C O L L O Q U I U M M A T H E M A T I C U M VOL. 82 1999 NO. 2

VOL. 82 1999 NO. 2

IMMERSIONS OF MODULE VARIETIES

BY

GRZEGORZ Z W A R A (TORU ´N)

Abstract. We show that a homomorphism of algebras is a categorical epimorphism if and only if all induced morphisms of the associated module varieties are immersions.

This enables us to classify all minimal singularities in the subvarieties of modules from homogeneous standard tubes.

1. Introduction and main results. Throughout the paper k will be a fixed algebraically closed field and all the algebras considered are associative k-algebras with identities. For any algebra A and d ≥ 1 we denote by M

(A) the algebra of square matrices of degree d with coefficients in A.

Let A be a finitely generated algebra, that is, A is isomorphic to the quo- tient of a finitely generated free (non-commutative) algebra khX

, . . . , X

i by a two-sided ideal I. For any natural number d we define an affine variety mod

(d) of (left) A-module structures on k

as follows:

mod

(d) = {m = (m

, . . . , m

) ∈ (M

(k))

: ̺(m

, . . . , m

) = 0, ̺ ∈ I}.

The general linear group Gl

(k) acts on mod

(d) by conjugation and the orbits of this action correspond to the isomorphism classes of d-dimensional left A-modules. The variety mod

(d) depends on the choice of the represen- tation khX

, . . . , X

i/I of A only up to a Gl

(k)-equivariant isomorphism.

Let ϕ : A → B be a homomorphism of finitely generated algebras. Then there are induced regular Gl

(k)-equivariant morphisms of affine varieties ϕ

: mod

(d) → mod

(d) for all d ≥ 1. An interesting problem is to find homomorphisms of algebras inducing regular morphisms of affine varieties with nice properties.

Examples of nice regular morphisms are immersions. Recall some defini- tions from [9]. By a variety we mean a quasi-affine variety, that is, a locally closed subset of an affine variety. We say that a regular morphism µ : X → Y of varieties is an immersion (respectively, closed immersion) if µ gives an isomorphism of X with a locally closed subset of Y (respectively, a closed subset of Y ).

1991 Mathematics Subject Classification: 14B05, 14E25, 14L30, 16B50, 16G10, 16G70.

[287]

On the other hand, well-known examples of homomorphisms of algebras are categorical epimorphisms. A homomorphism ϕ : A → B of rings with identity is called an epimorphism in the category of rings, or briefly a cat- egorical epimorphism, if for given homomorphisms of rings β

, β

: B → C such that β

ϕ = β

ϕ, we have β

= β

(see [14]). Our first main result explains connections between categorical epimorphisms of algebras and im- mersions of module varieties.

Theorem 1. Let ϕ : A → B be a homomorphism of algebras. As- sume that the algebra A is finitely generated and the algebra B is finite- dimensional. Then the following conditions are equivalent :

(i) ϕ is a categorical epimorphism.

(ii) ϕ

: mod

(d) → mod

(d) is an injective morphism for all d ≥ 1.

(iii) ϕ

is an immersion for all d ≥ 1.

This theorem is useful for problems concerning the geometry of module varieties like the types of singularities in closures of orbits. Following Hes- selink (see (1.7) in [10], (8.1) in [1] and (2.1) in [12]) we call two pointed varieties (X , x

) and (Y, y

) smoothly equivalent if there are smooth mor- phisms f : Z → X , g : Z → Y and a point z

∈ Z with f (z

) = x

, g(z

) = y

. This is an equivalence relation and equivalence classes will be denoted by Sing(X , x

). If Sing(X , x

) = Sing(Y, y

) then the variety X is regular or normal at x

if and only if the same is true for the variety Y at y

(see Section 17 in [8] for more information about smooth morphisms).

In the case of module varieties we wish to investigate the types of singularities Sing(Gl

(k)m, n) of closures of orbits, where n ∈ Gl

(k)m.

Especially, it is of interest to classify the types of minimal singularities, that is, Sing(Gl

(k)m, n), where n ∈ Gl

(k)m \ Gl

(k)m and there is no point p ∈ mod

(d) such that Gl

(k)n Gl

(k)p Gl

(k)m. In case A = k[X]/(X

) all types of minimal singularities in mod

(d) are known and we list them in Section 4.3. For some results concerning minimal singu- larities we refer to [15], [1], [12] and [5].

For any finite-dimensional algebra A one defines the Auslander–Reiten quiver Γ

of A, that is, a translation quiver whose vertices are the isomor- phism classes of indecomposable finite-dimensional left A-modules and the arrows correspond to irreducible maps (see [3] for more details). It will cause no confusion if we identify the isomorphism classes of modules with their representatives. A connected component Γ of the Auslander–Reiten quiver Γ

of A is said to be a homogeneous tube if it is isomorphic to the translation quiver (V

, V

, τ ) with the set of vertices V

= {v

: i ≥ 1}, the set of arrows V

= {α

: v

→ v

, β

: v

→ v

: i ≥ 1} and the translation τ : V

→ V

given by τ (v

) = v

for all i ≥ 1 (cf. Section 3.1 in [13]).

An A-module in Γ corresponding to v

is said to be quasi-simple.

v

1



1

,,

v

2 

1

H K N

ll

S V X

2 f

h

k

{{

p

s

v

v

3



2

;;

3

,,

v

4 3

H K N

ll

S V X

.

.

.

Homogeneous tubes are important components of the Auslander–Reiten quivers of tame algebras. A finite-dimensional algebra A is called tame if for any d ≥ 1 there is a finite number of A-k[X]-bimodules M

, . . . , M

which are free of rank d as right k[X]-modules and such that every indecomposable left A-module of dimension d is isomorphic to M

⊗

k[X]/(X −λ) for some 1 ≤ i ≤ n(d) and λ ∈ k. Crawley-Boevey proved in [6] that if the algebra A is tame, then all but finitely many isomorphism classes of indecomposable left A-modules of any dimension d lie in homogeneous tubes of Γ

. Furthermore, if A is a tame hereditary algebra, then all connected components of Γ

are standard and all but finitely many of them are homogeneous tubes. A connected component Γ of the Auslander–Reiten quiver Γ

of A is called standard if the category of A-modules from Γ is equivalent to the mesh category of Γ (see [13] for details).

Let mod

(d) denote the sum of orbits in mod

(d) corresponding to the modules from add(Γ ), for any component Γ of Γ

. Our second main result enables us to classify all minimal singularities in mod

(d), where Γ is a standard homogeneous tube.

Theorem 2. Let Γ be a standard homogeneous tube in the Auslander–

Reiten quiver of a finite-dimensional algebra A. Let d = d

e, where d

≥ 1, e = dim

T

and T

is a quasi simple module in Γ . Then:

(i) mod

(d) is a locally closed subset of mod

(d).

(ii) The types of singularities (respectively, minimal singularities) in the varieties mod

(d) and mod

(d

) coincide provided r ≥ d

.

Observe that mod

(d) is an empty set if d is not divisible by e = dim

T

. The rest of the paper is organized as follows. In Section 2 we consider al- gebraic conditions for ring homomorphisms to be categorical epimorphisms.

Section 3 is devoted to the proof of Theorem 1. In Section 4 we show some

applications of Theorem 1, in particular, we prove Theorem 2.

For background on the representation theory of algebras we refer to [3], [13] and for algebraic geometry to [9].

The author wishes to express his thanks to D. Simson for helpful com- ments concerning categorical epimorphisms.

2. Categorical epimorphisms. For any ring R (with an identity) we denote by Mod R the category of left R-modules and by Hom

(−, −) the functor of R-homomorphisms of left R-modules. Furthermore, any ring homomorphism ϕ : A → B induces a faithful functor ϕ

: Mod B → Mod A and we may consider any left B-module as a left A-module. In that way, Hom

(X, Y ) becomes a subset of Hom

(X, Y ) for any X, Y ∈ Mod B.

Lemma 3 collects some equivalent conditions for epimorphisms in the cate- gory of rings (see Section 1 in [11]).

Lemma 3. Let ϕ : A → B be a homomorphism of unitary rings. Then the following conditions are equivalent:

(i) ϕ is a categorical epimorphism.

(ii) Hom

(X, Y ) = Hom

(X, Y ) for all X, Y ∈ Mod B, that is, the functor ϕ

: Mod B → Mod A induced by ϕ is full.

(iii) The canonical surjective homomorphism of B-bimodules B ⊗

B

→ B, b ⊗ b

7→ bb

, is an isomorphism.

We now consider the category of algebras (over k). For an algebra A we have the canonical functor D = Hom

(−, k) : Mod A → Mod A

, where A

denotes the opposite algebra of A, or equivalently, Mod A

denotes the cat- egory of right A-modules. Since any algebra B is a left and a right B-module, the k-space Hom

(B, DB) is defined. Furthermore, for any homomorphism ϕ : A → B we may treat Hom

(B, DB) as a k-space containing the subspace Hom

(B, DB). We shall need the following fact.

Lemma 4. Let ϕ : A → B be a homomorphism of algebras. Then ϕ is a categorical epimorphism if and only if Hom

(B, DB) = Hom

(B, DB).

P r o o f. The homomorphism η : B ⊗

B → B, b ⊗ b

7→ bb

, is bijective if and only if so is the homomorphism D(η) : DB → D(B ⊗

B). Since the functor B ⊗

(−) is left adjoint to the functor Hom

(B

, −), we have the following isomorphisms:

D(B⊗

B) = Hom

(B⊗

B, k) ≃ Hom

(B, Hom

(B, k)) = Hom

(B, DB), and similarly,

DB ≃ D(B ⊗

B) ≃ Hom

(B, DB).

Observe that, applying the above isomorphisms, D(η) corresponds to the

inclusion Hom

(B, DB) ⊆ Hom

(B, DB). Hence, η is an isomorphism of

B-bimodules if and only if Hom

(B, DB) = Hom

(B, DB). Lemma 3 now

completes the proof.

3. Proof of Theorem 1. Throughout this section we fix a homomor- phism ϕ : A → B, where A is a finitely generated algebra and B is a finite-dimensional algebra. The implication (iii)⇒(ii) is obvious.

(ii)⇒(i). Let d = dim

(B ⊕ DB) < ∞. Take a point x ∈ mod

(d) such that its orbit Gl

(k)x corresponds to the isomorphism class of the left B-module B ⊕ DB. We set y = ϕ

(x), where ϕ

: mod

(d) → mod

(d) is the induced regular morphism. Then the orbit Gl

(k)y corresponds to the isomorphism class of the left A-module B ⊕ DB. Since the morphism ϕ

is injective we get a bijection Gl

(k)x → Gl

(k)y, and consequently, equality of the isotropy groups: Gl

(k)

= Gl

(k)

. Observe that the affine algebraic group Gl

(k)

is isomorphic to the group Aut

(B ⊕ DB) of B- module automorphisms of B ⊕ DB. Furthermore, Aut

(B ⊕ DB) is a non-empty open subset of the space End

(B ⊕ DB) of B-endomorphisms of B ⊕ DB. This implies that dim Aut

(B ⊕ DB) = dim

End

(B ⊕ DB).

Hence, dim Gl

(k)

= dim

End

(B ⊕ DB), and similarly, dim Gl

(k)

= dim

End

(B ⊕ DB). Consequently, End

(B ⊕ DB) = End

(B ⊕ DB).

Since Hom

(X, Y ) ⊆ Hom

(X, Y ) for any B-modules X, Y and Hom

(B, B) ⊕ Hom

(B, DB) ⊕ Hom

(DB, B) ⊕ Hom

(DB, DB)

= End

(B ⊕ DB) = End

(B ⊕ DB)

= Hom

(B, B) ⊕ Hom

(B, DB) ⊕ Hom

(DB, B) ⊕ Hom

(DB, DB), we get Hom

(B, DB) = Hom

(B, DB). Hence ϕ is a categorical epimor- phism by Lemma 4.

(i)⇒(iii). Let d ≥ 1. We divide the proof into several steps.

Step (a). If the homomorphism ϕ : A → B is surjective, then the induced regular morphism ϕ

: mod

(d) → mod

(d) is a closed immersion.

P r o o f. The algebra A is finitely generated, hence A ≃ khX

, . . . , X

i/I.

Observe that B ≃ khX

, . . . , X

i/J for some two-sided ideal J containing I.

By definition, the affine varieties mod

(d) and mod

(d) are the sets of points in (M

(k))

satisfying polynomial equations induced by J and I, respectively. Since J ⊇ I, mod

(d) becomes a closed subset of mod

(d), and ϕ

is an inclusion.

Step (b). We may assume that A is a subalgebra of B and ϕ is an inclusion.

P r o o f. By our assumptions and Lemma 3, the canonical homomorphism B ⊗

B → B is injective. Let A = A/ ker ϕ. Then the homomorphism ϕ : A → B factorizes through A and we get the induced regular morphisms of affine varieties:

mod

(d) → mod

(d) → mod

(d).

By (a), the latter is a closed immersion. Since the B-bimodules B ⊗

B and B ⊗

B are isomorphic to each other, we may assume that A = A, A ⊆ B and ϕ : A → B is an inclusion.

We choose a linear basis 1 = b

, b

, . . . , b

of B, where s = dim

B − 1, such that 1 = b

, b

, . . . , b

form a basis of A and t = dim

A − 1. We also choose the following representations of the algebras A and B:

A ≃ khX

, . . . , X

i/I and B ≃ khX

, . . . , X

i/J

for some two-sided ideals I and J, where the generator X

corresponds to b

for all 1 ≤ i ≤ s. Then ϕ

(x

, . . . , x

) = (x

, . . . , x

) for any point x = (x

, . . . , x

) ∈ mod

(d).

Let M

(C) denote the space of all matrices with α rows, β columns and coefficients in an algebra C.

Step (c). There are matrices Y

∈ M

(A), Y

∈ M

(A), Z ∈ M

(B), for some α ≥ 1, such that

Z · Y

= 1

, Y

· [b

, . . . , b

]

= Y

, (3.1)

Y

= [b

, . . . , b

|∗]

, Y

=

 1

0

∗ ∗



, Z =

 1

0 0 ∗

 (3.2)

where M

means the matrix transpose to M .

P r o o f. We quote Lemma 6.4 in [7] formulated for commutative rings.

This useful lemma remains true in the non-commutative case.

Lemma 5. Let M be a right A-module and N be a left A-module gen- erated by a family of elements {n

}

. Then an element P

l∈L

m

⊗ n

∈ M ⊗

N equals zero if and only if there exist elements {m

}

of M and elements {a

∈ R : j ∈ J, l ∈ L} such that

X

j∈J

m

a

= m

for all l ∈ L and X

l∈L

a

n

= 0 for all j ∈ J.

Since the canonical homomorphism B ⊗

B → B is injective, it follows that b

⊗ 1 = 1 ⊗ b

for all 1 ≤ i ≤ s. Applying Lemma 5 to the right A-module B, the left A-module B generated by elements {1, b

, . . . , b

}, and to the elements (−b

) ⊗ 1 + 1 ⊗ b

= 0 ∈ B ⊗

B, 1 ≤ i ≤ s, we get, for all 1 ≤ i ≤ s, a matrix Y

with s + 1 columns and coefficients in A and a matrix Z

with one row and coefficients in B such that

Z

· Y

= [−b

, 0, . . . , 0, 1, 0, . . . , 0] and Y

· [1, b

, . . . , b

]

= 0.

We may assume that Y

= [−b

, 0, . . . , 0, 1, 0, . . . , 0] and Z

= [1] for all

1 ≤ i ≤ t. We collect together the matrices Y

and Z

as follows:

Y =



  Y

Y

. . . Y



  and Z =



 

Z

0 . . . 0 0 Z

. . . 0 . . . . 0 0 . . . Z



  .

Then Y ∈ M

(A) and Z ∈ M

(B) for some α ≥ 1. Moreover,

Z · Y =



 

−b

1 0 . . . 0

−b

0 1 . . . 0 . . . .

−b

0 0 . . . 1



  and Y ·



  1 b

. . . b



  = 0.

Let Y = [−Y

, Y

], where Y

∈ M

(A) and Y

∈ M

(A). Then we obtain the formulas (3.1). Since Y

= [−b

, 0, . . . , 0, 1, 0, . . . , 0] and Z

= [1]

for all 1 ≤ i ≤ t, the matrices Y

, Y

and Z have the form (3.2).

Any point m of mod

(d) determines a homomorphism of algebras m : A → M

(k), for simplicity of notation denoted by the same letter. Fur- thermore, for any u, v ≥ 1, we may extend m to a map b m = b m(u, v) : M

(A) → M

(k) such that

b m









a

. . . a

. . . . a

. . . a







 =





m(a

) . . . m(a

) . . . . m(a

) . . . m(a

)



 .

Step (d). The set

C = {m ∈ mod

(d) : rank( b m(Y

)) = rank[ b m(Y

), b m(Y

)] = sd}

is locally closed in mod

(d) and im ϕ

⊆ C.

P r o o f. Indeed, the condition rank( b m(Y

)) = rank[ b m(Y

), b m(Y

)] = sd is equivalent to the vanishing and non-vanishing of some minors of the matrix [ b m(Y

), b m(Y

)]. This defines a locally closed subset of mod

(d).

Let n ∈ mod

(d) and m = ϕ

(n) ∈ mod

(d). The equalities (3.1) imply that b n(Z) · b n(Y

) = b n(1

) = 1

∈ M

(k). Therefore rank(b n(Y

)) = sd. Since Y

is a matrix with coefficients in A ⊆ B, we have b m(Y

) = b

n(Y

), and hence rank( b m(Y

)) = sd. The equalities (3.1) also imply that b

n(Y

) · b n([b

, . . . , b

]

) = b n(Y

) ∈ M

(k).

It follows that rank(b n(Y

)) = rank[b n(Y

), b n(Y

)], by classical linear algebra.

Since Y

and Y

have coefficients in A ⊆ B, we see that b m(Y

) = b n(Y

), b

m(Y

) = b n(Y

), hence rank( b m(Y

)) = rank[ b m(Y

), b m(Y

)], and finally

m ∈ C.

Step (e). There is an injective regular morphism η : C → (M

(k))

such that η(m) = (x

, . . . , x

) for all m ∈ C, where

(3.3) m(Y b

) · [x

, . . . , x

]

= b m(Y

).

P r o o f. The existence of a regular map η : C → (M

(k))

is a con- sequence of elementary linear algebra and the fact that rank( b m(Y

)) = rank[ b m(Y

), b m(Y

)] equals the number of columns of b m(Y

) ∈ M

(k).

The set C is a finite union of open subsets defined by the condition that fixed sd rows in b m(Y

) are linearly independent. From Cramer’s rule, η is a regular morphism on each of these open subsets. Therefore, η is a regular morphism.

Let m = (m

, . . . , m

) ∈ C and η(m) = (x

, . . . , x

). The combination of the formulas (3.2) and (3.3) yields

b m

 1

0

∗ ∗



· [x

, . . . , x

]

= b m([b

, . . . , b

|∗]

)

and hence b m([1

, 0]) · [x

, . . . , x

]

= b m([b

, . . . , b

]

). This implies that [1

, 0] · [x

, . . . , x

]

= [m(b

), . . . , m(b

)]

, hence that [x

, . . . , x

]

= [m

, . . . , m

]

. Thus the map η is injective.

Step (f). For any n ∈ mod

(d) we have η(ϕ

(n)) = n.

P r o o f. Let n = (n

, . . . , n

) ∈ mod

(d), m = ϕ

(n) = (n

, . . . , n

) ∈ mod

(d) and η(m) = (x

, . . . , x

) ∈ mod

(d). Since

b

m(Y

) · [n

, . . . , n

]

= b m(Y

) = b m(Y

) · [x

, . . . , x

]

and the rank of b m(Y

) equals the number of its columns, we find that n

= x

for all 1 ≤ i ≤ s. This implies that η(m) = n.

Step (g). The set D = η

(mod

(d)) is locally closed in mod

(d) and the morphisms ϕ

and η induce mutually inverse isomorphisms of mod

(d) and D.

P r o o f. The set D is closed in C. Hence D is a locally closed subset of mod

(d), by (d). Moreover, im ϕ

is contained in D, by (f). It remains to show that ϕ

(η(m)) = m for all m ∈ D. Indeed, this follows from the equality ηϕ

η(m) = η(m) and since the map η is injective, by (e) and (f).

The final step shows that ϕ

is an immersion, which completes the proof of Theorem 1.

4. Geometry of modules from standard homogeneous tubes

4.1. Finite tubes. Let A be an algebra and r ≥ 2. A finite tube over

A of height r is a sequence T = (T

, α

, β

: 1 ≤ i ≤ r, 1 ≤ j < r) such

that T

is a finite-dimensional A-module for any 1 ≤ i ≤ r, α

: T

→ T

,

β

: T

→ T

are A-homomorphisms for any 1 ≤ j < r, and the short sequences

0 → T

− −→ T

− − → T

→ 0, (4.1)

0 → T

(αi,β_i−1)^T

−−−−−−−→ T

⊕ T

(βi,−α_i−1)

−−−−−−−→ T

→ 0, 1 < i < r, (4.2)

are exact.

The Auslander–Reiten quiver of the algebra k[X]/(X

) is a finite tube of height r. The same conclusion holds for the algebras M

(k[X]/(X

)), e ≥ 1, which are Morita equivalent to k[X]/(X

).

Lemma 6. Let A = khX

, . . . , X

i/I be a finitely generated algebra. Let T = (T

, α

, β

) be a finite tube over A of height r and e = dim

T

. Then there are y

, . . . , y

∈ (M

(k))

such that the A-module T

corresponds to the point

t

=



 

 

y

y

y

. . . y

0 y

y

. . . y

0 0 y

. . . y

. . . . 0 0 0 . . . y



 

 

of mod

(ei) for all 1 ≤ i ≤ r.

P r o o f. By (4.1), α

is a monomorphism. If α

is a monomorphism then so is α

for all 1 < i < r, by (4.2). Therefore, α

is a monomorphism, and similarly, β

is an epimorphism for all 1 ≤ i < r. Moreover, the exactness of the sequences (4.1) and (4.2) leads to

(4.4) β

α

= 0, β

α

= α

β

, 1 < i < r.

Let γ = α

β

∈ End

(T

). Applying the formulas (4.4) several times we get

γ

= (α

β

)

= α

. . . α

β

. . . β

for all 1 ≤ i < r and γ

= 0. Observe that the short sequence 0 → T

α_r−1...αi

−−−−−−−→ T

β_r−i...β_r−1

−−−−−−−→ T

→ 0 is exact for all 1 ≤ i < r. Then T

is isomorphic to

ker γ

= ker(β

. . . β

) = im(α

. . . α

) = im γ

.

Let ε

, . . . , ε

be a basis of ker γ = im γ

. We inductively choose ε

satisfying γ(ε

) = ε

for all e < j ≤ er. Observe that the residue classes of the elements ε

in ker γ

/ker γ

= im γ

/im γ

, where e(i − 1) <

j ≤ ei and 1 ≤ i ≤ r, form a basis. Hence {ε

: 1 ≤ i ≤ er} is a basis of T

.

Furthermore, γ in this basis has the form

eγ =



 

 

0 1

0 . . . 0 0 0 1

. . . 0 . . . . 0 0 0 . . . 1

0 0 0 . . . 0



 

  .

Let t

be the point of mod

(er) corresponding to T

in the above basis.

We may consider t

as a homomorphism from A to M

(k). The map γ is a homomorphism of A-modules, thus the matrices eγ, t

(a) ∈ M

(k) commute for all a ∈ A. This implies that

t

=



 

y

y

. . . y

0 y

. . . y

. . . . 0 0 . . . y



 

for some y

∈ (M

(k))

, 1 ≤ i ≤ r. The epimorphism β

. . . β

: T

→ T

maps the set {ε

: e(r − i) < j ≤ er} into a basis B

of T

for all 1 ≤ i < r.

Let t

be the point of mod

(ei) determined by T

and the basis B

for all 1 ≤ i < r. Then t

is obtained from t

by removing the first e(r − i) rows and columns for all 1 ≤ i < r, which completes the proof.

The following proposition shows that finite tubes of height r are strongly related to the algebras M

(k[X]/(X

)), e ≥ 1.

Proposition 7. Let A be a finitely generated algebra and T = (T

, α

, β

) be a finite tube over A of height r. Let B = M

(k[X]/(X

)) with e = dim

T

. Then:

(i) There are, up to isomorphism, r indecomposable B-modules U

, . . . ,U

with dim

U

= ei for all 1 ≤ i ≤ r.

(ii) There is a homomorphism of algebras ϕ : A → B such that the A-module ϕ

(U

) is isomorphic to T

, where ϕ

: Mod B → Mod A is the functor induced by ϕ and 1 ≤ i ≤ r.

P r o o f. The first part is well known since the algebra B is Morita equivalent to k[X]/(X

). Take an epimorphism π : khX

, . . . , X

i → B such that x

= π(X

), . . . , x

= π(X

) form a basis of M

(k) ⊆ B and π(X

) = X ∈ B. Then B ≃ khX

, . . . , X

i/ker π and the point u

= (v

, . . . , v

) ∈ mod

(ei) satisfying

v

=



 

x

0 . . . 0 0 x

. . . 0 . . . .

0 0 . . . x



 , 1 ≤ j ≤ e

, v

=



 

 

0 1

0 . . . 0 0 0 1

. . . 0 . . . . 0 0 0 . . . 1

0 0 0 . . . 0



 

  ,

corresponds to the B-module U

for all 1 ≤ i ≤ r.

Applying Lemma 6 we deduce that the A-module T

determines in some basis a point t

of the form (4.3), for some y

, . . . , y

∈ (M

(k))

and n ≥ 1.

We may consider y

, . . . , y

as linear maps from A to M

(k). Then the map ϕ : A → B given by

ϕ(a) = y

(a) + y

(a) · X + . . . + y

(a) · X

, a ∈ A,

is easily seen to be a homomorphism of algebras. Furthermore, ϕ

(u

) = t

for all 1 ≤ i ≤ r. This implies that ϕ

(U

) is isomorphic to T

for all 1 ≤ i ≤ r, and the proposition follows.

Corollary 8. Let A be a finitely generated algebra and T = (T

, α

, β

) be a finite tube over A of height r. Let B=M

(k[X]/(X

)). If dim

End

(T

)

= r then the morphisms ϕ

: mod

(d) → mod

(d) are immersions for all d ≥ 1.

P r o o f. There is a B-module isomorphism B ≃ (U

)

. Moreover, U

is an indecomposable projective and injective B-module. Thus the left B-modules B and DB are isomorphic to each other. Furthermore, ϕ

(U

) ≃ T

, by Proposition 7. This implies that

dim

Hom

(B, DB) = dim

End

((T

)

)

= e

dim

End

(T

) = e

r = e

dim

End

(U

)

= dim

End

((U

)

) = dim

Hom

(B, DB).

Our claim follows from Lemma 4 and Theorem 1.

4.2. Proof of Theorem 2. Let Γ = (T

, α

, β

)

be a standard homoge- neous tube of the Auslander–Reiten quiver of A. Fix natural numbers r, d

satisfying r ≥ d

. Then the sequence T = (T

, α

, β

: 1 ≤ i ≤ r, 1 ≤ j < r) is a finite tube over A of height r. Let B = M

(k[X]/(X

)). Applying Propo- sition 7 we get a homomorphism ϕ : A → B of algebras such that im ϕ

is the union of orbits corresponding to modules from add(T

, . . . , T

). Since dim

T

= ei > d for any i > r, we get mod

(d) = im ϕ

.

Since Γ is a standard tube, dim

End

(T

) = r, and consequently, ϕ

is an immersion, by Corollary 8. This implies that mod

(d) is isomorphic to mod

(d) and is a locally closed subset of mod

(d).

The algebra B is Morita equivalent to the algebra C = k[X]/(X

).

In particular, there is an equivalence F : Mod C → Mod B of additive categories. Bongartz proved in [4] that there is a closed immersion µ : mod

(d

) → mod

(d) which induces a bijection between the set of Gl

(k)- orbits in mod

(d

) and the set of Gl

(k)-orbits in mod

(d). This bijection is compatible with F and preserves and reflects closures, inclusions and the types of singularities occurring in orbit closures (see also Section 4.2 in [5]).

Therefore, mod

(d) and mod

(d

) have the same types of singularities and

minimal singularities.

4.3. List of minimal singularities. We present here all types of minimal singularities in the module varieties over the algebra k[X]/(X

). They were investigated in [12] (for a field k of characteristic zero) and in [5] (in the general case). These types form two series:

• a

= Sing({nilpotent matrices in M

(k) of rank ≤ 1}, 0),

• A

= Sing({nilpotent matrices in M

(k)}, L

), where 1 ≤ n < r and

L

=



 

 

0 0 0 . . . 0 0 0 1 . . . 0 . . . . 0 0 0 . . . 1 0 0 0 . . . 0



 

 

is a nilpotent matrix of rank n − 1. In the above list all types are pairwise different except A

= a

.

Note that A

= Sing({X

+ Y Z = 0}, 0) is the type of one of the Kleinian (or simple) singularities. This can be obtained using transverse slices (see Section 5.1 in [15] and [2]).

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Faculty of Mathematics and Informatics Nicholas Copernicus University

Chopina 12/18 87-100 Toru´n, Poland

E-mail: gzwara@mat.uni.torun.pl

Received 8 June 1999;

revised 19 July 1999