Stable auto-equivalences for local symmetric algebras

Abstract

Abstract. We construct nontrivial auto-equivalences of stable module categories for elementary, local symmetric algebras over a field k. These auto-equivalences are modeled after the spherical twists of Seidel and Thomas and the P n -twists of Huybrechts and Thomas, which yield auto-equivalences of the derived category of coherent sheaves on a variety. For group algebras of p-groups in characteristic p we recover many of the auto-equivalences corresponding to endo-trivial modules. We also obtain analogous auto-equivalences for local algebras of dihedral and semi-dihedral type, which are not group algebras.

Some algebras that are not silting connected

“…The examples of spherical objects presented in this article were not discovered randomly, but rather correspond naturally to certain spherical stable twists introduced in [6]. While this correspondence is not necessary in the above exposition, it does present an alternative means of computing the actions of the spherical twist functors and also illustrates how further examples may be found.…”

“…When n = 2q is even, A is the Beilinson algebra (see [4]) of the dihedral algebra Λ = k x, y /(x 2 , y 2 , (xy) q − (yx) q ) with the usual grading that places x and y in degree 1. This connection provides another way to see that E is a spherical object and to study the action of the spherical twist Φ E using results of [6,7]. We will elaborate in the Appendix.…”

“…We denote the corresponding spherical twist in gr-Λ by Φ T . In [6], we defined a spherical stable twist σ y which is an auto-equivalence of the (ungraded) stable category mod-Λ. As both Φ T and σ y are defined using cones of right add(T )-approximations, it is clear that their actions agree on graded Λ-modules.…”

“…Likewise, the spherical object αE corresponds to T ′ = Λ/xΛ, and the spherical twist Φ T ′ on gr-Λ can be viewed as a graded version of the spherical stable twist σ x on mod-Λ. Now, as σ x σ y coincides with τ −1 on objects [6,Example 7.1], it follows that Φ ′ T Φ T also coincides with τ −1 on objects in gr-Λ. Carrying this information back to D b (A), we see that Φ αE Φ E also coincides with τ −1 on objects.…”

Deformations of complexes for finite dimensional algebras

“…Note that not every stable equivalence of Morita type between self-injective algebras is induced by a derived equivalence (see, for example, [13] and its references). Therefore, we next show that an arbitrary stable equivalence of Morita type between self-injective algebras preserves versal deformation rings.…”