Invariance of generalized Reynolds ideals under derived equivalence Alexander Zimmermann

Let k be an algebraically closed field of characteristic p > 0 and let A be a symmetric finite dimensional k-algebra with symmetrising bilinear form ( , ). Let KA be the k-vector space generated by the subset {xy −yx| x, y ∈ A} of A. This space was defined and used by R. Brauer in [1].

Külshammer defined in [3] and [4] for any integer n the spaces

T_{n}(A) := {x ∈ A | x^{p^{n}} ∈ KA}

and T_{n}(A)^{⊥} the orthogonal space to T_{n}(A) with respect to the symmetrising form ( , ). Then, T_{n}(A)^{⊥} is an ideal of the centre Z(A) of A.

Z(A) = T_{0}(A)^{⊥} ⊇ T_{1}(A)^{⊥} ⊇ T_{2}(A)^{⊥} ⊇ ⋯ ⊇ ⋂_{n∈ℕ} T_{n}(A)^{⊥} = soc(A) ∩ Z(A)

Héthelyi et al. show in [2] that Z_{0}A ⊆ (T_{1}A^{⊥})^{2} ⊆ HA, where HA is the Higman ideal, that is the image of the trace map of A, and where Z_{0}A is the sum of the centres of those blocks of A which are simple algebras. They show that for odd p the left inclusion is an equality, whereas for p = 2 one gets Z_{0}A = (T_{1}A^{⊥})^{3} = (T_{1}A^{⊥}) · (T_{2}A^{⊥}). Many more interesting properties of these ideals are given there.

The authors of [2] show that the ideals are invariant under Morita equivalence in the obvious sense and they ask if for derived equivalent symmetric algebras A and B there is an isomorphism φ : Z(A) ⟶ Z(B) with φ(T_{n}(A)^{⊥}) = T_{n}(B)^{⊥} for all n ∈ ℕ.

Let B be a k-algebra. By Rickard’s theory [5] given an equivalence D^{b}(A) ≃ D^{b}(B) as triangulated categories there is a complex X in D^{b}(B ⊗_{k} A^{op}) so that

X ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B)

is an equivalence, called “of standard type”. Now, for a symmetric algebra A, given such an equivalence the algebra B is symmetric as well again by [5], or in a more general context by [6]. Then, replacing X by a suitable isomorphic copy consisting of a complex formed of left and right projective A-modules,

X ⊗_{A} − ⊗_{A} Hom_{k}(X, k) : D^{b}(A ⊗_{k} A^{op}) ⟶ D^{b}(B ⊗_{k} B^{op})

is an equivalence. Moreover, this equivalence maps the bimodule A to B and therefore induces an isomorphism

HH^{n}(A) = Hom_{D^{b}(A⊗_{k}A^{op})}(A, A[n]) ≃ Hom_{D^{b}(B⊗_{k}B^{op})}(B, B[n]) = HH^{n}(B)

between the degree n Hochschild cohomology of A and B. Now, observe that HH^{0}(A) = Z(A) and HH^{0}(B) = Z(B).

Theorem. [7] Let k be an algebraically closed field of characteristic p > 0 and let A and B be finite dimensional k-algebras. Then, the isomorphism φ : ZA ⟶ ZB between the centres ZA of A and ZB of B induced by an equivalence D^{b}(A) ≃ D^{b}(B) of standard type has the property φ(T_{n}A^{⊥}) = T_{n}B^{⊥} for all positive integers n ∈ ℕ.

Remark: As an application of the theorem it is possible to distinguish the derived categories of certain algebras arising as blocks of group rings of tame representation type, which could not be distinguished otherwise. This will be subsequent joint work with Thorsten Holm.

The proof of the theorem uses first that the ideals T_{n}(A)^{⊥} are images of mappings ζ_{n} defined by the property

(ζ_{n}(x), y)^{p^{n}} = (x, y^{p^{n}}) ∀x ∈ Z(A)∀y ∈ A/KA .

Then, we study in detail the mapping ζ_{n} as a composition of mappings

Hom_{A⊗_{k}A^{op}}(A, A) ⟶ Hom_{A⊗_{k}A^{op}}(A, Hom_{k}(A, k))
⟶^{ψ} Hom_{A⊗_{k}A^{op}}(A, Hom_{k}(A, k))
⟶ Hom_{A⊗_{k}A^{op}}(A, A).

Here ψ is the composition of the n fold p-power mapping and the inverse of the Frobenius mapping. It is then possible to study the behaviour under a derived equivalence and this discussion gives the statement.

References

[1] Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. Zeitschr. 63 406-444 (1956)

[2] László Héthelyi, Erszébet Horváth, Burkhard Külshammer and John Murray, Central ideals and Cartan invariants of symmetric algebras, to appear in J. Alg.

[3] Burkhard Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, II, III and IV, J. Alg. 72 (1981) 1-7; 75 (1982) 59-69; 88 (1984) 279-291; 93 (1985) 310-323

[4] Burkhard Külshammer, Group-theoretical descriptions of ring theoretical invariants of group algebras, Progress in Mathematics 95 (1991) 425-441.

[5] Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991) 37–48.

[6] Alexander Zimmermann, Tilted symmetric orders are symmetric orders, Arch. Math. 73 (1999) 15-17

[7] Alexander Zimmermann, Invariance of generalised Reynolds ideals under derived equivalences, to appear in Mathematical proceedings of the Royal Irish Academy http://www.ria.ie/publications/journals/procai/index.html