Table Of ContentTENSOR PRODUCTS OF n-COMPLETE ALGEBRAS
7 ANDREAPASQUALI
1
0
2
n
Abstract. If A and B are n- and m-representation finite k-algebras, then
a
J their tensor product Λ = A⊗k B is not in general (n+m)-representation
finite. However, we prove that if A and B are acyclic and satisfy the weaker
2 assumption of n- and m-completeness, then Λ is (n+m)-complete. This
1 mirrors the fact that taking higher Auslander algebra does not preserve d-
representationfinitenessingeneral,butitdoespreserved-completeness. Asa
T] corollary, we get the necessary condition for Λ to be (n+m)-representation
finite which was found by Herschend and Iyama by using a certain twisted
R
fractionallyCalabi-Yauproperty.
.
h
t 1. Introduction
a
m
Higher Auslander-Reiten theory was developed in a series of papers [Iya07b],
[ [Iya07a],[Iya08]asatooltostudymodulecategoriesoffinite-dimensionalalgebras.
The idea is to replace all the homological notions in classical Auslander-Reiten
1
v theorywithhigher-dimensionalanalogs. Someearlyresultscanbe foundin[IO11],
5 [HI11b]. Thisapproachhasbeenfruitfulinthecontextofnoncommutativealgebraic
2 geometry, see for instance [AIR15], [HIO14], [HIMO14]. Higher Auslander-Reiten
3
theory is also deeply tied with d-homological algebra ([GKO13], [Jas16], [Jør15]).
3
A presentation of the theory from this point of view can be found in [JK16].
0
. In this setting, d-representation finite algebras were introduced in [Iya11] as a
1
generalisation of hereditary representation finite algebras. They are algebras of
0
7 global dimension at most d that have a d-cluster tilting module M. The category
1 addM has nice homological properties and behaves in many ways like the module
: category of a hereditary representation finite algebra. While classification of d-
v
i representationfinite algebrasseemsfar frombeing achieved,it makessenseto look
X
forexamples,andtotrytounderstandhowd-representationfinitenessbehaveswith
r respectto reasonableoperations. Notice that in this setting we havemore freedom
a
than in the hereditary case, since we are allowed to increase the global dimension
and still fall within the scope of the theory.
For instance, in [Iya11] Iyama investigates whether the endomorphism algebra
of the d-cluster tilting module (called the higher Auslander algebra) is (d + 1)-
representation finite. This turns out to be false in general, but a necessary and
sufficient condition is given: the only case where it is true is within the tower of
iterated higher Auslander algebras of the upper triangular matrix algebra, so this
construction gives only a specific family of examples. On the other hand, in the
same paper the weaker notion of d-complete algebra is introduced and studied. A
d-complete algebra is an algebra of global dimension at most d that has a module
which is d-cluster tilting in a suitable exact subcategory of the module category.
It turns out that this weaker notion is preserved under taking higher Auslander
algebras,thereby producing many examples of d-complete algebras for any d.
1
2 ANDREAPASQUALI
Another operation one might investigate is that of taking tensor products over
thebasefieldk. Indeed,ifkisperfectthengl.dimA⊗ B =gl.dimA+gl.dimB,so
k
itmakessense to askwhether the tensorproductofann- andanm-representation
finitealgebrasis(n+m)-representationfinite. Thisisfalseingeneral,andin[HI11a]
Herschend and Iyama give a necessary and sufficient condition (l-homogeneity)for
it to be true.
In this paper we prove that the same weaker notion of d-completeness which
is used in [Iya11] is preserved under tensor products, under the assumption of
acyclicity. Namely,ifAisn-completeandacyclicandB ism-completeandacyclic,
then A⊗ B is (n +m)-complete and acyclic. If we assume that A and B are
k
l-homogeneous, we recover the result by Herschend and Iyama. This gives a new
way of producing d-complete algebras for any d.
Theproofwegiveisstructuredasfollows. Weprovethatoverthetensorproduct
thereare(n+m)-almostsplitsequences(usingthesameconstructionasin[Pas17]),
and moreover that injective modules have source sequences. Then we use these
sequences,combinedwiththe assumptionofacyclicity,to provethatthe module T
inthedefinitionof(n+m)-completenessistilting. By[Iya11,Theorem2.2(b)],the
existence of the above sequences in T⊥ is equivalent to M being (n+m)-cluster
tilting in T⊥, which is the key point of (n+m)-completeness.
In Sections 2 we lay down notation, conventions, and preliminary definitions.
Section 3 contains the statement of our main result. Section 4 contains the results
about d-almost split sequences and tensor products which we want to use. Section
5 is dedicated to proving the main theorem, which amounts to checking that the
tensor product satisfies the defining properties of (n+m)-complete algebras. In
Section 6 we present some examples.
2. Notation and conventions
Throughout this paper, k denotes a fixed perfect field. All algebras are associa-
tive, unital, and finite dimensional over k. For an algebra Λ, modΛ (respectively
Λmod)denotesthecategoryoffinitelygeneratedright(left)Λ-modules. Wedenote
by D the duality D =Hom (−,k) between modΛ and Λmod (in both directions).
k
Subcategories are always assumed to be full and closed under isomorphisms, finite
direct sums and summands. For M ∈ modΛ, we denote by addM the subcate-
gory of modΛ whose objects are all modules isomorphic to finite direct sums of
summands of M. We write rad (−,−) for the subfunctor of Hom (−,−) defined
Λ Λ
by
rad (X,Y)={f ∈Hom (X,Y) | id −g◦f is invertible ∀g ∈Hom (Y,X)}.
Λ Λ X Λ
Moreover, for X,Y ∈ modΛ, we write top (X,Y) = Hom (X,Y)/rad (X,Y).
Λ Λ Λ
We often write Hom instead of Hom and similarly for rad and top when the con-
Λ
text allows it. We denote by Db(Λ) the bounded derived category of modΛ. For
a subcategory C of Db(Λ), we denote by thickC the smallest triangulated subcat-
egory of Db(Λ) containing C. If C = addM for some M ∈ modΛ ⊆ Db(Λ), we
write thickM = thick(addM). All tensor products are over k, even when the
specification is omitted to simplify the notation.
TENSOR PRODUCTS OF n-COMPLETE ALGEBRAS 3
Throughout this section, let gl.dimΛ ≤ d. Then we can define the higher
Auslander-Reiten translations by
τ =DExtd(−,Λ):modΛ→modΛ
d Λ
τ− =Extd (D−,Λ):modΛ→modΛ.
d Λop
We are interested in categories associated to tilting modules.
Definition 2.1. A Λ-module T is tilting if the following conditions are satisfied:
(1) Exti(T,T)=0 for all i6=0,
(2) there is an exact sequence 0→Λ→T →···→T →0 for some m with
0 m
T ∈addT for all i.
i
The second condition in the definition can be replaced by
thickT =Db(Λ).
For a tilting module T, we have an exact subcategory of modΛ
T⊥ = X ∈modΛ | Exti(T,X)=0 for every i6=0
We are interested in (cid:8)d-cluster tilting subcategories of T⊥. (cid:9)
Definition 2.2. Let T be a tilting module. A subcategory C of T⊥ is called d-
cluster tilting if
C = X ∈T⊥ | Exti(C,X)=0 for every 0<i<d =
=(cid:8)X ∈T⊥ | Exti(X,C)=0 for every 0<i<d(cid:9).
Wefollow[Iya11(cid:8),Definition1.11]anddefinethefollowingsubca(cid:9)tegoriesofmodΛ:
(1) M=M(Λ)=add τiDΛ | i≥0 ,
d
(2) P ={X ∈M | τ X =0},
d
(cid:8) (cid:9)
(3) M ={X ∈M | X has no nonzero summands in P}.
P
(4) M ={X ∈M | X has no nonzero summands in addDΛ}.
I
Let T be a basic module such that addT =P.
Λ Λ
Definition 2.3. An algebra Λ is d-complete if the following conditions hold:
(A ) T is a tilting module.
d Λ
(B ) M is a d-cluster tilting subcategory of T⊥,
d Λ
(C ) Exti(M ,Λ)=0 for every 0<i<d.
d P
Note that condition (A ) implies that τl = 0 for large l ([Iya11, Proposition
d d
1.12(d) and 1.3(c)]). Note moreover that if Λ is d-complete then since gl.dimΛ ≤
d it follows that gl.dimΛ ∈ {0,d}. This is a generalisation of the notion of d-
representation finiteness which we use in [Pas17]. Without loss of generality, from
now on we assume that Λ is basic. We write T for T when the context allows it.
Λ
Then [Iya11, Proposition 1.13] says that “d-representation finite” is the same as
“d-complete with T =Λ”.
If Λ is d-complete, then for every indecomposable injective I there is a unique
i
l ∈N such that τli−1I ∈P, and
i d i
T = τli−1I .
Λ d i
i
M
Definition 2.4 ([HI11a]). We say that a basic k-algebra Λ of global dimension d
is l-homogeneous if τl−1DΛ=T .
d Λ
4 ANDREAPASQUALI
If Λ is d-complete, this means that l =l for every i.
i
3. Main result
We now consider the case where A is n-complete, B is m-complete, and Λ =
A⊗ B. Sincekisperfect,wehavethatgl.dimΛ=gl.dimA+gl.dimB. Moreover,
k
bytheKu¨nnethformulawehaveτ X⊗Y =τ X⊗τ Y. Sinceindecomposable
n+m n m
injective Λ-modules are of the form X ⊗ Y, it follows that all indecomposable
modules in M are of this form. Our main result is the following:
Theorem 3.1. Let A,B be n- respectively m-complete acyclic k-algebras, with k
perfect. Then A⊗ B is (n+m)-complete and acyclic.
k
Note that as far as the author is aware, there are no known examples of d-
complete algebras which are not acyclic (this is Question 5.9 in [HIO14]).
This result can be applied inductively to construct d-complete algebrasstarting
for example from hereditary representation finite algebras and taking tensor prod-
ucts. In this sense, it is similar in spirit to [Iya11, Theorem 1.14 and Corollary
1.16], where Iyama constructs towers of d-complete algebras (with increasing d)
by taking iterated higher Auslander algebras. The algebra A⊗B is almost never
(n+m)-representationfinitebythecharacterisationgivenbyHerschendandIyama
in [HI11a]. Our result specialises to their characterisationin the acyclic case:
Corollary3.2. LetA,B ben-respectivelym-representationfiniteacyclick-algebras,
with k perfect. Then the following are equivalent:
(1) A⊗ B is (n+m)-representation finite.
k
(2) ∃l ∈N such that A and B are l-homogeneous.
Moreover, in this case A⊗ B is also l-homogeneous.
k
It should be noted that there is a choice involved in the definition we gave of
d-completeness, namely that we take M to be the τ -completion of addDΛ. We
d
might as well take M to be the τ−-completion of addΛ, and call Λ d-cocomplete
d
if it satisfies the dual conditions to (A ),(B ),(C ). Then Λ is d-complete if and
d d d
only if Λop is d-cocomplete. Notice that d-representation finite is the same as d-
complete and d-cocomplete with the same M. However, if A and B are n- and
m-representation finite, then A⊗B is (n+m)-complete and cocomplete, but in
general not with the same M.
4. Preparation
4.1. d-complete algebras. Following [Iya11], we make some observations about
d-complete algebras in general. Fix a finite-dimensional algebra Λ.
Lemma 4.1. If gl.dimΛ≤d, the following are equivalent:
(1) Exti(M ,Λ)=0 for 0<i<d
P
(2) Exti(M ,Λ)=0 for 0≤i<d.
P
Proof. The only direction to prove follows from [Iya11, Lemma 2.3(b)]. (cid:3)
Proposition 4.2. If Λ is d-complete, then
Hom(τiDΛ,τjDΛ)=0
d d
if i<j.
TENSOR PRODUCTS OF n-COMPLETE ALGEBRAS 5
Proof. This follows from [Iya11, Lemma 2.4(e)]. (cid:3)
We can define slices S(i) on M by saying that S(i)=addτiDΛ. Thus
d
M= S(i)
i≥0
_
(meaning that every object X ∈M can be written uniquely as X = X with
i≥0 i
X ∈S(i)) and moreoverHom(S(i),S(j)) =0 if i<j.
i
L
Lemma 4.3. If Λ is d-complete then τ± induce quasi-inverse equivalences M ↔
d P
M .
I
Proof. This is [Iya11, Lemma 2.4(b)]. (cid:3)
4.2. d-almostsplitsequences. InthespiritofgeneralisingAuslander-Reitenthe-
ory, it is natural to define the higher analog of almost split sequences as follows.
Definition 4.1 (Iyama). A complex with objects in a subcategory C of modΛ
C fd //C fd−1 // C fd−2 // ···
d d−1 d−2
is a source sequence (in C) of C if the following conditions hold:
d
(1) f ∈rad(C ,C ) for all i,
i i i−1
(2) The sequence of functors
···−◦fd−2//Hom(C ,−)−◦fd−1// Hom(C ,−)−◦fd // rad(C ,−) //0
d−2 d−1 d
is exact on C.
Dually we can define sink sequences. An exact sequence
0 //C //C //··· //C // C // 0
d+1 d−1 1 0
is an d-almost split sequence if it is a source sequence of C and a sink sequence
d+1
of C . We say that such d-almost split sequence starts in C and ends in C .
0 d+1 0
Definition 4.2. We say that M=M(Λ) has d-almost split sequences if for every
indecomposableX ∈M (respectivelyY ∈M )thereisand-almostsplitsequence
I P
in C
0→X →C →···→C →Y →0.
d 1
In this case we must have X ∼= τdY,Y ∼= τd−X. This holds for d-complete
algebras ([Iya11, Theorem 2.2(a)(i)]):
Theorem 4.4. If Λ is d-complete, then M has d-almost split sequences.
To apply the methods introduced in [Pas17], we need to rephrase Definition 4.1
asfollows: foranyindecomposableX ∈C wecandefineafunctorF oncomplexes
X
of radical maps by mapping
C = ··· fi+1 // C fi // ··· f1 // C f0 //···
• i 0
to
F (C )= ···fi+1◦−// Hom(X,C ) fi◦− //··· f1◦−// rad(X,C ) f0◦−// ···
X • i 0
6 ANDREAPASQUALI
(that is, F is the subfunctor of Hom(X,−) given by replacing Hom(X,C ) with
X 0
rad(X,C )). Similarly, we candefine a subfunctor G of the contravariantfunctor
0 X
Hom(−,X) by mapping C to
•
G (C )= ··· −◦f0 //Hom(C ,X) −◦f1 // ···−◦fd+1// rad(C ,X)−◦fd+2//···
X • 0 d+1
Lemma 4.5. Let C be a complex in C. Then
•
(1) If C =0 for all i>d+1, then C is a sink sequence if and only if F (C )
i • X •
is exact for every X ∈C.
(2) If C =0 for all i<0, then C is a source sequence if and only if G (C )
i • X •
is exact for every X ∈C.
(3) If C =0 for all i >d+1 and i< 0, then C is d-almost split if and only
i •
if F (C ) and G (C ) are exact for every X ∈C.
X • X •
Proof. Direct check using the definitions. (cid:3)
By additivity, in the above Lemma we can replace “every X ∈ C” by “every
indecomposable X ∈C”.
Notice that since d-almost split sequences come from minimal projective resolu-
tionsofafunctorrad(C ,−),theyareuniquelydeterminedbyC uptoisomorphism
0 0
of complexes. Moreover,we have
Lemma 4.6. Any map f : C → D between indecomposables in M induces a
0 0 0 P
map of complexes f :C →D , where
• • •
C = 0 //C gd+1 // ··· g1 // C //0,
• d+1 0
D = 0 // D hd+1 // ··· h1 // D // 0
• d+1 0
are the d-almost split sequences ending in C and D respectively, if these exist.
0 0
Proof. The map f g :C →D is a radical morphism, and since
0 1 1 0
Hom(C ,D ) h1◦− // rad(C ,D )
1 1 1 0
is surjective, there is a map f : C → D such that h f = f g . Now assume
1 1 1 1 1 0 1
we have constructed maps f : C → D that make all diagrams commute, for all
j j j
0≤j <i for some i≥2. We have that
Hom(C ,D ) hi◦−//Hom(C ,D )hi−1◦−//Hom(C ,D )
i i i i−1 i i−2
is exact in the middle term by assumption. Since h f g =f g g =0, we
i−1 i−1 i i−2 i−1 i
have that f g ∈ker(h ◦−)=im(h ◦−), that is there is a map f : C →D
i−1 i i−1 i i i i
such that f g =h f . The f ’s we have defined recursively give by construction
i−1 i i i i
a map of complexes f :C →D . (cid:3)
• • •
Thefollowingisaresultwhichappearedin[Pas17]inthesettingofd-representation
finitealgebras,andwhichcanbereformulatedinthesettingofd-completealgebras.
Theorem 4.7. Let Λ be d-complete. Let X ∈ S(i) with i > 0. Then the d-
almost split sequence starting in X is isomorphic as a complex to Coneϕ, where
ϕ:E →F is a map of complexes, such that:
• •
(1) All the maps appearing in E , F , and the components of ϕ are radical,
• •
(2) E ∈S(i) and F ∈S(i−1) for every j.
j j
TENSOR PRODUCTS OF n-COMPLETE ALGEBRAS 7
Proof. This is shown exactly as in [Pas17, Theorem 2.3]. Namely, one decomposes
the modules M appearing in the d-almost split sequence starting in X as M =
j j
M with M ∈S(i). One checks using Proposition 4.2 that in order for the
i≥0 ij ij
sequencetobed-almostsplit,alltheM mustbeinadd τiDΛ⊕τi−1DΛ forsome
L j d d
i. Now let E = M and F = M . Using that Hom(τi−1DΛ,τiDΛ) = 0
j i,j+1 j i−1,j (cid:0) d d(cid:1)
one can choose suitable differentials for E and F and a morphism ϕ : E → F
• • • • •
such that Coneϕ is the desired sequence. (cid:3)
We will need a technical lemma:
Lemma 4.8. Let
0 //C →fd+1 // C //··· // C f1 // C // 0
d+1 d 1 0
be a d-almost split sequence. Then for any choice of decomposition of the modules
C into indecomposbles, the corresponding matrices of the maps f have no zero
i i
column and no zero row.
Proof. We argue by contradiction. Assume f has a zero column for i < d. Then
i
there is a complex
f1
i+1
f2 [f1 0]
C i+1 //C1⊕C2 i // C
i+1h i i i i−1
such that
f1 ◦−
i+1
f2 ◦− Hom(C2,C1) [f1◦−0]
Hom(Ci2,Ci+1) h i+1 i // ⊕i i i //Hom(Ci2,Ci−1)
Hom(C2,C2)
i i
isexactinthemiddle,whichimpliesthatf2 ◦−issurjectiveonHom(C2,C2),and
i+1 i i
sothereish∈Hom(C2,C )suchthatf2 ◦h=id . Sincef2 ∈rad(C ,C ),
i i+1 i+1 C2 i+1 i+1 i
i
it follows that C2 = 0 and we are done. For proving the case i = d, just replace
i
Hom(C2,C ) with rad(C2,C ), and the argument goes through.
i i+1 i i+1
Thedualargument,usingthefactthatd-almostsplitsequencesaresource,yields
the second claim. (cid:3)
4.3. Tensor products. The main tool which allows us to perform homological
computations for tensor products is the Ku¨nneth formula over a field ([CE56,
VI.3.3.1]):
Lemma 4.9. If X ,Y are complexes, then there is a functorial isomorphism
• •
Hi(X•⊗Y•)∼= Hp(X•)⊗Hq(Y•).
p+q=i
M
Since tensor products of projective resolutions are projective resolutions, we
immediately get
Lemma 4.10. If M ,M ∈modA and N ,N ∈modB, then there is a functorial
1 2 1 2
isomorphism
Exti (M ⊗N ,M ⊗N )∼= Extp(M ,M )⊗Extq (N ,N ).
A⊗B 1 1 2 2 A 1 2 B 1 2
p+q=i
M
8 ANDREAPASQUALI
The total tensor product of complexes is a functor in a natural way, so we can
speak of tensor products of maps of complexes (for a very general treatment of
how this is done, see [CE56, IV.4 and IV.5]). An important result which is proved
in [Pas17] for d-representation finite algebras is also true for d-complete algebras,
namely:
Theorem 4.11. Let A,B be n- respectively m-complete algebras. Let Coneϕ and
Coneψ ben-respectivelym-almostsplitsequencesstartinginaddτiDArespectively
n
addτi DB for some common i>0. Then Cone(ϕ⊗ψ) is an (n+m)-almost split
m
sequence in M(A⊗B).
Proof. This is proved in the same way as in [Pas17, Section 3.3]. For convenience,
we present the main points of the proof. By definition Cone(ϕ⊗ψ) is a complex
bounded between 0 and n+m +1, it is exact by the Ku¨nneth formula, and it
is easy to check that all maps appearing are radical. Now ϕ : A0 → A1 and
• •
ψ :B0 →B1, and by assumptionwe have that A0 ∈addτiDA, A1 ∈addτi−1DA,
• • j n j n
B0 ∈ addτi DB and B1 ∈ addτi−1DB for every j since A ⊗B ∈ M(A⊗B).
j m j m j j
Let now M ⊗N be any indecomposable in M(A⊗B). We need to prove that
F (Cone(ϕ ⊗ ψ)) is exact. As in [Pas17, Section 2.3], for a radical map of
M⊗N
radical complexes η : A → B and a module X we can define F˜ (η) = η ◦− :
• • X
Hom(X,A )→F (B ). Then in our setting there is a commutative diagram
• X •
Hom(M,A0)⊗Hom(N,B0) ∼= //Hom(M ⊗N,A0⊗B0)
• • • •
F˜ (ϕ)⊗F˜ (ψ) F˜ (ϕ⊗ψ)
M N M⊗N
(cid:15)(cid:15) (cid:15)(cid:15)
F (A1)⊗F (B1) // F (A1⊗B1).
M • N • M⊗N • •
NowF (Cone(ϕ⊗ψ))isexactifandonlyifF˜ (ϕ⊗ψ)isaquasi-isomorphism.
M⊗N M⊗N
Theleftmapinthe diagramF˜ (ϕ)⊗F˜ (ψ)is aquasi-isomorphismsinceCone(ϕ)
M N
and Cone(ψ) are n- respectively m-almost split sequences. Then it is enough to
prove that the bottom map is a quasi-isomorphism, and this is done by showing
that its cokernel is isomorphic to
F (A1)⊗top(N,B1)⊕top(M,A1)⊗F (B1)
M • 0 0 N •
andthen by easyverificationthat the abovecokernelis exact. The computation of
the cokernel is done explicitly in [Pas17, Section 3.3, pp.660–662]. (cid:3)
Corollary 4.12. Let A,B be n-respectively m-complete algebras. Then M(A⊗B)
has (n+m)-almost split sequences.
NoticethattheabovetheoremdoesnotrequirethealgebraA⊗B tobe(n+m)-
representation finite (in which case we know a priori that (n + m)-almost split
sequences must exist). In the setting of [Pas17], this result is about describing the
structure of such sequences. In the setting of d-complete algebras, this result is
used to prove that (n+m)-almost split sequences exist, whereas it is a priori not
clear that they should.
One canalso say something about injective modules (which are not the starting
point of any d-almost split sequence).
TENSOR PRODUCTS OF n-COMPLETE ALGEBRAS 9
Proposition 4.13. Let A,B be n- respectively m-complete algebras, and let Λ =
A⊗B. Then for every injective Λ-module X ⊗Y there is a source sequence
X ⊗Y →E →···→E →0
n+m 1
in M(Λ).
Proof. SinceX andY areinjective,wehavesequencesinM(A)respectivelyM(B)
X =X →C →···→C →0
• n 1
Y =Y →D →···→D →0
• m 1
such that
0→Hom(C ,M)→···→Hom(X,M)→top(X,M)→0,
1
0→Hom(D ,N)→···→Hom(Y,N)→top(Y,N)→0
1
are exact for all indecomposables M,N. Now consider the homology of X ⊗Y .
• •
H (X )⊗H (Y ) if i=n+m+2
0 • 0 •
H (X ⊗Y )= H (X )⊗H (Y )=
i • • p • q •
(0 else.
p+q=i
M
So we have at least an exact sequence
X ⊗Y =X⊗Y →···→C ⊗D →0.
• • 1 1
Apply Hom(−,M ⊗N) to this sequence and compute homology.
H (Hom(X ⊗Y ,M ⊗N))=H (Hom(X ,M)⊗Hom(Y ,M))=
i • • i • •
= H (Hom(X ,M))⊗H (Hom(Y ,M))=
p • q •
p+q=i
M
top(X,M)⊗top(Y,N) if i=0
=
(0 else.
We will be done if we prove that X ⊗Y is source, which amounts now to prove
• •
that
top(X ⊗Y,M ⊗N)=H (Hom(X ⊗Y ,M ⊗N))=top(X,M)⊗top(Y,N).
0 • •
By tensoring the complexes
0→rad(X,M)→Hom(X,M)
and
0→rad(Y,N)→Hom(Y,N)
and looking at homology, one finds an exact sequence
0→rad(X,M)⊗Hom(Y,N)+Hom(X,M)⊗rad(Y,N)→
→Hom(X,M)⊗Hom(Y,N)→top(X,M)⊗top(Y,N)→0.
Now the middle term is isomorphic to Hom(X⊗Y,M⊗N), and this isomorphism
induces an isomorphism between the first term and rad(X ⊗Y,M ⊗N), hence by
10 ANDREAPASQUALI
looking at cokernels we get
Hom(X ⊗Y,M ⊗N)
top(X ⊗Y,M ⊗N)∼=
rad(X ⊗Y,M ⊗N)
Hom(X,M)⊗Hom(Y,N)
∼
=
rad(X,M)⊗Hom(Y,N)+Hom(X,M)⊗rad(Y,N)
∼=top(X,M)⊗top(Y,N)
and we are done. (cid:3)
Lemma 4.14. Let A,B be n- respectively m-complete algebras. Then the following
are equivalent:
(1) TA⊗B ∼=TA⊗TB.
(2) ∃l ∈N such that A and B are l-homogeneous.
Proof. (2)⇒(1) is clear by definition.
To prove (1) ⇒ (2), assume it does not hold, that is TA⊗B ∼= TA ⊗ TB but
there are i,j such that l 6= l for the corresponding indecomposable injectives
i j
E ∈ addDA and F ∈ addDB. We can assume that l > l , otherwise the proof
i j i j
is similar. Call Xij =τnli−1Ei⊗τmlj−1Fj ∈addTA⊗B. Then
τm−+ljn+1(Xij)=τnli−ljEi⊗Fj
is not injective, since by assumption τli−ljE is not injective. On the other hand,
n i
modules in M(A⊗B) which satisfy τ X =0 are precisely the injective A⊗B-
m+n
modules, and so τ−lj+1(X ) is not in M, contradiction. (cid:3)
m+n ij
4.4. Acyclicity. We need to discuss what we mean by acyclic algebras. Let
M ∈ modΛ, and let C = addM. We want to define a preorder on the inde-
composable objects indC of C. For X,Y ∈ indC, we say X < Y if there is a
sequence (X = X ,X ,...,X = Y) for some m ≥0, such that X ∈indC and
0 1 m+1 i
rad (X ,X ) 6= 0 for all i. This defines a transitive relation < on indC. Notice
Λ i i+1
that we can replace rad (X ,X )6=0 with rad (X ,X )6=0 above.
Λ i i+1 C i i+1
Definition 4.3. The category C is directed if < is antisymmetric, that is if no
indecomposable module X ∈ C satisfies X < X. If C = addM, we say that M is
directed. We call the algebra Λ acyclic if Λ is directed.
Λ
Lemma4.15. ThemoduleΛ isdirectedifandonlyifthemoduleD Λisdirected.
Λ Λ
Proof. The Nakayama functor induces an equivalence ν :addΛ →addD Λ, and
Λ Λ
the definition of directedness is invariant under equivalence. (cid:3)
Lemma 4.16. Let Λ be d-complete. Then Λ is acyclic if and only if M is directed.
Proof. IfMisdirected,thensoisaddDΛ⊆M. ByLemma4.15,Λisthenacyclic.
Conversely, if Λ is acyclic then addDΛ is directed by Lemma 4.15, and then so
is addτiDΛ for any i by Lemma 4.3. Any nonzero map between indecomposables
d
in M is either within a slice S(i) = addτiDΛ or from S(i) to S(j) with j < i.
d
Therefore there can be no cycles within a slice nor cycles that contain modules
from different slices and M is directed. (cid:3)
The relationwehaveintroducediswellsuitedtostudy d-almostsplitsequences.
