HOCHSCHILD COHOMOLOGY OF STRATIFIED ALGEBRAS

HIROSHI NAGASE

1. Introduction

When studying Hochschild cohomology it is natural to try relating cohomology of an algebra B to that of an ’easier’ or ’smaller’ algebra A. One such situation is that of B being a one-point extension of A, which has been studied by Happel[5]. More recently, Happel’s long exact sequence has been generalized to triangular matrix algebras, for example by Michelena and Platzeck [6], C.Cibils, E.Marcos, M.J.Redondo and A.Solotar [2] and E.L.Green and O.Solberg [4].

We would like to suggest to try further generalizing these results. Natural generalizations of directed or triangular algebras are stratified algebras (when just keeping good homological connections between B and its quotient A).

2. Preliminaries

Let k be a field. Throughout this paper, all algebras are finite dimensional k-algebras and all modules are left modules unless otherwise stated. For any algebra A, we denote by A^{e} the enveloping algebra A ⊗_{k} A^{op}. We prepare the following lemma for the next section.

Lemma 1. Let X be a A-B-bimodule, Y a B-C-bimodule and Z a A-C-bimodule. We have the following isomorphisms:

(1) If Tor_{i}^{B}(X, Y) = 0 and Ext_{C}^{i}(Y, Z) = 0 for all i ≥ 1 then, for any n ≥ 0,

Ext_{A−C}^{n}(X ⊗_{B} Y, Z) ≅ Ext_{A−B}^{n}(X, Hom_{C}(Y, Z)).

(2) If Tor_{i}^{B}(X, Y) = 0 and Ext_{A}^{i}(X, Z) = 0 for all i ≥ 1 then, for any n ≥ 0,

Ext_{A−C}^{n}(X ⊗_{B} Y, Z) ≅ Ext_{B−C}^{n}(Y, Hom_{A}(X, Z)).

Proof. See Cartan-Eilenberg’s book [1].

Lemma 2. Let I be an ideal of an algebra B. If Ext_{B^{e}}^{i}(I, B/I) = 0 for all i ≥ 0, then we have the following two long exact sequences:

(1) ⋯ → Ext_{B^{e}}^{n}(B, I) → Ext_{B^{e}}^{n}(B, B) → Ext_{B^{e}}^{n}(B/I, B/I) → Ext_{B^{e}}^{n+1}(B, I) → ⋯;

(2) ⋯ → Ext_{B^{e}}^{n}(B/I, I) → Ext_{B^{e}}^{n}(B, B) → Ext_{B^{e}}^{n}(B/I, B/I) ⊕ Ext_{B^{e}}^{n}(I, I) → Ext_{B^{e}}^{n+1}(B/I, I) → · · ·.

The detailed version of this paper will be submitted for publication elsewhere.

Proof. We denote B/I by A, the inclusion I → B by f and the surjection B → A by g. We consider the following commutative diagram with exact rows and columns:

Ext_{B^{e}}^{i−2}(I, A) ⟶ Ext_{B^{e}}^{i−1}(I, I) ⟶^{f_{i−1}^{I}} Ext_{B^{e}}^{i−1}(I, B) ⟶ Ext_{B^{e}}^{i−1}(I, A)
↓  ↓^{δ_{I}^{i−1}}  ↓  ↓
Ext_{B^{e}}^{i−1}(A, A) ⟶^{δ_{i−1}^{A}} Ext_{B^{e}}^{i}(A, I) ⟶ Ext_{B^{e}}^{i}(A, B) ⟶ Ext_{B^{e}}^{i}(A, A)
↓^{g_{A}^{i−1}}  ↓^{g_{I}^{i}}  ↓  ↓^{g_{A}^{i}}
Ext_{B^{e}}^{i−1}(B, A) ⟶^{δ_{i−1}^{B}} Ext_{B^{e}}^{i}(B, I) ⟶^{f_{i}^{B}} Ext_{B^{e}}^{i}(B, B) ⟶^{g_{i}^{B}} Ext_{B^{e}}^{i}(B, A)
↓  ↓  ↓^{f_{B}^{i}}  ↓
Ext_{B^{e}}^{i−1}(I, A) ⟶ Ext_{B^{e}}^{i}(I, I) ⟶^{f_{i}^{I}} Ext_{B^{e}}^{i}(I, B) ⟶ Ext_{B^{e}}^{i}(I, A).

Since Ext^{n}(I, A) = 0 for all n ≥ 0, we have that f_{n}^{I} and g_{A}^{n} are isomorphic for all n ≥ 0. It is not diffecult to show that the followig two sequences are exact:

Ext_{B^{e}}^{i−1}(A, A) ⟶^{δ_{i−1}^{B}g_{A}^{i−1}} Ext_{B^{e}}^{i}(B, I) ⟶^{f_{i}^{B}} Ext_{B^{e}}^{i}(B, B) ⟶^{(g_{A}^{i})^{−1}g_{i}^{B}} Ext_{B^{e}}^{i}(A, A)

and

Ext_{B^{e}}^{i−1}(A, A) ⊕ Ext_{B^{e}}^{i−1}(I, I) ⟶^{(−δ_{i−1}^{A},δ_{I}^{i−1})} Ext_{B^{e}}^{i}(A, I) ⟶^{f_{i}^{B}g_{I}^{i}} Ext_{B^{e}}^{i}(B, B)

\underrightarrow{\binom{(g^{i}_{A})^{−1}g^{B}_{i}}{(f^{I}_{i})^{−1}f^{i}_{B}}} Ext_{B^{e}}^{i}(A, A) ⊕ Ext_{B^{e}}^{i}(I, I) ⟶^{(−δ_{i}^{A},δ_{I}^{i})} Ext_{B^{e}}^{i+1}(A, I).

3. Stratifying ideals

Definition 3 (Cline, Parshall and Scott [3], 2.1.1 and 2.1.2). Let B be a finite dimensional algebra and e = e^{2} an idempotent. The two-sided ideal J = BeB generated by e is called a stratifying ideal if the following equivalent conditions (A) and (B) are satisfied: (A) (a) The multiplication map Be ⊗_{eBe} eB → BeB is an isomorphism.

(b) For all n > 0: Tor_{n}^{eBe}(Be, eB) = 0. (B) The epimorphism B → A := B/BeB induces isomorphisms

Ext_{A}^{∗}(X, Y) ≃ Ext_{B}^{∗}(X, Y)

for all A-modules X and Y.

Example 4. (1) Any ideal generated by a central idempotent is a stratifying ideal.

(2) If an algebra A has an idempotent e such that eA(1−e) = 0, then AeA and A(1−e)A are both stratifying ideals, namely, triangular matrix algebras have stratifying ideals.

(3) Heredity ideals (used to define quasi-hereditary algebras) are examples of stratifying ideals.

Proposition 5. Let B be an algebra with a stratifying ideal BeB. We denote by A the factor algebra B/BeB. For any i ≥ 0 and finite dimensional A^{e}-module M, the induced morphism g_{M}^{i} : Ext_{A^{e}}^{i}(A, M) → Ext_{B^{e}}^{i}(B, M) is isomorphic.

Proof. It is enough to show that the induced morphisms g_{M}^{i} is isomorphic for any simple A^{e}-module M. For any A-modules X and Y, there exists the following commutative diagram:

Ext_{A^{e}}^{i}(A, Hom_{k}(X, Y)) ⟶^{∼} Ext_{A}^{i}(X, Y)
↓^{g_{Hom_{k}(X,Y )}^{i}}  ↓
Ext_{B^{e}}^{i}(B, Hom_{k}(X, Y)) ⟶^{∼} Ext_{B}^{i}(X, Y),

where these isomorphisms in rows are induced from Lemma1. Any simple A^{e}-module has the form of Hom_{k}(X, Y) for some simple A-modules X and Y. Hence, by using the condition (B) of stratifying ideals, it is shown that g_{M}^{i} is isomorphic for any simple A^{e}-module M.

Proposition 6. Let B be an algebra with a stratifying ideal BeB. We denote by A the factor algebra B/BeB. For any i ≥ 0, the following hold:

(1) Ext_{B^{e}}^{i}(BeB, A) = 0;

(2) Ext_{B^{e}}^{i}(BeB, BeB) ≅ Ext_{eBe^{e}}^{i}(eBe, eBe);

(3) Ext_{B^{e}}^{i}(A, A) ≅ Ext_{A^{e}}^{i}(A, A).

Proof. By Lemma1 and the condition (A) of stratifying ideals, for any B^{e}-module X, we have that

Ext_{B^{e}}^{i}(BeB, X) ≅ Ext_{B^{e}}^{i}(Be ⊗_{eBe} eB, X)
≅ Ext_{B−eBe}^{i}(Be, Hom_{B}(eB, X))
≅ Ext_{B−eBe}^{i}(Be ⊗_{eBe} eBe, Xe)
≅ Ext_{eBe^{e}}^{i}(eBe, Hom_{B}(Be, Xe))
≅ Ext_{eBe^{e}}^{i}(eBe, eXe).

Hence (1) and (2) hold.

By Proposition5, Ext_{A^{e}}^{i}(A, A) ≅ Ext_{B^{e}}^{i}(B, A). By (1) above, Ext_{B^{e}}^{i}(B, A) ≅ Ext_{B^{e}}^{i}(A, A). Hence (3) holds.

Theorem 7. Let B be an algebra with a stratifying ideal BeB. We denote by A the factor algebra B/BeB. There exist the following two long exact sequences:

(1) ⋯ → Ext_{B^{e}}^{n}(B, BeB) → HH^{n}(B) → HH^{n}(A) → ⋯;

(2) ⋯ → Ext_{B^{e}}^{n}(A, BeB) → HH^{n}(B) → HH^{n}(A) ⊕ HH^{n}(eBe) → · · · .

Proof. By Lemma2 and Proposition6 (1), we have the following two long exact sequence:

⋯ → Ext_{B^{e}}^{n}(B, BeB) → Ext_{B^{e}}^{n}(B, B) → Ext_{B^{e}}^{n}(A, A) → ⋯;

and

⋯ → Ext_{B^{e}}^{n}(A, BeB) → Ext_{B^{e}}^{n}(B, B) → Ext_{B^{e}}^{n}(A, A) ⊕ Ext_{B^{e}}^{n}(BeB, BeB) → · · · . –35–

By Proposition6 (2) and (3), we have the following two long exact sequences:

⋯ → Ext_{B^{e}}^{n}(B, BeB) → Ext_{B^{e}}^{n}(B, B) → Ext_{A^{e}}^{n}(A, A) → ⋯;

and

⋯ → Ext_{B^{e}}^{n}(A, BeB) → Ext_{B^{e}}^{n}(B, B) → Ext_{A^{e}}^{n}(A, A) ⊕ Ext_{eBe^{e}}^{n}(eBe, eBe) → · · · .

References

1. H. Cartan and S. Eilenberg: Homological algebra, Princeton Landmarks in Mathematics (1973). Originally pubulished 1956.

2. C.Cibils, E.Marcos, M.J.Redondo and A.Solotar: Cohomology of split algebras and of trivial extensions, preprint, available from Cibils’ homepage.

3. E.Cline, B.Parshall and L.Scott: Stratifying endomorphism algebras, Memoir AMS 591 (1996), i-vi, 1-119.

4. E.L.Green and O.Solberg: Hochschild cohomology rings and triangular rings, preprint.

5. D. Happel, Hochschild cohomology of finite dimensional algebras , Lecture Note in Math. 1404 (1989), 108–112.

6. S.Michelena and M.I.Platzeck: Hochschild cohomology of triangular matrix algebras, J. Algebra, 233 (2000), 502-525.

Nara National College of Technology 22 Yata-cho, Yamatokoriyama, Nara, 639-1080, JAPAN

  E-mail address: nagase@libe.nara-k.ac.jp