sequence e Sτ −1 (B), ek Sτ (B) ⊕ Sk , which will cover our
claim.
Acyclicity of Q yields eΛei = 0 for any vertex ei with ei Λek 6= 0. In light of a = ek a, we
deduce that eJ l B ∼
= eJ l G canonically for all l ≤ L, whence e Sl (B) = e Sl (G). Moreover,
we find ek Sτ (B) = ek Sτ (G) because a ∈
/ J τ G and ek JA = 0.
b ⊆ e Sτ −1 (G), while ek Sτ (G)
b % ek Sτ (G).
Due to our choice of τ and k, we have e Sτ −1 (G)
b
b
Therefore realizability of e Sτ −1 (G), ek Sτ (G) implies realizability of
e Sτ −1 (G), ek Sτ (G) ⊕ Sk = e Sτ −1 (B), ek Sτ (B) ⊕ Sk
as required.
Claim 3. Let χ ∈ Ext1Λ (B, A/JA). If the canonical image of χ in Ext1Λ (J τ B, A/JA) is
zero, then χ lifts to a class in Ext1Λ (B, A).
Proof. Let
χ: 0
/ A/JA
/X
23
h
/B
/0
be an extension which maps to zero in Ext1Λ (J τ B, A/JA). The Λ-structure of the simple
module A/JA ∼
= Sk coincides with its K {e1 ,...,en } -structure, and the latter evidently boils
down to the K-structure induced by the k-th factor of the direct product of copies of K; we
identify the Λ-submodule A/JA of X with Ka and choose a splitting X = Ka ⊕ B 0 of χ
over K {e1 ,...,en } . As a K {e1 ,...,en } -module, B is then isomorphic to B 0 , and our hypothesis
provides us with a Λ-submodule Bτ of J τ X such that Ka + J τ X = Ka ⊕ Bτ . Clearly, the
map Bτ → J τ B induced by h is a Λ-isomorphism, and we may choose B 0 to be contained
in Bτ .
Next we define a Λ-module structure on the K {e1 ,...,en } -module M = A ⊕ B 0 , now viewing
A/JA = Ka as a K {e1 ,...,en } -submodule of A. If, for α ∈ Q1 , we denote by fα the action
of α on the submodule A of G and by gα the action on X, the following definition yields a
well-defined KQ-module structure on M :
α(a0 + b0 ) = fα (a0 ) + gα (b0 )
for a0 ∈ A, b0 ∈ B 0 .
We verify that this is in fact a Λ-module structure: Indeed, let p = p2 p1 be a path in Q,
where p1 is a path of length τ and p2 a path of length L+1−τ . We obtain p1 (a0 +b0 ) = a00 +bτ
for some a00 ∈ A and bτ ∈ Bτ . Then p2 a00 = 0 by the construction of A (see Claim 1), and
p2 bτ = 0, given that Bτ is a Λ-submodule of J τ X. Therefore p(a0 + b0 ) = 0 as required.
Our construction clearly entails that the extension
0→A→M →B→0
is a lift of χ.
The next step is based on Claim 2. It will provide us with a suitable non-split extension
χ ∈ Ext1Λ (B, A/JA) satisfying the hypothesis of Claim 3.
Claim 4. The canonical map E : Ext1Λ (B/J τ B, A/JA) −→ Ext1Λ (J τ −1 B, A/JA) is nonzero.
Proof. Let P → B/J τ +1 B be a projective cover with kernel C. By realizability of the pair
(Sτ −1 (B), Sτ (B) ⊕ Sk ), the semisimple sequence
S0 (B), . . . , Sτ −1 (B), Sτ (B) ⊕ (A/JA)
is in turn realizable, which provides us with a submodule C 0 ⊂ C such that C/C 0 = Sk ,
Sj (P/C 0 ) = Sj (B) for j < τ , and Sτ (P/C 0 ) = Sτ (B) ⊕ (A/JA). Starting with the exact
sequence
0 → Sτ (B) ⊕ (A/JA) → P/C 0 → B/J τ B → 0
(3)
we pull back along the inclusion ι : J τ −1 B/J τ B → B/J τ B , to obtain the induced extension
0 → Sτ (B) ⊕ (A/JA) → J τ −1 (P/C 0 ) → J τ −1 B/J τ B → 0.
(4)
Observe that (4) is non-split, since by construction Sτ (B) ⊕ (A/JA) = J τ (P/C 0 ) is the
radical of J τ −1 (P/C 0 ). In fact, we even obtain that the exact sequence
0 → A/JA → M 0 → J τ −1 B/J τ B → 0
(5)
which results from (4) by pushing out along the projection Sτ (B) ⊕ (A/JA) → A/JA , does
not split either. Performing the pull-back and push-out operations that led from (3) to (5)
24
in reverse order results in an extension equivalent to (5). Hence the class of (5) is the image
under Ext1Λ (ι, A/JA) of the class of some extension
0 → A/JA → M → B/J τ B → 0.
That the latter does not belong to the kernel of E, follows from non-splitness of (5). Indeed,
the canonical map Ext1Λ (B/J τ B, A/JA) −→ Ext1Λ (J τ −1 B/J τ B, A/JA) factors through
E.
By Claim 4, we may choose a class χ0 : Ext1Λ (B/J τ B, A/JA) with nonzero image in
Ext1Λ (J τ −1 B, A/JA). We set χ = Ext1Λ (π, A/JA)(χ0 ), where π : B → B/J τ B is canonical.
Since the image of Ext1Λ (π, A/JA) in Ext1Λ (B, A/JA) is the kernel of the natural map
Ext1Λ (B, A/JA) → Ext1Λ (J τ B, A/JA),
Claim 3 guarantees that χ lifts to a class ξ in Ext1Λ (B, A). By construction, the image of
ξ in Ext1Λ (J τ −1 B, A/JA) is nonzero. Hence, for general t, the image of the class η + tξ is
nonzero in Ext1Λ (J τ −1 B, A/JA). Consider the one-parameter family of extensions
gt
0 −→ A −→ Gt −→ B −→ 0
corresponding to η+tξ, respectively, and note that G0 = G. It is straightforward to translate
the family (Gt )t∈K into a curve A1 → Repd (Λ). As a consequence, we find that G belongs
to Rep S(Gt ) for general t, which implies S(Gt ) ≤ S(G). In a final step, we will check that
S(Gt ) < S for general t.
Claim 5. S(Gt ) < S(G) for general t. More specifically, dimK (J τ G) < dimK (J τ Gt ).
Proof. For any t ∈ K, the inverse image of J τ B under gt is A+J τ Gt . Hence dimK (A+J τ Gt )
is constant in t. Let t be such that η + tξ has nonzero image ηt in Ext1Λ (J τ −1 B, A/JA), and
consider the extension
0 → A/JA → (A + J τ −1 Gt )/JA → J τ −1 B → 0
representing ηt . Non-splitness forces the simple module A/JA into the radical of the middle
term, that is, A ⊆ JA+J τ Gt . This means that J τ Gt contains an element in A\JA. Since JA
is the unique maximal submodule of A, we conclude that A ⊆ J τ Gt , i.e., J τ Gt = A + J τ Gt .
On the other hand, J τ G $ A + J τ G by Claim 1. Thus dimK (J τ G) < dimK (J τ Gt ) as
claimed.
Now let e
S be the generic radical layering of the family (Gt ). Then e
S < S = S(G) by Claim
5, and our curve places G into the closure of Rep e
S. Since G is generic for Rep S, this
shows Rep S to be contained in the irreducible subvariety Rep e
S of Repd (Λ). In light of
S(X) ≥ S for all X ∈ Rep S, this containment is proper. Thus Rep S indeed fails to be an
irreducible component of Repd (Λ), which completes the argument.
Remark 6.1. It is only in the proof of Claim 2 that we make use of acyclicity of Q. A
slight refinement of our argument shows that, in fact, it suffices to assume Q to be acyclic
after removal of all loops. Here is some detail: In case there is a loop at ek , the role played
by the idempotent e is taken over by the sum ee of all vertices different from ek that arise
25
as starting points of nontrivial paths ending in ek . We observe that the relaxed
hypothesis
yields realizability of the semisimple sequence ee Sτ −1 (G), ek Sτ (G) ⊕ Sk as before, the
same being true for the equality ee Sτ −1 (G) = ee Sτ −1 (B). Hence, to verify realizability of
ee Sτ −1 (B), ek Sτ (B)⊕Sk , it suffices to observe that ek Sτ (B) ⊆ ek Sτ (G). This last inclusion
is always true, because J τ B/J τ +1 B ∼
= J τ G/ (J τ +1 G + A) ∩ J τ G is clearly an epimorphic
τ
τ +1
image of J G/J
G.
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