Abstract:
Stable equivalences of Morita type preserve many interesting properties and is proved to be the appropriate concept to study for equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology.

Abstract:
It is well-known that derived equivalences preserve tensor products and trivial extensions. We disprove both constructions for stable equivalences of Morita type.

Abstract:
For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this paper, we first show that each derived equivalence $F$ between the derived categories of Artin algebras $A$ and $B$ arises naturally a functor $\bar{F}$ between their stable module categories, which can be used to compare certain homological dimensions of $A$ with that of $B$; and then we give a sufficient condition for the functor $\bar{F}$ to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classic result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras which are both derived-equivalent and stably equivalent of Morita type, thus they share many common invariants.

Abstract:
Motivated by understanding the Brou\'e's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants of both derived equivalences and stable equivalences. Especially, Auslander-Reiten conjecrure is true for stable equivalences of Morita type between Frobenius-finite algebras without semisimple direct summands. Examples of such a class of algebras are abundant, including Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we further show that, for a Nakayama-stable idempotent element $e$ in an algebra $A$ over an arbitrary field, each tilting complex over $eAe$ can be extended to a tilting complex over $A$ that induces an almost $\nu$-stable derived equivalence studied in the first paper of this series. Moreover, we demonstrate that our techniques are applicable to verify the Brou\'{e}'s abelian defect group conjecture for several cases mentioned by Okuyama.

Abstract:
In this note we study Morita contexts and Galois extensions for corings. For a coring $\QTR{cal}{C}$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\QTR{cal}{M}^{\QTR{cal}{C}}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called \QTR{em}{cleft}$C$\QTR{em}{-Galois extensions} over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) $H$-Galois extensions.

Abstract:
We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*-algebras which behave like the proper actions on C*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on induced algebras yield isomorphic equivalences. A similar analysis of the symmetric imprimitivity theorem for graph algebras gives new information about the amenability of actions on graph algebras.

Abstract:
Let $A$ and $B$ be finite-dimensional $k$-algebras over a field $k$ such that $A/\rad(A)$ and $B/\rad(B)$ are separable. In this note, we consider how to transfer a stable equivalence of Morita type between $A$ and $B$ to that between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and in $B$, respectively. In particular, if the Auslander algebras of two representation-finite algebras $A$ and $B$ are stably equivalent of Morita type, then $A$ and $B$ themselves are stably equivalent of Morita type. Thus, combining a result with Liu and Xi, we see that two representation-finite algebras $A$ and $B$ over a perfect field are stably equivalent of Morita type if and only if their Auslander algebras are stably equivalent of Morita type. Moreover, since stable equivalence of Morita type preserves $n$-cluster tilting modules, we extend this result to $n$-representation-finite algebras and $n$-Auslander algebras studied by Iyama.

Abstract:
In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $\Phi$ and any generator $X$ of the $A$-module category, the $\Phi$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and $B$, respectively. Consequently, for self-injective algebras $A$ and $B$ over a field without semisimple direct summands, and for any $A$-module $X$ and $B$-module $Y$, if the $\Phi$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one finite admissible set $\Phi$, then so are the $\Psi$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $\Psi$. Moreover, two representation-finite algebras over a field without semisimple direct summands are stably equivalent of Morita type if and only if so are their Auslander algebras. As another consequence, we construct an infinite family of algebras of the same dimension and the same dominant dimension such that they are pairwise derived equivalent, but not stably equivalent of Morita type. This answers a question by Thorsten Holm.

Abstract:
Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

Abstract:
If two cluster-tilting objects of an acyclic cluster category are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent [Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a simple module". This result has been generalised by D. Yang, using a result of P-G. Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi--Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly-Morita equivalent and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by results of Buan-Marsh, we also describe our result in terms of localisations.