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Register a new userInvariance of the restricted $p$-power map on integrable derivations under stable equivalences
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Authors
Lleonard Rubio y Degrassi
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We show that the $p$-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic $p$ commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the $p$-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.
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We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
It is well-known that derived equivalences preserve tensor products and trivial extensions. We disprove both constructions for stable equivalences of Morita type.
This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Gradel, and Wied Pakusa, published at ICALP 2019 and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences $equiv^text{IM}_{k,Q}$ on certain variants of Cai-Furer-Immerman (CFI) structures. These $equiv^text{IM}_{k,Q}$-equivalences, for a natural number $k$ and a set of primes $Q$, refine the well-known Weisfeiler-Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs $G equiv^text{IM}_{k,Q} H$ cannot be distinguished by iterative refinements of equivalences on $k$-tuples defined via linear operators on vector spaces over fields of characteristic $p in Q$. In the first paper it has been shown that for a prime $q otin Q$, the $equiv^text{IM}_{k,Q}$ equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field $mathbb{F}_q$. In the second paper, a similar but not identical construction for CFI-structures over the rings $mathbb{Z}_{2^i}$ has been shown to be indistinguishable with respect to $equiv^text{IM}_{k,{2}}$. Together with earlier work on rank logic, this second result suffices to separate rank logic from polynomial time. We show here that the two approaches can be unified to prove that CFI-structures over the rings $mathbb{Z}_{2^i}$ are indistinguishable with respect to $equiv^text{IM}_{k,mathbb{P}}$, for the set $mathbb{P}$ of all primes. This implies the following two results. 1. There is no fixed $k$ such that the invertible-map equivalence $equiv^text{IM}_{k,mathbb{P}}$ coincides with isomorphism on all finite graphs. 2. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.
We study resource similarity and resource bisimilarity -- congruent restrictions of the bisimulation equivalence for the (P,P)-class of Process Rewrite Systems (PRS). Both these equivalences coincide with the bisimulation equivalence for (1,P)-subclass of (P,P)-PRS, which is known to be decidable. While it has been shown in the literature that resource similarity is undecidable for (P,P)-PRS, decidability of resource bisimilarity for (P,P)-PRS remained an open question. In this paper, we present an algorithm for checking resource bisimilarity for (P,P)-PRS. We show that although both resource similarity and resource bisimilarity are congruences and have a finite semi-linear basis, only the latter is decidable.
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