Conditional closure of stable equivalences of Morita type under tensor products, and completeness of the syzygy-algebra invariant $B(\Omega_{C^e};C)$

Rickard (1998) asked whether stable equivalences of Morita type between finite-dimensional algebras are compatible with tensor products, in analogy with his theorem that derived equivalences are closed under tensor products. Mroczyńska and Pogorzały (2015) studied the functor-level form of this question — given stable equivalences of Morita type between $A$ and $E$ and between $B$ and $F$, does the tensor-induced functor $M\otimes_A-\otimes_BS$ give a stable equivalence between $A\otimes_KB^{op}$ and $E\otimes_KF^{op}$ — reducing it to smaller subcategories for self-injective algebras and constructing new invariants for arbitrary finite-dimensional algebras. The unconditional question has since been answered negatively in full generality: Liu–Zhou–Zimmermann (2017) for general algebras, and Bouc–Zimmermann (2017, published online in 2016) already for symmetric algebras arising from principal blocks of the finite projective special unitary groups $PSU(3,p^r)$. This record formulates the surviving open core: (1) an exact characterization — necessary and sufficient, effectively checkable conditions on the input bimodules — under which the tensor-induced functor is a stable equivalence of Morita type, consistent with the derived-induced sufficient condition and the known counterexamples; and (2) the completeness question for the Mroczyńska–Pogorzały invariant algebra $B(\Omega_{C^e};C)$: does it classify stable equivalence classes of Morita type, or does it admit a nontrivial kernel realizable by explicit finite-dimensional examples?

Background

Throughout, $K$ is a field and all algebras are finite-dimensional $K$-algebras; $\mathrm{mod}\,A$ denotes the category of finite-dimensional left $A$-modules. The stable module category $\underline{\mathrm{mod}}A$ is the quotient of $\mathrm{mod}\,A$ by the ideal of morphisms that factor through a projective module, and two algebras are called stably equivalent when their stable module categories are equivalent — a classical notion going back to Auslander and Reiten for which no Morita-type generator description is known (by contrast with Morita equivalence, an equivalence of module categories induced by tensoring with a suitable bimodule). In practice one works with a much more tractable subclass introduced by Broué in the study of blocks of group algebras (a block is an indecomposable two-sided direct summand of a group algebra; the principal block is the block containing the trivial module): $A$ and $E$ are stably equivalent of Morita type if there are an $E$-$A$-bimodule $M$ and an $A$-$E$-bimodule $N$, each projective as a left and as a right module, such that $M\otimes_AN\cong E\oplus Q$ and $N\otimes_EM\cong A\oplus P$ as bimodules for some projective bimodules $Q$ and $P$; then $M\otimes_A-$ and $N\otimes_E-$ are mutually inverse stable equivalences. An algebra is self-injective if every projective module is injective; group algebras, their block algebras, and symmetric algebras (algebras isomorphic to their $K$-dual as bimodules) are of this kind. A derived equivalence is an equivalence of bounded derived module categories; Rickard proved that these are generated by tilting complexes (bounded complexes of projective modules that generate the homotopy category and have no shifted self-extensions) and that a derived equivalence between self-injective algebras induces a stable equivalence of Morita type. He also proved that derived equivalences are closed under tensor products: if $A$ and $B$ are derived equivalent, and $C$ and $D$ are derived equivalent, then $A\otimes_KC$ and $B\otimes_KD$ are derived equivalent. In his 1998 survey (Question 3.8) Rickard asked whether the analogous statement holds for stable equivalences of Morita type between indecomposable, non-semisimple self-injective algebras — the restriction is needed since $A$ and $A\times K$ are trivially stably equivalent of Morita type. The question is motivated by modular representation theory, the representation theory of finite groups over fields of positive characteristic, where the Green correspondence — a standard correspondence between indecomposable modules of a block and indecomposable modules of its Brauer correspondent, the associated block of the normalizer of a defect group (a maximal $p$-subgroup controlling the block) — supplies stable equivalences of Morita type between a block and its Brauer correspondent for which no derived equivalence is known. Rickard himself expected the answer to be negative in general, noting that the simplest potential counterexamples were already quite complicated; he also observed that when both stable equivalences come from derived equivalences the answer is positive. Mroczyńska and Pogorzały (2015) studied the functor-level form of this question: given stable equivalences of Morita type defined by bimodules $M$ (between $A$ and $E$) and $S$ (between $B$ and $F$), decide whether $M\otimes_A-\otimes_BS$ is a stable equivalence between $A\otimes_KB^{op}$ and $E\otimes_KF^{op}$. They reduced the problem to smaller subcategories in the self-injective case and, for arbitrary finite-dimensional algebras, constructed new invariants of stable equivalence of Morita type: to an algebra $C$ one associates the graded algebra $B(\Omega_{C^e};C)=\bigoplus_{i\ge0}\underline{\underline{\mathrm{Hom}}}_{C^e}(\Omega_{C^e}^i(C),C)$, where $\Omega_{C^e}^i(C)$ is the $i$-th syzygy of $C$ viewed as a module over its enveloping algebra $C^e=C\otimes_KC^{op}$ (so $\Omega_{C^e}^0(C)=C$, and $\Omega_{C^e}^i(C)$ is the kernel of a projective cover of $\Omega_{C^e}^{i-1}(C)$) and $\underline{\underline{\mathrm{Hom}}}_{C^e}(X,Y)$ denotes $\mathrm{Hom}_{C^e}(X,Y)$ modulo the maps that factor through finite direct sums of tensor-generated bimodules ${}_CU\otimes_KV_C$; the product of homogeneous elements $f\in\underline{\underline{\mathrm{Hom}}}_{C^e}(\Omega_{C^e}^i(C),C)$ and $g\in\underline{\underline{\mathrm{Hom}}}_{C^e}(\Omega_{C^e}^j(C),C)$ is $f\cdot g=f\circ\Omega_{C^e}^i(g)$, under which $B(\Omega_{C^e};C)$ is an associative unital $K$-algebra with unit the class of $\mathrm{id}_C$. The construction is invariant under stable equivalence of Morita type. The state of the art is that the unconditional question is settled negatively: Liu, Zhou and Zimmermann (2017) disproved it for general finite-dimensional algebras using triangular matrix algebras and trivial extensions (the algebras $A\ltimes D(A)$ obtained by adjoining the $K$-dual $D(A)=\mathrm{Hom}_K(A,K)$ of $A$ as a square-zero ideal), and Bouc and Zimmermann (2017) disproved Rickard's original self-injective version using the principal block of $\overline{\mathbb{F}}_pPSU(3,p^r)$ — the group algebra, over an algebraic closure $\overline{\mathbb{F}}_p$ of the field with $p$ elements, of the finite projective special unitary group $PSU(3,p^r)$ — and its Brauer correspondent: these two blocks are stably equivalent of Morita type, yet after tensoring both with $\overline{\mathbb{F}}_p[X]/X^p$ the resulting symmetric algebras are not stably equivalent of Morita type for $p^r\in\{3,4,5,8\}$, as their centres (which here coincide with the stable centres, the centre modulo the ideal of endomorphisms factoring through a projective bimodule) have radical series of different shape — the descending chains of powers of the Jacobson radical (the intersection of all maximal left ideals) have layers of different dimensions — verified by computer algebra. What remains, and what this record states, is the conditional classification problem and the completeness question for the new invariant.

Problem Statement

Let $K$ be a field and let $A,E,B,F$ be finite-dimensional $K$-algebras. Suppose the $E$-$A$-bimodule $M$ (with inverse $A$-$E$-bimodule $N$) and the $B$-$F$-bimodule $S$ (with inverse $F$-$B$-bimodule $T$) define stable equivalences of Morita type between $A$ and $E$ and between $B$ and $F$, respectively: all four bimodules are projective as left and as right modules, and $M\otimes_AN\cong E\oplus Q$ and $N\otimes_EM\cong A\oplus P$ as bimodules for some projective bimodules $Q$ and $P$, with analogous decompositions $S\otimes_FT\cong B\oplus P'$ and $T\otimes_BS\cong F\oplus Q'$.

(1) Find necessary and sufficient, effectively checkable conditions on the input bimodules $M$ and $S$ (and their inverses) under which the tensor-induced functor

$$M\otimes_A-\otimes_BS:\mathrm{mod}(A\otimes_KB^{op})\longrightarrow\mathrm{mod}(E\otimes_KF^{op})$$

is a stable equivalence of Morita type between $A\otimes_KB^{op}$ and $E\otimes_KF^{op}$. The conditions must be consistent with the known boundary of the problem: they must be satisfied when both input stable equivalences are induced by derived equivalences between self-injective algebras, that is, by equivalences of bounded derived module categories (where the conclusion then holds, by Rickard's tensor-closure theorem for derived equivalences together with the fact that a derived equivalence between self-injective algebras induces a stable equivalence of Morita type), and they must fail in the counterexamples of Bouc–Zimmermann (the principal blocks of the group algebras $\overline{\mathbb{F}}_pPSU(3,p^r)$ and their Brauer correspondents, each tensored with the truncated polynomial algebra $\overline{\mathbb{F}}_p[X]/X^p$) and of Liu–Zhou–Zimmermann (the construction sending a self-injective algebra $A$ to the triangular matrix algebra $T_2(A)\cong A\otimes_KT_2(K)$, where $T_2(K)$ is the algebra of $2\times2$ lower triangular matrices over $K$).

(2) Determine whether the invariant algebra $B(\Omega_{C^e};C)$ constructed by Mroczyńska and Pogorzały for an arbitrary finite-dimensional algebra $C$ from the syzygies of $C$ over its enveloping algebra $C^e=C\otimes_KC^{op}$ (defined in Background above) is complete for stable equivalence of Morita type: is every pair of algebras with isomorphic invariants linked by a stable equivalence of Morita type, or does the invariant admit a nontrivial kernel, realizable by an explicit pair of finite-dimensional algebras that are not stably equivalent of Morita type yet have isomorphic invariants?

The two components share one objective: a complete description of what the tensor-product construction and its associated syzygy-algebra invariant detect about stable equivalences of Morita type — component (1) fixes the exact closure boundary of the construction, and component (2) decides whether the accompanying invariant can see that boundary at all. The unconditional form of (1) ('does the tensor-induced functor always give a stable equivalence?') is not part of this problem: it has a negative answer in the literature, in the self-injective case and in general.

References

1. Karolina Mroczyńska and Zygmunt Pogorzały (2015). On Stable Equivalence of Tensor Products of Algebras. Communications in Algebra 43(3), 916–934. DOI: 10.1080/00927872.2013.852007, https://doi.org/10.1080/00927872.2013.852007 (Zbl 1318.16012)
2. Jeremy Rickard (1998). Some recent advances in modular representation theory. In: Algebras and Modules I (Trondheim, 1996), CMS Conference Proceedings 23, American Mathematical Society, Providence, RI, pp. 157–178. ISBN 0-8218-0850-8. Zbl 0914.20010, https://zbmath.org/0914.20010
3. Yuming Liu, Guodong Zhou and Alexander Zimmermann (2017). Stable equivalences of Morita type do not preserve tensor products and trivial extensions of algebras. Proceedings of the American Mathematical Society 145, 1881–1890. DOI: 10.1090/proc/13448, https://doi.org/10.1090/proc/13448 (preprint arXiv:1409.1821, https://arxiv.org/abs/1409.1821)
4. Serge Bouc and Alexander Zimmermann (2017). On a Question of Rickard on Tensor Products of Stably Equivalent Algebras. Experimental Mathematics 26(1), 31–44, published online 5 July 2016. DOI: 10.1080/10586458.2015.1107869, https://doi.org/10.1080/10586458.2015.1107869 (preprint arXiv:1501.01461, https://arxiv.org/abs/1501.01461)
5. Michel Broué (1994). Equivalences of Blocks of Group Algebras. In: Finite Dimensional Algebras and Related Topics, V. Dlab and L. L. Scott (eds.), Kluwer Academic Publishers, Dordrecht, pp. 1–26. DOI: 10.1007/978-94-017-1556-0_1, https://doi.org/10.1007/978-94-017-1556-0_1
6. Zygmunt Pogorzały (2003). A new invariant of stable equivalences of Morita type. Proceedings of the American Mathematical Society 131, 343–349. DOI: 10.1090/S0002-9939-02-06553-X, https://doi.org/10.1090/s0002-9939-02-06553-x
7. Alexander Zimmermann (2019). On equivalences between categories of representations. AIP Conference Proceedings 2192, 020002. DOI: 10.1063/1.5139120, https://doi.org/10.1063/1.5139120