Ejercicios

HOCHSCHILD COHOMOLOGY AND HOMOLOGY OF ALGEBRAS
———————–
EXERCISES
PETTER ANDREAS BERGH
Exercise 1. (a) Show that a sequence
0
/M
f
/N
of abelian groups and homomorphisms is exact if and only if f is injective.
(b) Show that a sequence
f
M
/N
/0
of abelian groups and homomorphisms is exact if and only if f is surjective.
Exercise 2. Recall that a chain complex (or just “complex” for short) of abelian groups
is a sequence
M:
···
d3
/ M2
d2
/ M1
d1
d0
/ M0
/ M°1
d °1
/ ···
of groups and homomorphisms satisfying d n ± d n+1 = 0 for all n 2 Z. The nth homology
group of this complex is defined as
def
Hn (M) = Ker d n / Im d n+1 ,
which makes sense because Im d n+1 is a subgroup of Ker d n . Show that M is exact
(acyclic) if and only if Hn (M) = 0 for all n 2 Z.
Exercise 3. (a) Let p be a prime number. Compute the homology of the complex
···
·p
/ Z/(p 2 )
·p
/ Z/(p 2 )
·p
/ Z/(p 2 )
·p
/ Z/(p 2 )
·p
/ ···
Try to generalize this to the case when p is not necessarily a prime number.
(b) Compute the homology of the complex
···
·4
/ Z/(16)
·8
/ Z/(16)
where
dn =
Ω
·4
·8
·4
/ Z/(16)
if n is odd,
if n is even.
1
·8
/ Z/(16)
·4
/ ···
2
PETTER ANDREAS BERGH
Exercise 4. Recall that a chain map f : M ! L between two complexes is a collection
{ f n : M n ! L n | n 2 Z}
of group homomorphisms such that the diagram
M:
···
d3
d2
/ M2
f
f2
✏
L:
···
/ M°1
f0
✏
±2
/ L2
d0
/ M0
f1
✏
±3
d1
/ M1
±1
/ L1
✏
±0
/ L0
✏
d °1
/ ···
f °1
/ L °1
±°1
/ ···
commutes:
±n ± f n = f n°1 ± d 0
for all n 2 Z. The chain map is an isomorphism if all the homomorphisms f n are isomorphisms.
(a) Show that
f n (Ker d n )
f n (Im d n+1 )
µ
µ
Ker ±n
Im ±n+1
for all n 2 Z.
(b) Show that such a chain map induces a well defined homomorphism
Hn ( f ) : Hn (M)
!
m + Im d n+1
7!
Hn (L)
f n (m) + Im ±n+1
for all n 2 Z.
(c) Show that if the chain map f is an isomorphism, then Hn ( f ) is an isomorphism
for all n 2 Z.
Exercise 5. Let k be a commutative ring and A a k-algebra. Recall that the opposite
algebra A op of A has the same underlying set and group structure as A, but with multiplication reversed:
a1 · a2 = a2 a1 .
The enveloping algebra of A is defined as A e = A ≠k A op . Show that a k-module B is
a bimodule over A if and only if it is a left A e -module (and if and only if it is a right
A e -module).
Exercise 6. Take a commutative ring k, a k-algebra A and a bimodule B .
(a) Recall that we used the sequence
HB : 0
d 1 =0
/B
d0
/ Homk (A, B )
d °1
/ Homk (A ≠2 , B )
d °2
/ Homk (A ≠3 , B )
d °3
/ ···
to define Hochschild cohomology: the nth Hochschild cohomology group of A, with
coefficients in B , is defined as
def
HHn (A, B ) = Ker d °n / Im d 1°n .
The maps in HB are given by
(d 0 b)(a) = ab ° ba
EXERCISES
3
and
(d °n f )(a 1 ≠ · · · ≠ a n+1 )
=
a 1 f (a 2 ≠ · · · ≠ a n+1 )
n
X
+ (°1)i f (a 1 ≠ · · · ≠ a i a i +1 ≠ · · · ≠ a n+1 )
i =1
+(°1)n+1 f (a 1 ≠ · · · ≠ a n )a n+1
for n ∏ 1. Show that HB is a complex.
(b) Now recall that we used the sequence
d4
HB : · · ·
/ B ≠k A ≠3
d3
/ B ≠k A ≠2
d2
/ B ≠k A
d1
/B
d 0 =0
/0
to define Hochschild homology: the nth Hochschild homology group of A, with coefficients in B , is defined as
def
HHn (A, B ) = Ker d n / Im d n+1 .
The maps in HB are given by
d n (b ≠ a 1 ≠ · · · ≠ a n )
ba 1 ≠ a 2 ≠ · · · ≠ a n
n°1
X
+
(°1)i b ≠ a 1 ≠ · · · ≠ a i a i +1 ≠ · · · ≠ a n
=
i =1
+(°1)n a n b ≠ a 1 ≠ · · · ≠ a n°1
for n ∏ 1. Show that HB is a complex.
Exercise 7. Take a commutative ring k, a k-algebra A and a bimodule B .
(a) Show that
HH0 (A, B ) ' {b 2 B | ab = ba for all a 2 A}
and
HH0 (A, B ) ' B /K ,
where K is the k-submodule of B generated by the set
{ab ° ba | a 2 A, b 2 B }.
(b) Define the k-module of derivations of A on B as
def
Derk (A, B ) = {d 2 Homk (A, B ) | d (a 1 a 2 ) = a 1 d (a 2 ) + d (a 1 )a 2 for all a 1 , a 2 2 A},
and the k-module of inner derivations as
def
Derik (A, B ) = {d 2 Derk (A, B ) | there exists b 2 B with d (a) = ab ° ba for all a 2 A}.
Show that
HH1 (A, B ) ' Derk (A, B )/ Derik (A, B ).
4
PETTER ANDREAS BERGH
Exercise 8. Let k be a commutative ring and A = k[x] the polynomial ring in one variable. Since this is a commutative ring, there are no inner derivations on A except the
trivial one: Derik (A, A) = 0. Therefore HH1 (A, A is isomorphic to Derk (A, A) by Exercise
7(b).
(a) Let d 2 Derk (A, A). Show that
°
¢
d f (x) = f 0 (x) · d (x)
for all f (x) 2 A, where f 0 (x) denotes the ordinary derivative of the polynomial f (x). Note
the familiar “illegal formula”
°
¢
d f (x)
= f 0 (x).
d (x)
(b) For a polynomial p(x) 2 A, define a map d p : A ! A by
°
¢
d p f (x) = f 0 (x) · p(x).
Show that d p 2 Derk (A, A).
(c) Define a map √ : A ! Derk (A, A) by
°
¢
√ p(x) = d p .
Show that this is a k-module isomorphism, and that consequently HH1 (A, A) ' A.
Exercise 9. Let k be a field, Q the quiver
4O
∞
1
Æ
/2
Ø
/3
and A = kQ its path algebra. Show that HH1 (A, A) = 0.
Exercise 10. Read Dieter Happel’s paper Hochschild cohomology of finite-dimensional
algebras, in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année
(Paris, 1987/1988), 108–126, Lecture Notes in Math., 1404, Springer, Berlin, 1989. You
can find the paper below.
Hochschild c o h o m o l o g y o f finite-dimensional algebras
DIETER HAPPEL
[ This paper is in final form, and no version of it will be submitted for publication
elsewhere] .
The aim of this article is to report on some recent results on computing Hochschild cohomology groups of finite-dimensional algebras.
For this let k be an algebraically closed field and A a finite-dimensional k-algebra (associative,
with unit). By modA we denote the category of finitely generated left A-modules.
Let A X A be a finitely generated A-bimodule. The Hochschild cohomology groups H i ( A , X )
(i > 0) were introduced by Hochschild [Ho] (for a definition see section 1). The lowdimensional groups (i _< 2) have a very concrete interpretation of classical algebraic structures
such as derivations and extensions. It was observed by Gerstenhaber [Ge] that there are also
connections to algebraic geometry. In fact, H2(A, A) controls the deformation theory of A.
And it was shown that the algebras A which satisfy H2(A, A) = 0 are rigid. For a similar
approach we also mention an article of Gabriel [Ga 1].
Despite this very little was done in actual computations for particular classes of finitedimensional algebras. In section i we briefly review the fundamental definitions of Hochschild
cohomology and include an alternative description which one often uses for direct computations. In section 2 we present some computations. This includes a report on results due to
Cibils [C1], {C2], [C3] and Gerstenhaber and Schack [GS]. For some of these results we have
included proofs and some examples. In section 3 we deal with deriwations.
In the remaining two sections we outline how recently emerged methods in the representation
theory of finite-dimensional algebras yield information on the Hochschild cohomology.
1. H o c h s c h i l d c o h o m o l o g y g r o u p s .
This section contains the basic definitions of the Hochschild cohomology groups. We will
also give an alternative description which one often uses for direct computations. Moreover
this section contains some elementary examples. The exposition follows closely the original
approach in [Ho] and [CE]. We omit the proofs of these standard results.
1.1. Let k be an algebraically closed field and A a basic and connected finite-dimensional
k-algebra. Let A X A be an A-bimodule which is finite-dimensional over k. We define the
Hochschild complex C" = (C i, di)i¢z associated with this data as follows:
C i = O,d i = 0 f o r i < O , C ° = A X A , C i = Homk(A®~,X) for i > 0 (where A ®i denotes the
/-fold tensor product over k of A with itself), d°: X ~ Homk(A, X) with (d°x)(a) = a x - x a
for x E X and a E A, d i : C i ~ C i+l with
(di f ) ( a l ® " " ® ai+l) = a l f ( a 2 @ . . . ® ai+l)
i
+ ~-~(--1)Jf(al @...@ajaj+l
j=l
+ ( - 1 ) i + l f ( a l ® " " @ ai)ai+l
for f E C i and al . . . . ,ai+l E A.
®...®ai+l)
109
It is a direct verification that this indeed gives a complex.
So we may define H i ( A , X ) = H i ( c °) = ker d i / i m d i-~ and we call it the i - t h cohomology
group of A with coefficients in the bimodule X.
Of particular interest to us is the example A X A : A A A . In this case H ~ ( A , A ) is simply
denoted by H i ( A ) .
1.2. Let us pause for ~ moment to recall the interpretation of the low-dimensionM groups.
Clearly H ° ( A , X ) = Z A = {x E Z I ax = z a Va • A}. In particular, H ° ( A ) coincides with
the center of A.
Let D e r ( A , X ) = {5 • H o m k ( A , Z ) I 5(ab) = aS(b) + 5(a)b}be the k-vectorspace of
derivations of A on X. By D e r ° ( A , X ) we denote the subspace of inner derivations. Thus
D e r ° ( A , X ) = {5, : A ---* X I 5,(a) = ax - xa, x • Z } . It follows immediately from the
definition that H I ( A , X ) = D e r ( A , X ) / D e r ° ( A , X ) . Let 5 • D e r ( A , Z ) . Then 5 is called an
outer derivation if the residue class of 5 in H I ( A , X ) is different from zero.
For example let A = k[x]/(x 2) and A X A = AAA. Then 5 : A ~ A given by 5 ( , \ ' I A + # g ) = # g
is an outer derivation.
Let f • Homk(A ®2, X), then we may form A ~< X (the extension of A by X along f). As
f
a k-vectorspace this is simply A @ X. The multiplication is defined by (a, x) • (a', x') =
(ha', ax' + xa' + f ( a ® a')). If f E ker d 2 then A ~< X is an associative algebra with unit.
f
The following is an easy verification.
LEMMA. Let f , g • ker d 2. Then A ~<X - ~ A ~<X i f ] = y in H2(A, X ) .
f
g
1.3. A different way of approaching the cohomology groups is to consider the envelopping
algebra A ~ = A ® A*, where A* is the opposite algebra. For a • A we denote by a' the
k
corresponding element in A*. An A-bimodule A X A c a n be considered as a left A¢-module
b y ( a ® b I) . x = axb for a, b • A and x • X .
In particular A is an A-bimodule and we will therefore consider A always as a left A~-module.
We will now construct a projective resolution of A over A% For each integer i > - 1 , let
Si(A) denote the (i + 2)-fold tensor product over k of A with itself. Then Si(A) becomes
a left A~-module by setting (a ® b') • (ao ® ... ® hi+l) = a . ao ® al ® ... ® ai ® ai+ib for
a, b • A and ao ® . . . ® ai+ ] • Si( A ). We define an A~-homomorphism 5i : Si(A) ~ Si-I ( A )
(for i _> 0) by
i
5i(ao ® . . . ® ai+l) = ~~.(-1)J ao ® . . . ® ajaj+l ® " . ® hi+].
j=o
Then So = (Si(A),hi)i>_o is a projective resolution of A over A% Note that S - I ( A ) = A
and 5o : A ® A --* A is surjective. It is called the standard resolution [CE, p.174]. It is
frequently convenient to write Si(A) in the form Si(A) = A ® Si(A) ® A = A ~ ® Si(A), where
k
k
k
SI(A) = A ®i.
In computing the cohomology groups we use
HomAo(Si(A),X) = HomA,(A¢~Si(A),X)=
Homk(Si(A),X)=
Homk(gi(A),X)=
Ci
110
It is easy to see that the two complexes H o m k ( S . , X ) and C" can be identified• In particular
we see that ExtiA,(A, X) = Hi(A, X).
1.4. The last observation can be used to define the cohomology algebra H ( A ) = (~ H~(A).
iEl
The multiplication is induced by the Yoneda product. In this way H(A) is a / - g r a d e d algebra.
We will see in 1.6 below that H(A) is a finite--dimensional k-algebra if A has finite global
dimension (gl. dim A < oo).
The converse seems to be not known•
1.5. We want to construct a minimal projective resolution of A over A% For this let e l , . . . , e,~
be a complete set of primitive orthogonal idempotents in A. By P(i) we denote the indecomposable projective A-module Aei. Note that in this way we obtain a complete set of
representatives from the isomorphism classes of indecomposable projective A-modules. We
let S(i) = top P(i) be the corresponding simple A-module.
Then ei ® e~ (1 _< i , j <_ n) is a complete set of primitive orthogonal idempotents in A ¢. We
denote by P(i, f ) the indecomposable projective A~-module A~(ei ® e~). And we denote by
S(i, j') = top P(i, j') the corresponding simple A ~-module•
Observe that S ( i , j ' ) "~ Hom,(S(i), S(j)).
LEMMA. Let
... R,~ --* R,,-1 ~ "" ----*R1 ~ Ro ~ A ---* 0
be a minimal projective resolution of A over A ~. Then
R. = (~P(i,j')
i,j
PRooF: Let R,, = ~ . P ( i , f ) r o .
dim
ExtnA(S(1),S(j)).
Then by definition we have that
*J3
rlj = dim Ext~o(A, S ( i , j ' ) ) = dim Ext~°(A, Homk(S(i), S(j)))
= dim g n ( A , Homk(S(i), S(j)))
= dim Ext~(S(i), S(j)).
The last equality follows from corollary 4.4, p.170 of [CE].
In particular we see that pdA, A = gl. dim A.
We will use this projective resolution in section 2 for some direct computations•
1.6. As an illustration we will compute the cohomology groups of finite-dimensional heredi•
---+
tary k-algebras. Our assumptions imply that there exists a finite connected qmver A without
oriented cycles such that A-~k--~, where k ~ denotes the path algebra of ~ . The set of vertices is denoted by Ao and the set of arrows is denoted by A 1. For an arrow a in ~ we
denote by s(a) the starting point and by e(a) the end point of a. The vertices of ~ when
considered as trivial paths form a complete set of primitive orthogonal idempotents in k ~ .
111
and let u(a) = dim~s(a)(k-~)e(a). Moreover let n be the number
Let a be an arrow in
of vertices in A.
PROPOSITION.
H°(k-~) = k, dim Hl(k-~) = 1 - n + ~
u(a),H'(k~)
= 0 for i > 2.
aE~
PROOF: Note that dim ExtZ_(S(i), S(j)) coincides with the number of arrows from i to j.
kA
Clearly H°(k--~) = k and Hi(k-~) -=- 0 for i _> 2, for gl. dim k ~ _< 1.
Let 0 --* R1 --+ Ro --* k A --. 0 be the minimal projective resolution constructed above. Then
Ro = ~]~ P(i,i') and R1 = ~ P(s(a),e(a)'). Applying H o m ( k x ) o ( - , k ~ ) to the above
iEA°
oeEAI
exact sequence yields:
0 --* H o m ( k x ) , ( k - ~ , k ~ ) ~
-----+
Hom(kx)~(Ro , k ~ ) -~ Hom(kx),(R1 , k X ) ~ 0
~
--4
Clearly HOm(k~)~(kA , k A ) - ~ k and Hom(kx)~)(Ro, k A ) - ~ k
and S o m ( k - E ) , ( R l , k ~ ) = {~ k ~(~). Thus dim H l ( k ~ ) =
oeEAi
n
1 - n + ~ u(a).
aEAi
COROLLARY. Let -~ be a l~nite quiver without oriented cycle. Then Hl(k-~) --= 0 if and
only if -~ is a tree.
1.7. There is the dual concept of Hochschild homology. Again let AXA be a finitelygenerated A-bimodule. Then the Hochschild homology groups H i ( A , X ) can be defined
as Tor A" (X, A). We denote by D the standard duality on rood A. Then it is well-known that
Hi(A, X) ~_ HI(A, D(X)) [CE].
Suppose that A is a factor-algebra of a finite-dimensional hereditary k-algebra. Then it is
an easy consequence from 1.5 that Hi(A, A) = 0 for / > 0. This was first shown in [C4].
2. Survey of results.
In this section we report on some results on computing the Hochschild cohomology groups
for particular classes of algebras. Some proofs are given.
2.1 Incidence algebras [C3], [GS].
Let P = ( P , < ) be a finite partially ordered set. We may assume that P = 1 , 2 , . . . , n is a
labelling of the vertices. With P we may associate its incidence algebra I(P). This is by
definition the subalgebra of the algebra of (n x n)-matrices over k with elements (X~j) E
Mn(k) satisfying Xij = 0 if i ~ j. An equivalent definition is given as follows. Let ~ be the
112
following quiver attached to P. ~ has n vertices and there is an arrow from s to t in A if
t < s and there is no u E P with t < u < s. Let w,w ~ be two paths in ~ . We say that w,w'
are parallel if w and w ~ have the same starting and endpoint. Let I be the two sided ideal in
k ~ generated by the differences of parallel paths. Then it is easy to see that I ( P ) = k - ~ / I .
With P we may associate a simplicial complex ~'~p : (Ci, dl), where
Ci = {so > sl > ... > si 1~i 6 P }
and di is the obvious boundary map. The cohomology of Ep with coefficients in k is the
cohomology of the following complex:
0~
Hom(kCo, k) bl
Hom(kCl,k) b2
Hom(kCi, k) ~-~ Hom(kCi+l, k) ----~ ...
where
i+1
( b i + l f ) ( s o , ' " ,Si+l) = E ( - 1 ) J
f(so, "'" ,g j , " " ,si+l).
j=o
THEOREM.
H i (~p, k) ~ H'(I(P))
Moreover, let E be an arbitrary finite simplicial complex and Px: the corresponding partiMly
ordered set. So the elements of P~ are the simplices of E and the partial order is defined
by inclusion. Then Hi(E, k ) ~ Hi(I(P~)). This follows immediately from the theorem above
and the observation that Ep~ is the barycentric subdivision of E.
Example: Consider the following partially ordered set P
p =
Then the incidence algebra I ( P ) is given by:
i(p)=
Thus dim I ( P ) = 18.
kkkk
0
0
0
0
k
0
0
0
0
k
0
0
k
k
k
0
113
And Ep is the octahedron which is displayed below.
2.2 N a r r o w q u i v e r s [C1].
In this subsection we give a slight generalisation of a result of Cibils [C1]. Let A be a finite
connected quiver without oriented cycles. Let I be a two-sided ideal in k ~ generated by
.____+
paths of length at least two. Let A = k A / I . We will further assume that A is schurian
(i.e. dim H o m A ( P , P ' ) _< 1 for P, P ' indecomposable projective A-modules) and that A is
semicommutative (i.e. if w , w r are parallel paths in A with w ~ I then w t C I). In 1.6 we
have introduced a valuation v on the arrows of a quiver A . Note that our assumptions imply
that v ( a ) = 1 for a E A1. Let IZXol (resp.lAll)denote the number of vertices of A (resp. the
--~
..._¢
number of arrows of A ). The Euler characteristic of A is defined by X ( X ) = 1 - I A ~ l+ IA11.
..-+
THEOREM. Let A = k ~ l Z
Hi(A) = 0 for i > 2.
5e as above. T h e , H ° ( A ) = k, dim H i ( A ) = g ( - ~ ) and
PROOF: Since ~ does not contain an oriented cycle we infer that H ° ( A ) = k. Also note
that dimkExtl(S, T) < 1 for any pair S, T of simple A-modules and that E x t , ( S , T) ¢ 0 if
___+
.--+
and only if there is an arrow from the vertex of A corresponding to S to the vertex of A
corresponding to T. Let
"" Rn
~
Rn-1
--'* " " ~
R1 ~
Ro ~
A -* 0
be the minimal projective resolution of A over A e constructed in 1.5.
Applying ( - , A) = HomA~(--, A) to this exact sequence yields:
0 ---* (A, A) ~ (Ro, A) ~ (.R1, A)
---¢
(R2, A) 4 . . .
Clearly we have that (A, A) = k, (Ro, A) = klaol and by the considerations above we infer
that ( R 1 , A ) = klzxll (note that A is supposed to be schurian). By the lemma below we infer
that (Ri, A) = 0 for i _> 2). So the assertion follows.
114
LEMMA. Let A = k'-~/I be as above. Let ei, ej be primitive orthogoned idempotents corresponding to the vertices i, j • Ao. If EXttA ( S( i ), S(j ) ) ¢ 0 for t _> 2 then eiAej = O.
PROOF: If Ext~(S(i), S(j)) 7~ 0 for t _> 2 then P(j) occurs in a minimal projective resolution
of S(i), yielding w from i to j in A . It is a straightforward verification that our assumptions
imply that w • I. Using again that A is semicommutative we infer that eiAej = O.
We say that a finite connected quiver ~ is narrow if there is at most one path between any
pair of vertices of ~ .
For an arbitrary quiver ~ we denote by ~ +
the two-sided ideal in
k ' ~ generated by the arrows of A . Finally we recall that a two-sided ideal I in k-~ is said
to be admissible if ( ~ + ) r n C I C ( A + )
2
for some rn > 2.
COROLLARY [C1]. Let --~ be a narrow quiver and I be an amissibie ideal in k--~.
Then
H ° ( k N / I ) = k, dimHl(k--~ /I) = X(--~) and H i ( k N ) = 0 for i _> 2.
Remark: The class of narrow quivers clearly includes those quivers whose underlying graph
is a tree. But it also contains for example the following quiver:
C)
2.3
2-nilpotent algebras.
Let A be a finite-dimensional k-algebra. The radical of A is denoted by radA. We say that
A is t-nilpotent if (radA)* -- 0 but (radA) *-1 # O.
If A is a basic, connected 2-nilpotent algebra, then there exists a finite and connected quiver
~
such that A-*
k( "A- ~ ) / ( A" "++ )
2
.
.-.+
PROPOSITION. Let A = k-~ /(-~+ ) 2 with A connected. The following statements are equiv-
alent:
O) Hi( A ) = O fori > o
(ii) H i ( A ) = 0
(iii) ~ is a tree.
PROOF: (i) =~ (ii) trivial and (iii) ~
(i) follows from 2.2.
So it remains to show (ii)
(iii). Suppose ~ is not a tree and let a C A~ be an arrow belonging to a cycle of A (the
underlying graph of ~ ) . We will construct 5 C Der(A) such that the residue class of ~ in
H i ( A ) is different from zero. Since A is 2-nilpotent there exists a k-basis of A induced
by the vertices and arrows of A . We denote this basis by e l , . . . , en (for the vertices) and
a l , . . . , a t (for the arrows). We may assume that a = al. Define 5 : A ~ A by 5(ei) = 0
for 1 _< i _< n, 5 ( a l ) = a~ and 5(aj) = 0 for 2 _< j _< r. An easy computation shows
115
that $ E Der(A). Suppose that 6 = 8z for some x E A where ~, is the inner derivation
associated with x. Then x = E'~=lAiei + E jr = l # j .a 3. for some Ai,#j E k. Since ~(al) =
OL1 = 6 z ( O f l )
= XO/1 -- OLIX =
As(al)•
1 - - )~e(al)O~l it follows that As(a1) - Ae(~l) = 1. Since
5(ai) = 0 = ~z(aj) = x a i - a j x = As(~,~)aj- A~(,~,)aj f o r j > 2 it follows that As(~) = A¢(a~).
Since A \ { a l } is connected we infer that A1 . . . . .
An. Thus the residue class of ~ in H~(A)
is different from zero. This finishes the proof.
---4"
THEOREM [C2]. Let A = k - ~ /(--~+ ) 2 with A connected.
equivalent:
The following statements are
(i) H2(A) = 0
(ii) --~ does not contain a loop, --~ does not contain a triangle, and --~ is not O "
Note that a loop is an arrow a with s(a) = e(a) and a triangle is a quiver of the following
form
z%
We indicate the easy direction ((i) =~ (ii)) and refer for the converse to [C21.
Suppose ~ contains a loop a. As above we may choose a basis of A induced by the vertices
and arrows, say xl = a, x 2 , . . . ,xm. We define f E H o m k ( A ® A , A ) by f ( x i N x j ) - xl for
i = 1 = j and zero otherwise. It is an easy verification to show that f E kerd 2 (compare 1.1)
and that the residue class of f in H2(A) is different from zero.
Suppose that ~ contains a triangle
,y
Choose a basis X l , . . - , x m of A as above w i t h x l = a, x2 = fl, x3 = 7. So we may define
f E Homk(A @ A , A ) by f(x~ ® xj) = x3 for i = 1 and j = 2 and equal to zero otherwise.
Again f E kerd 2, but f {[ i m d 1.
Finally suppose that A is the following quiver
X2
Xl~[~X3
X4
Choose a basis of A as indicated. Define f E Homk(A®A, A) by f(x2@x4) = Xl, f ( x 4 Q x 2 ) =
x3 and zero otherwise. Again f E ker d 2, but f ~_ im d 1. So in all three cases we see that
H2(A) is different from zero.
116
3. D e r i v a t i o n s .
In this section we will concentrate on computing derivations• Again let A be a finitedimensional k-algebra. For expository reasons we will assume that the characteristic of
k is zero. Let e l , ' " , en be a complete set of primitive orthogonal idempotents in A. Let
A = ~ e i A e i be the two-sided Pierce decomposition. Recall that we have denoted by
%./
Der(A, A) = Der(A) the vectorspace of derivations of A. We denote by Der°(A) the vectorspace of i n n e r derivations.
3.1.
LEMMA. Let ~ E Der(A). T h e n there exists ~5' E D e r ° ( A ) such that 8(ei) - ¢5'(e,) = 0
forl <i <n.
PROOF: Since ,5(ei) = ~5(e~) = ei~5(ei) + ~5(ei)ei, we infer that ei~5(ei)ei = O. Thus ei~5(ei) C
eiA(1 - el) a n d ~5(ei)ei E (1 - ei)Aei. Moreover we have for i # j that 0 = ~5(eiej) =
eia(ej) + a(ei)ej. Thus there exists rij E eiAey, rii = 0 s u c h that eia(ei) = E]=I rij and
,5(ei)ei = - P']=I rji, thus ~5(ei) = 2jn_l rij -- rji. Let r = Ei,j ri,j and let ~' = ~_~ the inner
derivation attached to - r . Thus (5_r = - r e i + ei r = _~nl=l rli + 2nl=l ril = (5(ei).
We denote by Der'~(A) = {~ E Der(A) J $(ei) = 0) the subspace of normalized derivations.
Let $ E Der"(A) be a normalized derivation. T h e n * respects the two-sided Pierce decomposition. In fact, let a E e i A e j , then 5(a) = 5(eic~ej) = c i S ( a e j ) = ciS(a)ej E eiAe i. By
Der "'° we denote the subspace Der~(A) Cl Der°(A). It is now an immediate consequence of
the l e m m a above that Dern(A)/Der'~,°(A)-:+HI(A).
----4
----}
---4
3.2. Let A be a finite quiver and k A the corresponding path algebra. We will consider k A
as a graded algebra, where the grading is induced by assigning to a path w in LX as degree
the length of w.
THEOREM. Let I be an admissible ideal in k A which is h o m o g e n e o u s with respect to the
---4
grading. Let A = k A / I .
I f H 1 (A) = 0 then -~ is directed.
PROOF: F o r w e k Z ~ we denote by I(w) the length of w. We define fi : k ~
--* k ~
by
6(w) = l ( w ) w .
T h e n ~ E D e r n ( k ~ ) . Since I is homogeneous, we infer that ~ induces
a normalized derivation on A again denoted by ~. By assumtion there exists a E A such
that 6 = ~a. Since ~ E Dern(A) we must have that a = Ein=l#iei -]-y for #i E k and
--.+
y E ~ e i ( r a d A ) e i . Let a be an arrow in A and ~- the residue class of a in A.
s
T h e n ~ = fi(~) = ~ ( ~ ) = aa - aa =/*s(~)N - #~(~)~ + z , where z E rad2A.
Thus/*s(a) - #e(a) = 1. Now suppose that
1-2-!+2 a2~ 3 . - . r - 1 - - - - ~ r = l
is an oriented cycle in "~. T h e n we must have that # i - #i+1 = 1 for 1 < i < n and
#n - #1 = 1. It is easily seen that in char k = 0 this system of linear equations has no
solution, a contradiction.
117
3.3. Recall that in 2.3 we have defined t-nilpotent algebras.
COROLLARY. Let A =
rected.
kXIS be 3-nilpotent
and suppose that H i ( A ) -~ O, then -~ is di-
PROOF: It is well-known and easy to see that a 3-nilpotent algebra k - A / I is isomorphic to
its associated graded algebra. With I we have denoted an admissible ideal in k A.
3.4. For the next corollary we need some additional terminology. Let A be a representationfinite algebra (i.e. up to isomorphism there exist only finitely many indecomposable Amodules.) Let X 1 , - " , X r be a list of representatives from the isomorphism classes of indecomposable A-modules and let X = ~)Xi. Then A = End X is called the Auslander-algebra
i
of A [Ga2]. Recall that A is said to be standard if A is isomorphic to its associated graded
---4
algebra. For different characterizations we refer to [BrG]. Let A -- k A / I , then A is called
representation-directed if ~ is directed. For an equivalent definition we refer to [Ri].
(Note that ~ is the Auslander-Reiten quiver of A and I is the mesh-ideal.)
COROLLARY. Let A be a representation-t~nite, standard k-algebra and A be the Auslander
algebra of A. If H i ( A ) = 0 then A is representation-directed.
We will come back to the computation of the Hochschild cohomology of Auslander algebras
in a subsequent publication.
We conclude this section by some examples. We leave out the rather elementary computations.
a) Let A -- klx]f(x" ) and A be the Auslander algebra of A. Then d i m H ° ( A ) = n,
dim H i ( A ) -- n - 1 and d i m H 2 ( h ) = n - 1.
(Note that g / . d i m h _< 2, so Hi(A) = 0 for i > 2.)
b)
Let A =
kR/swith ~
=aQ
~
~ ) 7 and I = < a2,afl, flT,7 2 >.
Let A be the Auslander algebra of A. Then the Auslander-Reiten quiver of A is given by:
I
I
I
I
I
I
I
I
I
I
The identification is along the vertical dotted lines and the horizontal dotted lines indicate
the Auslander-Reiten translation. Then dim H°(A) = 2,dimHl(A) = 2 and H2(A) = 0.
118
4.Tilting invariance.
4.1. Let A be a finite-dimensional k-algebra. An A - m o d u l e A M is called an r-tilting module
if (i) p d A M < r, (ii) Ext/a(M, M ) = 0 for 1 < i < r, (iii) there exists an exact sequence
0 ~A
A--* M ° ~
M 1 ~
" " "--* M r -'* 0
W i t h a d d M we have denoted the additive category generated by M.
For some information about tilting modules as well as for the proofs of the results we will
use here we refer to [ C P S I , [ H a l l , [ H a 2 ] .
We say that a finite-dimensional k-algebra B is tiltable to A if there exists a family
( A i , M i, Ai+l = End Mi)o_<i<m such that A o = B,A,-,, = A and M i is an r-tilting module
over Ai.
Let us give an easy example. Let A = k ~ , where
/~
= 0
(
1
0
2
+----- O "
3
••
0
n--I
+----- 0
n
Note that A is just the algebra of all n × n - u p p e r triangular matrices over k. Let B = k ' - ~ / I .
Then B is tiltable to A if and only if ~ is a full and connected subquiver of the infinite quiver
"~ defined below having n vertices containing the central vertex and I is generated by all
possible paths c~/3 and/3~. Below we have defined the infinite quiver -~. The arrows on the
diagonals pointing down are labelled with ~ and those on the diagonals pointing upwards are
labelled with/3.
119
Let Db(A) be the derived category of bounded complexes over mod A [V]. Recall that mod A
is fully embedded into Db(A) by sending a module A X to a complex which is concentrated
in degree zero. We will identify rood A with its image in Db(A). Let T be the translation
functor on Db(A). Then HomD,(A)(X, T i Y ) ~_ EXtiA(X,Y)forX, Y 6 m o d A and i 6 Z.
Let B M be an r-tilting module and A = E n d B M . Then there exists a triangle-equivalence
F : Db(A)
~ Db(B) such that F ( B M ) = AA. In fact F is the right derived functor of
HomB(M, -).
4.2. In the following result we show that Hochschild cohomology is invariant under tilting.
THEOREM. Let A and B be flnite-dimensional k-algebras. If B is tiltable to A then H(B)
and H ( A ) are isomorphic as Z-graded algebras.
PROOF: Let B M be an r-tilting module and A -- E n d B M . Clearly it is enough to show
that H ( B ) and H ( A ) are isomorphic as ][-graded algebras. We will construct below a
triangle-equivalence F : Db(B ~)
, Db(A e) such that F ( B , B ) = A*A. From this the
assertion follows immediately. In fact, Hi(A) = E x t ~ , ( A , A ) = HomDb(A~)(A,T'A) ~--
HomD~(B~)(B, TiB) = Hi(B).
Since _~ preserves the composition of morphisms, we see that F induces an isomorphism of
the ][-graded algebras H ( A ) and H ( B ).
We now turn to the construction of /~. Recall that A ~ = A ® A*,B e = B ® B*. Let
k
k
C = B ® A*. Note that BMA is a left C-module. It is easily checked that X = M ® A* is
k
k
an r-tilting module over C such that E n d c X = A% Thus we obtain a triangle-equivalence
F : Db(c) --, Db(A e) such that F ( c X ) = A,A e. It follows from the description above that
F ( c M ) = A,A. Similarly, let Y = B ® M. Then Y is an r-tilting module over C such
k
that E n d c Y = B e. Thus we obtain a triangle-equivalence G : Db(C) --* Db(B ~) such that
G ( c Y ) = B,B e. Moreover it follows that G ( c M ) = BoB. Now let G' be a quasi-inverse to
G. Then F = FG' is the required triangle-equivalence from Db(B e) to Db(A~).
4.3.
COROLLARY. Let -~ be a finite quiver without oriented cycle and let B be a finite-
dimensional k-algebra which is tiltable to k-~. Then Hi(B) = O, for i >_ 2,H°(B) = k and
dimHl(B) = dimHl(k-~). In particular B is rigid. II A is not a tree and B = k-~ /I, then
F is not a tree.
PROOF: The first assertions follow from 4.2 and 1.6. The second statement follows from this
and 2.2.
4.4. Exaznples:
1) Let B = k'-~/I where F'~ and I are given as follows:
F=
o; I =
<~f16-~6e>
120
T h e n B is tiltable to k ~
(in one step) where
A___
In p a r t i c u l a r H i ( B ) = O. Let
A I --
so A I = k A-~ is a factor algebra of B generated by a primitive idempotent. By 1.6 we infer
t h a t d i r n H l ( A ~) = k. This shows t h a t even for well-behaved factors it is not always possible
to extend n o n - t r i v i a l o u t e r derivations.
2) Let A = k--~/I where
a~d I = ( ~ + ) ~ .
T h e n A satisfies the assumptions of 2.2, so H°(A) = k, HI(A) = k a n d Hi(A) = 0 for i > 2.
We choose as indecomposable s u m m a n d s of a 1-tilting module AM the indecomposable modules which correspond to the vertices marked with • in the A u s l a n d e r - R e i t e n quiver of A
Then B = EndAM =
k-~/I where
e
l"=
3'
and I =
<o~7, flS, T e > .
121
So in particular we see that H2(B) = 0 despite the fact that B admits a factor algebra C by
an ideal generated by primitive idempotents such that H2(C) ~ O.
4.5. In the final part of this section we want to report'on some joint work with M. Schaps
[HS] which is related to the theorem above. The proof of the theorem below will be omitted.
Let Algd be the affine variety of algebra structures (associative, with unit) on a vector space
V of dimension d over the field k. This variety is the set of bilinear maps from V × V to V
that are associative and admit a unit; for a formal definition and some of its properties the
reader is referred to [Gal]. To a point p of Algd we associate the finite-dimensional k-algebra
Ap. Let G = Gl(V). The canonical linear operation of G on V induces an operation of G on
Algd. For p E Algd we denote by Op the orbit of p under G. Let p, q E Algd. We say that Aq
deforms to Ap (or Ap degenerates to Aq) if q E Op, the Zariski-closure of Op. The algebra
Ap is then called an algebra deformation of Aq.
For example M2(k) (the matrix algebra of (2 × 2)-matrices over k) is an algebra deformation
of k--~/I where
O~
-~-= ~
and I-= < aj3,~o~ > .
Z
If Ap is a non-trivial algebra deformation of Aq (i.e. gp ~ dq), then
H2(dq) ~
0 [Ge].
4.6. THEOREM. Let A be a finite-dimensional basic k-algebra and let .4 be an algebra
deformation of A. Let A M be a 1-tilting module with endomorphism a/gebra E n d A M = B.
Then there exists a 1-tilting module il~f such that B = End i M is an algebra deformation
of B.
We point out that even in the case where A M is multiplicity-free (i.e. each indecomposable
direct summand of A M occurs with multiplicity one) the tilting module AJl~/will usually not
be multiplicity-free. In other words, /~ will usually not be a basic k-algebra. An example
for this will be given below.
4.7. Examples:
1) Let A = k A / I , where
A=
and I = < a~ > .
Then A admits a non-trivial deformation ft. = k--~/I ~ where I ~ = < aft - 75 >. We choose as
indecomposable summands of a 1-tilting module A M the indecomposable modules which correspond to the vertices marked with • in the Auslander-Reiten quiver of A (the identification
is along the vertical dotted lines).
122
i
I
I
t
*
t
Then EndAM = k'-~/J where
and J = < oLj3 > .
P=
Then AM of the theorem above corresponds to the vertices marked with • in the AuslanderReiten quiver of A:
So End ~.~/= k ~ where
<
which clearly degenerates to EndAM.
2) Let A = k ~ / I ,
where ~
= 0~
and I = < c~2, c~fl >. Then A admits a
non-trivial deformation A = kA
--~ where
ol
3ex
02
123
(note that A t is not connected).
We choose as indecomposable summands of a 1-tilting module AM the indecomposable modules which correspond to the vertices marked with * in the Auslander-Reiten quiver of A (the
identification is along the vertical dotted lines).
|
t
Then EndAM = k--~/J where
O~
"-~= ~
and J = < aria >
and ~i~/of the theorem above is P(1)@P(1)@P(2)@P(3). So E n d 2 . ~ / = M2(k)x ( ~
5.
~).
One-point extensions.
5.1. In this section we present for a special class of algebras a way of computing the
Hochschild cohomology of a given algebra by knowing the cohomology of a particular factor
algebra. To be more precise we have to recall the definition of a one-point extension algebra.
For more details and the representation-theoretic tools available in this context we refer to
[Ri]. Let A be a finite--dimensional k-algebra and M E mod A. The one-point extension
algebra A[M] of A by M is by definition the finite-dimensional k-algebra
AIM] = (A M
with multiplication
m
t
where a, a t E A, m, rn I e M and •, A' E k.
For example let Tr,(k) be the algebra of n x n - upper triangular matrices over k. Then
T,(k) operates on an n-dimensional k-vector space by left multiplication. We denote this
T,(k)-module by M. Then an easy verification shows that T,(k)[M] ~- T,,+](k).
From the definition of the one-point extension we see that a necessary condition for an
algebra B to be of the form AIM] for some algebra A and an A-module M is that there is a
simple injective B-module. This clearly is sufficient. Indeed, if an algebra B admits a simple
injective module S. Let P(S) be a projective cover of S and let e e B be an idempotent such
that P(S) = Be. Let A = B/(e) (where (e) denotes the two-sided ideal in B generated by
124
the idempotent e) and let M = radP(S). Then M is an A-module and it is easily checked
that B "~ A[M].
There is the dual concept of a one-point coextension. We denote by M* the k-dual of M.
Then the one-point coextension [M]A of A by M is by definition the finite-dimensional
k-algebra
with the obvious multiplication.
We will state our results for one-point extensions and leave it to the reader to derive the
corresponding statements for one-point coextensions.
5.2. Let A be a finite-dimensional k-algebra and let M E modA. We denote by B the
one-point extension of A by M. Let e E B be a primitive idempotent such that M -- tad Be.
Further let I = (e) be the two-sided ideal generated by e. Then we have that A = B / I .
In the following lemma we have collected some useful homological information about this
data. As the proofs are easy we omit them.
LEMMA. With the above notation we have:
(i)
(ii)
(iii)
(iv)
(v)
B,I ~-- P(e, e') ~_ Homk(S(e), P(e))
ExtJA~(A,A) ~_ ExtJ~(B,B)
Ext~(S(e), P(e)) _~ E x t y ~ ( M , M) for i _> 2
ExtlB(S(e), P(e)) _ H o m A ( M , M ) / k
HomB(S(e),P(e)) = 0
5.3 THEOREM. Let B = A[M] be as above. Then there exists the following long exact
sequence connecting the Hoehschild cohomotogy of A and B.
0 ~ H°(B)
~ H°(A) ~
HomA(M,M)/k
~ H i ( B ) ~ H i ( A ) --* EXtlA(M,M) 4 . . .
• .. ~ ExtiA(M, M) ~ H'+I(B) ~ Hi+I(A) --* EXtiA+l(M, M ) 4 . . .
PROOF:
(*) O ~ I ~ B ~ A ~ O
By construction and 5.2(i) we infer that E x t ~ ( I , A) = 0 for i > 0. Applying H o m E . ( - , A)
to (*) then yields E x t , . (A, A)-~ E x t , , (B, A) for i > 0. Note that E x t , , (B, I ) = H ' ( B , I) =
Hi(B, Homk(S(e), P(e))) = Ext~(S(e), P(e)) which can be computed by 5.2 (iii),(iv) and
(v). Apply H o m B , ( B , - ) to (*). This yields the long exact sequence
0--~ HomBo(B,B) --. H o m B , ( B , A ) ~ E x t ~ . ( B , I ) ~ Ext]~.(B,B) -~ E x t ~ , ( B , A ) 4 . . .
which by the identifications above gives the assertion.
5.4. For the next corollaries we have to recall the definition of the S-condition [BrG],[BLS].
Let B = k - ~ / Y with ~ directed. An indecomposable projective B-module P(i) for i E A0
is said to have a separated radical if distinct indecomposable summands of radP(i) have
125
----+
support on different connected components of Ai, where ~ is the full subquiver of A with
vertices j such that there is no path from j to i. Finally we say that B satisfies the S-condition
if all indecomposable projective B - m o d u l e s have separated radical.
COROLLARY. If B is representation-directed then H I ( B ) = 0 for i > 2.
PROOF: We proceed by induction on the number of distinct simple B-modules. We may
assume that B = A[M] and A = A1 x -.. x An with Ai being representation-directed for
r
1 < i < r. Since B is representation-directed we infer that M = ]-L=z Mi, with Mi E m o d A i
satisfies E x t J ( M i , M i ) = 0 for j > 1. By induction we have that H J ( A i ) = 0 for j > 2 and
1 < i < r. The assertion now follows from 5.3.
,.5.5. COROLLARY. Let B be representation-directed.
and only if H 1( B ) = O.
Then B satisfies the S-condition if
PROOF: Suppose that B = A[M] and A -- A1 x ... x An, M = I_[i~=zMi with Mi E m o d A i .
If B satisfies the S-condition so do A 1 , - . . ,An. By induction we infer that H i ( A ) = O.
Moreover we see that iV// is indecomposable. As Ai is representation-directed we have that
E n d M i = k. Clearly H ° ( B ) = k, H ° ( A ) = k ~ and dim H o m A ( M , M ) / k = r - 1, so
H i ( B ) = O.
Conversely, if H i ( B ) = 0 and B = A[M] we see as above that E x t ~ ( M , M ) -- 0, for B is
representation-directed. By induction we infer that A satisfies the S-condition. As above
let A = A1 x .-. x An and M = H ir= l M~. If H i ( B ) = 0 then dim HomA(M, M ) / ' k = r - 1.
Thus dim E n d Mi = 1 and in particular we see that the Mi are indecomposable. Hence B
satisfies the S-condition.
5.6. COROLLARY. Let A be a connected finite-dimensional k-algebra with H ° ( A ) = k and
H i ( A ) = 0 for i > 0. Let M E m o d A and B = A[M] the corresponding one-point extension.
Then H ° ( B ) = k, H Z ( B ) ~_ H o m A ( M , M ) / k and H I ( B ) ~_ EXtiA-Z(M,M) for i > 1.
We conclude by giving some examples.
Let A = k ~ and A a Dynkin diagram of type ~ n , D , , E 6 , E T , E a and let M E m o d A be
indecomposable. Then ttI(A[M]) = 0 for i >__1.
Let A = k ~ and A an a ~ n e diagram of type Dn,I~6, ET, E8 and let M C m o d A be indecomposable such that E x t ~ ( M , M ) ___ k. Then H I ( A [ M ] ) = 0 and H2(A[M]) ~- k. Note
that such modules exist. This example includes the canonical tubular algebras of type I)4,~
of Ringel [Ri].
126
REFERENCES
[BLS] Bautista, R.; Larrion, F.; Salmeron, L., On simply connected algebras,, J. London Math. Soc. 27
(1983), 212-220.
[BrG] Bretscher, O.; Gabriel, P., The standard form of a representation-finite algebra, Bull. Soc. Math.
France 111 (1983), 21-40.
[C1] Cibils, C., Cohomologie de Hochschild d'alg~bres de dimension finie, preprint.
[C2] Cibils, C., $~nilpotent and rigidfinite-dimensional algebras, J. London Math. Soc. 36 (1987), 211-218.
[C3] Cibils, C., Cohomology of incidence algebras and simplicial complexes, preprint.
[C4] Cibils, C., Hochschild homology of an algebra whose quiver has no oriented cycles, Springer Lecture
Notes, Heidelberg 1177 (1986), 55-59.
ICE] Cartan, H.; Eilenberg, S., Homological Algebra, Princeton University Press (1956).
[CPS] Cline, E.; Parshall, B.; Scott, L., Derived categories and Morita theory, 3. Algebra 104 (1986),
397-409.
[Gal] Gabriel, P., Finite representation type is open, Springer Lecture Notes, Heidelberg 488 (1975),
132-155.
[Ga2] Gabriel, P., Auslander-Reiten sequences and representation-finite algebras, Springer Lecture Notes,
Heidelberg 831 (1980), 1-71.
[Gel Gerstenhaber, M., On the deformations of rings and algebras, Ann. of Math. 79 (1964), 59-103.
[GS] Gerstenhaber, M.; Schack, S.P., Simplieial homology is Hochschild cohomology, J. Pure and Appl.
algebra 30 (1983), 143-156.
[Half Happel, D., On the derived category oI a finite-dimensional algebra, Comment. Math. Helv. 62
(1987), 339-389.
[Ha2] Happel, D., LMS, Cambridge University Press 119 (1988), "Triangulated categories in the representation theory of finite--dimensionalalgebras,".
[HS] Happel, D.; Schaps, M., Deformations of tilting modules, preprint.
[Ho] Hochschild, G., On the cohomology groups of an associative algebra, Ann. of Math. 46 (1946), 58-67.
[Ri] Ringel, C.M., Tame algebras and integral quadratic forms, Springer Lecture Notes, Heidelberg 1099
(1984).
[V] Verdier, J.L., CatEgories d~riv~es, ~ta~ O, Springer Lecture Notes, Heidelberg 569 (1977), 262-311.
Fakult£t f/ir Mathematik, Universit~.t Bielefeld, Postfach 8640, 4800 Bielefeld 1, FRG

Download Report