Support $\tau$ -tilting posets and Hochschild reconstruction for matrix centralizer algebras

Jiangsheng Hu¹, Yu-Zhe Liu² and Tiwei Zhao^3,‡

¹School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China
E-mail: hujs@hznu.edu.cn

²School of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China
E-mail: liuyz@gzu.edu.cn

³School of Artificial Intelligence, Jianghan University, Wuhan 430056, P. R. China
E-mail: tiweizhao@jhun.edu.cn

^‡Corresponding author

Abstract: Let $R$ be a field, and let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$ -algebra of Loewy length $\ell$ with $\operatorname{rad}\mathcal{O}=(\pi)$ . For a nonempty set $P=\{p_{1}<\cdots<p_{s}\}\subseteq\{1,\ldots,\ell\}$ , set $\Lambda_{\mathcal{O}}(P):=\End_{\mathcal{O}}(\bigoplus_{p\in P}\mathcal{O}/(\pi^{p}))$ . We prove that the support $\tau$ -tilting poset of $\Lambda_{\mathcal{O}}(P)$ is isomorphic to the left weak order on $\Sigma_{s+1}$ and hence recovers precisely $s$ . We also show that $Z(\Lambda_{\mathcal{O}}(P))\simeq\mathcal{O}/(\pi^{p_{s}})$ and, as a module over this center, $HH_{0}(\Lambda_{\mathcal{O}}(P))\simeq\bigoplus_{i=1}^{s}\mathcal{O}/(\pi^{g_{i}})$ , where $g_{1}=p_{1}$ and $g_{i}=p_{i}-p_{i-1}$ for $i\geq 2$ . Applied blockwise, these formulas compute the corresponding invariants of matrix centralizer algebras over arbitrary fields. Finally, a polynomial primary block is Morita equivalent to a split string algebra if and only if its defining irreducible polynomial is linear and its exponent set is $\{p\}$ or $\{p,p+1\}$ . It is Morita equivalent to a split gentle algebra if and only if the polynomial is linear and the exponent set is $\{1\}$ , $\{2\}$ , or $\{1,2\}$ . These criteria yield Morita reconstruction within the corresponding classes.

2020 Mathematics Subject Classification: Primary 16G10, 16E40, 15A27; Secondary 16E35, 15A20, 16D90.

Keywords: matrix centralizer algebra; support $\tau$ -tilting theory; elementary divisor; central reduction; degree-zero Hochschild homology; string algebra; gentle algebra.

1 Introduction

Reconstruction from tilting-theoretic posets asks which algebraic data are determined by an order. Happel–Unger reconstructed certain acyclic path algebras from their tilting posets [HU09]. Aihara–Kase and Kase obtained support $\tau$ -tilting reconstruction results for other classes, including tree-quiver and weak-order cases [AK18, Kas17, Kas24]. Support $\tau$ -tilting modules correspond to functorially finite torsion classes and basic two-term silting complexes [AIR14, Theorems 2.7 and 3.2]. Therefore, their posets provide a common reconstruction invariant. We study endomorphism algebras of sums of cyclic modules over local principal ideal algebras and their matrix-centralizer realizations. Our invariants do not detect elementary-divisor multiplicities; they recover the number of distinct exponents and the gap multiset described below.

Let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$ -algebra, with $\operatorname{rad}\mathcal{O}=(\pi)$ and $\operatorname{LL}(\mathcal{O})=\ell$ . Let $P=\{p_{1}<\cdots<p_{s}\}\subseteq\{1,\ldots,\ell\}$ be nonempty, and put $\Lambda_{\mathcal{O}}(P):=\End_{\mathcal{O}}(\bigoplus_{i=1}^{s}\mathcal{O}/(\pi^{p_{i}}))$ . Put $g_{1}=p_{1}$ , $g_{i}=p_{i}-p_{i-1}$ for $i\geq 2$ , and $\mathcal{O}_{P}:=\mathcal{O}/(\pi^{p_{s}})$ . For an $R$ -algebra $A$ , write $\mathrm{s}\tau\text{-}\mathrm{tilt}A$ for the poset of isomorphism classes of basic support $\tau$ -tilting $A$ -modules, and put $HH^{0}(A):=Z(A)$ and $HH_{0}(A):=A/[A,A]$ , where $HH_{0}(A)$ is endowed with its natural $Z(A)$ -module structure. For $q\geq 2$ , let $\operatorname{Weak}(\Sigma_{q})$ denote the left weak order on the symmetric group $\Sigma_{q}$ .

The central-reduction theorem of Eisele–Janssens–Raedschelders is stated for basic algebras over algebraically closed fields [EJR18, Theorem 11]. We prove the form needed over an arbitrary field. We use their Hom-space calculation and the arbitrary-field uniqueness of $g$ -vectors for $\tau$ -rigid pairs [AIR14, Theorem 5.5]. Iyama–Zhang obtained the weak-order formula for the Auslander algebra corresponding to $P=\{1,\ldots,s\}$ [IZ20, Corollary 1.4]. Thus the following theorem extends their formula to every exponent set in the Loewy range. It also gives the complementary gap reconstruction.

Theorem 1.1.

Under the above assumptions, there are isomorphisms

$\displaystyle\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{\mathcal{O}}(P)$	$\displaystyle\simeq\operatorname{Weak}(\Sigma_{s+1})$	$\displaystyle\text{as posets},$	(1.1)
$\displaystyle Z(\Lambda_{\mathcal{O}}(P))$	$\displaystyle\simeq\mathcal{O}_{P}$	$\displaystyle\text{as $R$-algebras},$
$\displaystyle HH_{0}(\Lambda_{\mathcal{O}}(P))$	$\displaystyle\simeq\bigoplus_{i=1}^{s}\mathcal{O}_{P}/(\pi^{g_{i}})$	$\displaystyle\text{as $\mathcal{O}_{P}$-modules}.$

Consequently, the abstract support $\tau$ -tilting poset recovers $s=|P|$ but no finer exponent data, while the algebra–module pair $(Z(\Lambda_{\mathcal{O}}(P)),HH_{0}(\Lambda_{\mathcal{O}}(P)))$ determines the gap multiset $\{\!\{g_{1},\ldots,g_{s}\}\!\}$ .

The support $\tau$ -tilting statements in Theorem 1.1 are proved in Corollary 3.2. The center and Hochschild statements are proved in Theorem 3.4.

Quotienting by the central endomorphism induced by $\pi$ gives the doubled line on $s$ vertices in which every immediate reversal is zero, so the first isomorphism follows from central reduction. The center is computed using the faithful longest summand, whereas the description of $HH_{0}$ follows from the diagonal-commutator calculation in Lemma 3.3. Neither argument requires a coefficient field in $\mathcal{O}$ , so both are valid over an arbitrary field. In particular, if two local endomorphism algebras of this form are $R$ -linearly derived equivalent, then their centers are isomorphic, their gap multisets agree under this isomorphism, and their support $\tau$ -tilting posets are isomorphic; see Corollary 3.5.

Let $M_{n}(R)$ be the full $n\times n$ matrix algebra over $R$ . For $c\in M_{n}(R)$ , we call $S_{n}(c,R):=\{a\in M_{n}(R)\mid ac=ca\}$ the matrix centralizer algebra of $c$ . We apply (1.1) through the identification $S_{n}(c,R)\simeq\End_{R[x]}(R^{n})$ , where $x$ acts on $R^{n}$ as $c$ . Brenner proved that every finite-dimensional algebra over a field is isomorphic to the common centralizer of two matrices [Bre72, Lemma 2]. A direct classification of such common centralizers is generally inaccessible. Therefore, centralizers of a single matrix form a natural testing ground for reconstruction, and we restrict throughout to this class. Let $\operatorname{Irr}(c)$ be the set of monic irreducible factors of the minimal polynomial of $c$ . For $f\in\operatorname{Irr}(c)$ , let $P_{c}(f):=\{a\geq 1\mid f^{a}\text{ is an elementary divisor of }c\}$ , $r_{c}(f):=\max P_{c}(f)$ , and $\kappa_{c}(f):=|P_{c}(f)|$ , with repetitions omitted. Set $U_{c}(f):=R[x]/(f^{r_{c}(f)})$ and $T_{R}(c):=\{\!\{\kappa_{c}(f)\mid f\in\operatorname{Irr}(c)\}\!\}$ . Each primary block of $S_{n}(c,R)$ is Morita equivalent to an algebra of the form considered in Theorem 1.1. Morita invariance of support $\tau$ -tilting posets therefore gives the following exact reconstruction statement.

Theorem 1.2.

There is an isomorphism of posets

\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\prod_{f\in\operatorname{Irr}(c)}\operatorname{Weak}(\Sigma_{\kappa_{c}(f)+1}).

In particular, $S_{n}(c,R)$ is $\tau$ -tilting finite and $|\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)|=\prod_{f\in\operatorname{Irr}(c)}(\kappa_{c}(f)+1)!$ . If $c\in M_{n}(R)$ and $d\in M_{m}(R)$ , then the following are equivalent:

(1)

$T_{R}(c)=T_{R}(d)$ ;
(2)

$\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}S_{m}(d,R)$ as posets;
(3)

$\operatorname{tors}S_{n}(c,R)\simeq\operatorname{tors}S_{m}(d,R)$ as lattices, where $\operatorname{tors}A$ denotes the lattice of torsion classes.

Since every matrix centralizer is $\tau$ -tilting finite, every torsion class is functorially finite [DIJ19, Theorem 3.8]; hence (2) and (3) are equivalent. The connected components of the atom-interaction graph of the product poset have sizes $\kappa_{c}(f)$ . Thus the poset determines exactly $T_{R}(c)$ .

The center recovers complementary block data. By the classical double-centralizer theorem [Jac85, Chapter III], $Z(S_{n}(c,R))=R[c]$ as subalgebras of $M_{n}(R)$ , and $R[c]\simeq\prod_{f\in\operatorname{Irr}(c)}U_{c}(f)$ as $R$ -algebras. This classical formula, also used in [XZ21, XZ22], identifies the local coefficient algebras acting on $HH_{0}$ . For a finite set $P=\{p_{1}<\cdots<p_{s}\}$ , write $H(P):=\{\!\{g_{1},\dots,g_{s}\}\!\}$ , where $g_{1}:=p_{1}$ and $g_{i}:=p_{i}-p_{i-1}$ for $i\geq 2$ . For monic irreducible $f\in R[x]$ , set $U_{f}:=R[x]/(f^{p_{s}})$ and $\Lambda_{f}(P):=\End_{U_{f}}(\bigoplus_{p\in P}U_{f}/\operatorname{rad}^{p}U_{f})$ . Morita invariance of the center, of $HH_{0}$ , and of the center action on $HH_{0}$ now gives the following blockwise statement from the last two isomorphisms in (1.1).

Theorem 1.3.

There is an isomorphism of $Z(S_{n}(c,R))$ -modules

HH_{0}(S_{n}(c,R))\cong\bigoplus_{f\in\operatorname{Irr}(c)}\ \bigoplus_{g\in H(P_{c}(f))}U_{c}(f)/\operatorname{rad}^{g}U_{c}(f).

Here $Z(S_{n}(c,R))\simeq\prod_{f\in\operatorname{Irr}(c)}U_{c}(f)$ acts on each summand through the projection to its $f$ -factor. The algebra–module pair $(HH^{0},HH_{0})$ recovers the gap multiset of each primary block. Moreover, $\dim_{R}HH_{0}(S_{n}(c,R))=\sum_{f\in\operatorname{Irr}(c)}\deg(f)r_{c}(f)=\dim_{R}Z(S_{n}(c,R))$ . If $S_{n}(c,R)$ and $S_{m}(d,R)$ are derived equivalent as $R$ -algebras, then there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that, for every $f\in\operatorname{Irr}(c)$ ,

U_{c}(f)\simeq U_{d}(\sigma(f))\quad\text{as $R$-algebras},\qquad H(P_{c}(f))=H(P_{d}(\sigma(f))).

Since finitely generated $U_{c}(f)$ -modules have unique cyclic decompositions, the module structure determines the individual gaps. Its $R$ -dimension determines only their sum. The cap-product action of $HH^{0}$ on $HH_{0}$ is derived invariant [AK17, Theorem 2.2]. Therefore, the algebra–module pair shows directly that derived equivalence preserves the local coefficient algebras and gap multisets. Conversely, Li–Xi proved that a blockwise bijection preserving the local coefficient algebras and gap multisets implies derived equivalence [LX25, Theorem 1.1]. Thus their classification supplies the reverse implication. Our contribution is the intrinsic realization of their gap data and the direct forward implication.

Subsection 4.4 establishes the strict implication chain among seven reconstruction relations. Here T records the support $\tau$ -tilting type, TZ adds the center separately, bTZ pairs these data blockwise, and gTZ further records $K_{0}(D_{\mathrm{sg}}(A))$ , where $D_{\mathrm{sg}}(A):=D^{\mathrm{b}}(\operatorname{mod}A)/K^{\mathrm{b}}(\operatorname{proj}A)$ denotes the singularity category of $A$ . Thus gTZ retains only its Grothendieck group; it does not encode singular equivalence. The Li–Xi relations M, AD and D correspond to Morita, almost $\nu$ -stable derived and derived equivalence [LX25, Theorem 1.1]. Their definitions give $\mathrm{M}\Rightarrow\mathrm{AD}\Rightarrow\mathrm{D}$ . Theorem 4.15 proves that

\mathrm{D}\Rightarrow\mathrm{gTZ}\Rightarrow\mathrm{bTZ}\Rightarrow\mathrm{TZ}\Rightarrow\mathrm{T}

is strict over every field. Explicit examples after that theorem show that $\mathrm{M}\Rightarrow\mathrm{AD}$ and $\mathrm{AD}\Rightarrow\mathrm{D}$ are also strict.

We finally determine when these data recover the Morita class. We use the split bound-quiver notions of string and gentle algebras [BR87, AS87]. Over a perfect field, the Morita-string condition is strictly more restrictive than representation-finiteness. For example, if $r\geq 4$ , then $P=\{1,r\}$ gives a representation-finite block that is not Morita-string; see [LX25, Lemma 3.3] and [DM06, Theorem 2.1(i)]. Chan–Marczinzik classify representation-finite gendo-symmetric biserial algebras over algebraically closed fields by Brauer-tree data [CM19, Theorem 3.9]. Here we give an exponent-set criterion for split string and gentle primary centralizer blocks over an arbitrary field.

Theorem 1.4.

Let $f\in R[x]$ be monic and irreducible, and let $P$ be a nonempty finite set of positive integers.

(1)

The primary block $\Lambda_{f}(P)$ is Morita equivalent to a split string $R$ -algebra if and only if $f$ is linear and $P=\{p\}$ or $P=\{p,p+1\}$ for some $p\geq 1$ .
(2)

The primary block $\Lambda_{f}(P)$ is Morita equivalent to a split gentle $R$ -algebra if and only if $f$ is linear and $P\in\{\{1\},\{2\},\{1,2\}\}$ .
(3)

For matrix centralizers within the Morita-string class,

$\mathrm{M}\Longleftrightarrow\mathrm{AD}\Longleftrightarrow\mathrm{D}\Longleftrightarrow\mathrm{gTZ}\Longleftrightarrow\mathrm{bTZ},$

whereas TZ is strictly weaker than bTZ and T is strictly weaker than TZ. Within the Morita-gentle class, TZ determines the Morita class.

Parts (1) and (2) of Theorem 1.4 are proved in Theorem 5.8. Part (3) is proved in Theorem 5.10.

For a Morita-string block, the center determines $r=\max P$ , while the poset determines $|P|$ . Hence the paired block data distinguish $P=\{r\}$ from $P=\{r-1,r\}$ , and bTZ determines the Morita class. For a Morita-gentle block, $P\in\{\{1\},\{2\},\{1,2\}\}$ ; consequently the separate center and poset data already determine the Morita class.

The arbitrary-field formulation is essential. Scalar extension can split a primary label and therefore change the support $\tau$ -tilting poset. If $f$ is inseparable, then $R[x]/(f^{a})\twoheadrightarrow R[x]/(f)$ need not admit an $R$ -algebra section. Hence the proofs use neither scalar extension nor coefficient fields. Nonsplit modulated string structures over the residue field lie outside the scope of the split classification.

The paper is organized as follows. Section 2 proves the arbitrary-field central reduction used in the proof of Theorem 1.1. Section 3 completes the proof of Theorem 1.1 by establishing the local weak-order and Hochschild formulas. Section 4 proves Theorems 1.2 and 1.3 and compares the resulting reconstruction relations. Section 5 proves Theorem 1.4, namely the string and gentle rigidity criteria and their Morita-reconstruction consequences.

Throughout this paper, $R$ denotes a field. By an algebra we always mean a finite-dimensional associative algebra over $R$ with identity, and modules mean finitely generated left modules. All algebra homomorphisms are assumed to preserve identities. Unless stated otherwise, isomorphisms and equivalences of $R$ -algebras are understood to be $R$ -linear.

2 Central reduction for support $\tau$ -tilting posets

This section recalls the support $\tau$ -tilting conventions used throughout the paper and proves the central-reduction result needed in Section 3. This central-reduction theorem allows us to quotient by centrally generated radical ideals without changing the support $\tau$ -tilting poset. Together with the uniform quotient in Section 3, it reduces the local centralizer problem to a weak-order calculation.

2.1 Support $\tau$ -tilting posets

For an algebra $A$ , let $A_{b}$ stand for a basic algebra in the Morita class of $A$ . By $\operatorname{tors}A$ we denote the lattice of torsion classes in $A\text{-}\mathrm{mod}$ , and by $\operatorname{brick}A$ the set of isomorphism classes of $A$ -modules whose endomorphism rings are division rings. We write $\mathrm{s}\tau\text{-}\mathrm{tilt}A$ for the set of isomorphism classes of basic support $\tau$ -tilting $A$ -modules, ordered by $M\leq N$ if and only if $\operatorname{Fac}M\subseteq\operatorname{Fac}N$ . Here $\operatorname{Fac}M$ is the full subcategory of factor modules of finite direct sums of copies of $M$ . Write $\proj A$ for the category of finitely generated projective $A$ -modules. A two-term complex $C=(P^{-1}\longrightarrow P^{0})\in K^{\mathrm{b}}(\proj A)$ is presilting if $\Hom_{K^{\mathrm{b}}(\proj A)}(C,C[i])=0$ for every $i>0$ , and it is silting if, in addition, $\operatorname{thick}(C)=K^{\mathrm{b}}(\proj A)$ . Its $g$ -vector is $[P^{0}]-[P^{-1}]\in K_{0}(\proj A)$ .

The original support $\tau$ -tilting correspondences are due to Adachi–Iyama–Reiten [AIR14]. For the arbitrary-field form used here, see [DIJ19, Theorem 2.3 and Corollary 2.8]. Since we work with left modules, we apply the right-module statements to $A^{\mathrm{op}}$ . They identify $\mathrm{s}\tau\text{-}\mathrm{tilt}A$ with the poset of basic two-term silting complexes in $K^{\mathrm{b}}(\proj A)$ , and the order defined above agrees with the silting order.

For the lattice of all torsion classes, see [DIRRT23]. The quotient reduction below should not be confused with Jasso’s reduction at a fixed $\tau$ -rigid object [Ja15].

For a positive integer $s$ , let $\operatorname{Weak}(\Sigma_{s+1})$ denote the left weak order on the symmetric group $\Sigma_{s+1}$ ; we use the standard Coxeter group conventions of [BB05]. Note that the left and right weak orders are isomorphic by inversion. Thus this choice does not affect the poset-isomorphism statements below.

2.2 Central reduction

Theorem 11 of [EJR18] is stated under the standing assumptions that the ground field is algebraically closed and that the algebra is basic. Its Hom-space comparison uses only projectivity, centrality, and nilpotence. Thus this comparison remains valid over an arbitrary field. The remaining injectivity follows from the arbitrary-field uniqueness of $g$ -vectors for $\tau$ -rigid pairs [AIR14, Theorem 5.5]. Therefore, we obtain the following version of the reduction theorem.

Proposition 2.1.

Let $A$ be a finite-dimensional algebra over an arbitrary field, and put $J_{A}:=\bigl(Z(A)\cap\operatorname{rad}A\bigr)A$ . If $I$ is an ideal of $A$ such that $I\subseteq J_{A}$ , then reduction modulo $I$ induces an isomorphism of posets $\mathrm{s}\tau\text{-}\mathrm{tilt}A\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}(A/I)$ .

Proof.

Since $I\subseteq J_{A}\subseteq\operatorname{rad}A$ , reduction induces a bijection between the isomorphism classes of indecomposable projective $A$ -modules and those of $A/I$ . Thus it identifies their Grothendieck groups and preserves $g$ -vectors.

We first consider $I=(z)$ , where $z\in Z(A)\cap\operatorname{rad}A$ and $z^{2}=0$ , and put $\bar{A}:=A/(z)$ . Since we use left modules, we apply the right-module argument in [EJR18, Theorem 11] to $A^{\mathrm{op}}$ . This is valid because $Z(A^{\mathrm{op}})=Z(A)$ and $\operatorname{rad}(A^{\mathrm{op}})=\operatorname{rad}A$ . Let $C$ and $D$ be two-term complexes of projective $A$ -modules, and put $\bar{C}:=\bar{A}\otimes_{A}C$ and $\bar{D}:=\bar{A}\otimes_{A}D$ . The Hom-space calculation in [EJR18, Theorem 11] uses only projectivity, the centrality of $z$ , and the equality $z^{2}=0$ . Therefore, it is valid over an arbitrary field and gives

\Hom_{K^{\mathrm{b}}(\proj A)}(C,D[1])=0\iff\Hom_{K^{\mathrm{b}}(\proj\bar{A})}(\bar{C},\bar{D}[1])=0.

Every projective $\bar{A}$ -module lifts to a projective $A$ -module. Moreover, morphisms between projective $\bar{A}$ -modules lift to morphisms between their projective lifts. Hence every two-term complex over $\bar{A}$ lifts to a two-term complex over $A$ . If the complex over $\bar{A}$ is presilting, then the displayed equivalence shows that its lift is presilting. Thus reduction is surjective on the isomorphism classes of two-term presilting complexes.

By [AIR14, Theorem 5.5], $\tau$ -rigid pairs over an arbitrary field are determined by their $g$ -vectors. Via the correspondence between support $\tau$ -rigid pairs and two-term presilting complexes, the same is true for two-term presilting complexes. Since reduction preserves $g$ -vectors, it is also injective. Thus reduction induces a $g$ -vector-preserving bijection on two-term presilting complexes. It also preserves and reflects direct sums. Therefore, it induces a bijection on basic two-term silting complexes.

We next let $z\in Z(A)\cap\operatorname{rad}A$ be arbitrary. Since $A$ is finite-dimensional, $z$ is nilpotent. Choose $N$ such that $z^{N}=0$ . For $2\leq r\leq N$ , the kernel of $A/(z^{r})\longrightarrow A/(z^{r-1})$ is generated by the central radical element $z^{r-1}+(z^{r}),$ whose square is zero. Therefore, the square-zero case applies successively to $A=A/(z^{N})\longrightarrow A/(z^{N-1})\longrightarrow\cdots\longrightarrow A/(z)$ . Hence reduction modulo $(z)$ has the same properties.

Finally, choose $z_{1},\ldots,z_{m}\in Z(A)\cap\operatorname{rad}A$ such that $J_{A}=(z_{1},\ldots,z_{m})$ . Applying the principal case successively to the images of the $z_{i}$ gives a bijection from the two-term presilting complexes over $A$ to those over $A/J_{A}$ .

Since $I\subseteq\operatorname{rad}A$ , we have $\operatorname{rad}(A/I)=\operatorname{rad}A/I$ . Thus the images of the $z_{i}$ in $A/I$ are central radical elements and generate $J_{A}/I$ . Applying the same argument to $A/I$ gives a bijection from the two-term presilting complexes over $A/I$ to those over $(A/I)/(J_{A}/I)=A/J_{A}$ . The reduction maps factor as $A\longrightarrow A/I\longrightarrow A/J_{A}.$ Since the composite map and the second map are bijective, the first map is bijective as well. The same factorization shows that it preserves and reflects the relevant Hom-vanishing conditions. Thus it preserves and reflects the silting order. The correspondence between two-term silting complexes and support $\tau$ -tilting modules now gives $\mathrm{s}\tau\text{-}\mathrm{tilt}A\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}(A/I)$ . ∎

3 Reconstruction over local principal ideal algebras

The central-reduction theorem of Section 2 provides the mechanism for removing centrally generated radical ideals without changing the support $\tau$ -tilting poset. We now apply that mechanism to endomorphism algebras of sums of cyclic modules over a commutative local principal ideal algebra and prove Theorem 1.1.

Adachi classified support $\tau$ -tilting modules over Nakayama algebras [Ada16], while we consider endomorphism algebras of sums of uniserial modules. Iyama–Zhang treated the saturated split case, namely the Auslander algebra corresponding to $P=\{1,\ldots,s\}$ [IZ20]; Kase gave more general quiver-theoretic criteria for weak-order support $\tau$ -tilting posets of finite-dimensional bound-quiver algebras [Kas17]. Our contribution here is an explicit central quotient, valid for arbitrary exponent sets and residue fields, together with the complementary recovery of the gap multiset from the center action on $HH_{0}$ .

Let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$ -algebra. Write its maximal ideal as $(\pi)$ , its Loewy length as $\operatorname{LL}(\mathcal{O})=\ell$ , and its residue field as $k:=\mathcal{O}/(\pi)$ . Thus $\mathcal{O}=(\pi^{0})\supset(\pi)\supset\cdots\supset(\pi^{\ell})=0$ and $\pi^{\ell-1}\neq 0$ . Let $P=\{p_{1}<\cdots<p_{s}\}\subseteq\{1,\ldots,\ell\}$ be nonempty, and put $M(i):=\mathcal{O}/(\pi^{p_{i}})$ , $M_{P}:=\bigoplus_{i=1}^{s}M(i)$ , and $\Lambda_{\mathcal{O}}(P):=\End_{\mathcal{O}}(M_{P})$ . Let $z_{\pi}$ act as multiplication by $\pi$ on every summand of $M_{P}$ . Since scalar multiplication commutes with every $\mathcal{O}$ -linear endomorphism, $z_{\pi}$ is central. Moreover, $z_{\pi}^{\ell}=0$ , and hence $(1-az_{\pi})^{-1}=\sum_{r=0}^{\ell-1}(az_{\pi})^{r}$ for every $a\in\Lambda_{\mathcal{O}}(P)$ . Therefore, $z_{\pi}\in Z(\Lambda_{\mathcal{O}}(P))\cap\operatorname{rad}\Lambda_{\mathcal{O}}(P)$ .

Let $Q_{s}$ be the doubled line

1\underset{\beta_{1}}{\overset{\alpha_{1}}{\rightleftarrows}}2\ \rightleftarrows\ \cdots\ \rightleftarrows\ s-1\underset{\beta_{s-1}}{\overset{\alpha_{s-1}}{\rightleftarrows}}s,

where $\alpha_{i}\colon i\to i+1$ and $\beta_{i}\colon i+1\to i$ . We compose maps from right to left; thus $\beta_{i}\alpha_{i}$ means first $\alpha_{i}$ , then $\beta_{i}$ . Set $I_{s}:=\langle\beta_{i}\alpha_{i},\alpha_{i}\beta_{i}\mid 1\leq i<s\rangle$ and $B_{s}(k):=kQ_{s}/I_{s}$ . Every nonmonotone path contains an immediate reversal, while for each ordered pair of vertices there is a unique monotone path. Hence these monotone paths form a $k$ -basis of $B_{s}(k)$ and $\dim_{k}B_{s}(k)=s^{2}.$ For $s=1$ , this means $B_{1}(k)=k$ .

The following theorem shows that the quotient depends only on the residue field and the number of summands.

Theorem 3.1.

The quotient $\Lambda_{\mathcal{O}}(P)/(z_{\pi})$ carries a canonical $k$ -algebra structure for which $k$ acts centrally, and

\Lambda_{\mathcal{O}}(P)/(z_{\pi})\simeq B_{s}(k)

as $k$ -algebras. In particular, the quotient depends on $\mathcal{O}$ and $P$ only through $k$ and $s=|P|$ .

Proof.

Put $A:=\Lambda_{\mathcal{O}}(P)$ , $J:=(z_{\pi})$ and $\overline{A}:=A/J$ . Scalar multiplication gives a central homomorphism $\mathcal{O}\to Z(A)$ . Since multiplication by $\pi$ becomes zero in $\overline{A}$ , this homomorphism induces a unital map $k=\mathcal{O}/(\pi)\longrightarrow Z(\overline{A}).$ Since $k$ is a field, the map is injective. Thus $\overline{A}$ has the canonical $k$ -algebra structure induced by this central embedding.

For $1\leq i,j\leq s$ , define $u_{ji}\colon M(i)\to M(j)$ by $u_{ji}(1):=\pi^{(p_{j}-p_{i})_{+}}$ , where $(a)_{+}:=\max\{a,0\}$ . Since every ideal of $\mathcal{O}$ is a power of $(\pi)$ , evaluation at $1$ gives $\Hom_{\mathcal{O}}(M(i),M(j))=\mathcal{O}u_{ji}\simeq\mathcal{O}/(\pi^{\min\{p_{i},p_{j}\}})$ . Let $e_{i}$ be the projection onto $M(i)$ followed by its inclusion into $M_{P}$ . Since $z_{\pi}$ is central, $e_{j}Je_{i}=\pi\Hom_{\mathcal{O}}(M(i),M(j))$ . Therefore, the single element $\overline{u}_{ji}$ forms a $k$ -basis of the $(j,i)$ -corner of $\overline{A}$ . Consequently, $\overline{A}=\bigoplus_{1\leq i,j\leq s}k\overline{u}_{ji}$ and $\dim_{k}\overline{A}=s^{2}$ .

For $i,j,k\in\{1,\ldots,s\}$ , direct evaluation at $1$ gives $u_{kj}u_{ji}=\pi^{\epsilon(i,j,k)}u_{ki}$ , where $\epsilon(i,j,k):=(p_{j}-p_{i})_{+}+(p_{k}-p_{j})_{+}-(p_{k}-p_{i})_{+}$ . Since $p_{1}<\cdots<p_{s}$ , we have $\epsilon(i,j,k)=0$ if and only if $i\leq j\leq k$ or $i\geq j\geq k$ . Otherwise, $\epsilon(i,j,k)\geq 1$ . Thus, modulo $(z_{\pi})$ , monotone compositions survive and every composition containing a change of direction vanishes.

Sending $e_{i}\mapsto\overline{u}_{ii}$ , $\alpha_{i}\mapsto\overline{u}_{i+1,i}$ , and $\beta_{i}\mapsto\overline{u}_{i,i+1}$ defines a homomorphism $kQ_{s}\to\overline{A}$ . The preceding composition rule shows that it factors through $B_{s}(k)$ . Moreover, every $\overline{u}_{ji}$ is the image of the monotone path from $i$ to $j$ . Hence the induced map $B_{s}(k)\to\overline{A}$ is surjective. Since both algebras have dimension $s^{2}$ over $k$ , it is an isomorphism. ∎

Together with Proposition 2.1, this uniform presentation yields the following corollary. It proves the first isomorphism in (1.1) of Theorem 1.1 and the assertion there that the abstract support $\tau$ -tilting poset recovers $s=|P|$ .

Corollary 3.2.

Let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$ -algebra of Loewy length $\ell$ , and let $P=\{p_{1}<\cdots<p_{s}\}\subseteq\{1,\ldots,\ell\}$ be nonempty. Then

\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{\mathcal{O}}(P)\simeq\operatorname{Weak}(\Sigma_{s+1}).

The weak order $\operatorname{Weak}(\Sigma_{s+1})$ has exactly $s$ atoms, namely the simple transpositions. Hence the abstract poset determines $s=|P|$ . The displayed formula also shows that the poset depends only on $s$ ; in particular, it is independent of the residue field, the Loewy length of $\mathcal{O}$ and the values of the integers in $P$ .

Proof.

By Proposition 2.1 and Theorem 3.1, $\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{\mathcal{O}}(P)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}B_{s}(k)$ . Let $A_{s}:=k[t]/(t^{s})$ , $G_{s}:=\bigoplus_{i=1}^{s}k[t]/(t^{i})$ , and $\Gamma_{s}(k):=\End_{A_{s}}(G_{s})$ . The modules $k[t]/(t^{i})$ , $1\leq i\leq s$ , are precisely the indecomposable $A_{s}$ -modules. Thus $\Gamma_{s}(k)$ is the Auslander algebra of $A_{s}$ . Applying Proposition 2.1 and Theorem 3.1 again gives $\mathrm{s}\tau\text{-}\mathrm{tilt}B_{s}(k)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}\Gamma_{s}(k)$ .

Since $k$ is finite-dimensional over $R$ and acts centrally, the finite-dimensional module categories of $B_{s}(k)$ and $\Gamma_{s}(k)$ are unchanged if these algebras are viewed over $R$ rather than over $k$ . Iyama–Zhang work over an arbitrary field and use right modules. The $k$ -dual of each $k[t]/(t^{i})$ is isomorphic to itself. Hence duality gives $\Gamma_{s}(k)^{\mathrm{op}}\simeq\Gamma_{s}(k)$ , so their right-module result applies to our left-module convention. By [IZ20, Corollary 1.4(3),(4)], the generation order is anti-isomorphic to the left weak order. With our convention $M\leq N$ if and only if $\operatorname{Fac}M\subseteq\operatorname{Fac}N$ , this gives the opposite left weak order. Since $w\mapsto ww_{0}$ is an order-reversing bijection of the left weak order, where $w_{0}$ is the longest element, it follows that $\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{\mathcal{O}}(P)\simeq\operatorname{Weak}(\Sigma_{s+1})$ . ∎

It remains to prove the second and third isomorphisms in (1.1), together with the recovery of the gap multiset. We begin with the required diagonal-commutator calculation. Put $g_{1}:=p_{1}$ , $g_{i}:=p_{i}-p_{i-1}$ for $2\leq i\leq s$ , and $\mathcal{O}_{P}:=\mathcal{O}/(\pi^{p_{s}})$ . The algebra $\mathcal{O}_{P}$ acts faithfully on $M_{P}$ , and hence centrally on $\Lambda_{\mathcal{O}}(P)$ and on its cocenter.

We first identify the diagonal commutators.

Lemma 3.3.

Put $A:=\Lambda_{\mathcal{O}}(P)$ . Via scalar multiplication, identify $e_{i}Ae_{i}=\End_{\mathcal{O}}(\mathcal{O}/(\pi^{p_{i}}))$ with $\mathcal{O}/(\pi^{p_{i}})e_{i}$ , and set $D:=\bigoplus_{i=1}^{s}e_{i}Ae_{i}$ . Then the inclusion $D\hookrightarrow A$ induces an isomorphism of $\mathcal{O}_{P}$ -modules $D/(D\cap[A,A])\xrightarrow{\sim}HH_{0}(A)$ . Moreover, $D\cap[A,A]$ is generated as an $\mathcal{O}_{P}$ -module by the following elements, indexed by pairs $i<j$ . If $d_{ij}:=p_{j}-p_{i}$ , they are

\pi^{w}(e_{j}-e_{i})\quad(d_{ij}\leq w<p_{i}),\qquad\pi^{w}e_{j}\quad(\max\{d_{ij},p_{i}\}\leq w<p_{j}),

(3.1)

where an empty range is omitted.

Proof.

The Peirce decomposition gives $A=D\oplus\bigoplus_{i\neq j}e_{j}Ae_{i}$ . If $y\in e_{j}Ae_{i}$ and $i\neq j$ , then $e_{j}y=y$ and $ye_{j}=0$ . Thus $y=[e_{j},y]$ , so the entire off-diagonal summand lies in $[A,A]$ . It follows that projection onto $D$ induces the asserted isomorphism. The diagonal component of a commutator is a sum of commutators inside the diagonal corners and paired-corner commutators $xy-yx$ , where $x\in e_{j}Ae_{i}$ , $y\in e_{i}Ae_{j}$ , and $i<j$ . Conversely, every such paired-corner commutator belongs to $D\cap[A,A]$ . The diagonal corners are commutative under the scalar-multiplication identification above, so their internal commutators vanish. Hence $D\cap[A,A]$ is generated by the paired-corner commutators.

Fix $i<j$ and put $d:=d_{ij}$ . The $\mathcal{O}_{P}$ -modules $e_{j}Ae_{i}=\Hom_{\mathcal{O}}(M(i),M(j))$ and $e_{i}Ae_{j}=\Hom_{\mathcal{O}}(M(j),M(i))$ are cyclic: choose generators $u_{ji}$ with $u_{ji}(1)=\pi^{d}$ and the canonical projection $v_{ij}\colon M(j)\twoheadrightarrow M(i)$ . For $a,b\in\mathcal{O}_{P}$ , direct evaluation at $1$ gives $(au_{ji})(bv_{ij})=ab\pi^{d}e_{j}$ and $(bv_{ij})(au_{ji})=ab\pi^{d}e_{i}$ , where each equality is interpreted in its corresponding truncated diagonal corner. As $ab$ ranges through $\mathcal{O}_{P}$ (take $b=1$ ), the resulting commutators are generated by $\pi^{d+t}(e_{j}-e_{i}),\ 0\leq t<p_{i}.$ For $d+t<p_{i}$ both diagonal terms survive. For $d+t\geq p_{i}$ the $e_{i}$ -term is zero, while the $e_{j}$ -term survives until $d+t=d+p_{i}=p_{j}$ . Splitting the range accordingly gives exactly the two families in (3.1). Summing over all pairs $i<j$ proves the result. ∎

The preceding calculation yields the following Hochschild reconstruction formula. This theorem proves the second and third isomorphisms in (1.1) and the assertion in Theorem 1.1 that the algebra–module pair $(Z,HH_{0})$ recovers the gap multiset.

Theorem 3.4.

Let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$ -algebra of Loewy length $\ell$ , and let $P=\{p_{1}<\cdots<p_{s}\}\subseteq\{1,\ldots,\ell\}$ be nonempty. Then scalar multiplication induces an $R$ -algebra isomorphism $\mathcal{O}_{P}\xrightarrow{\sim}Z(\Lambda_{\mathcal{O}}(P))$ , and there is an isomorphism of $\mathcal{O}_{P}$ -modules

HH_{0}\bigl(\Lambda_{\mathcal{O}}(P)\bigr)\simeq\bigoplus_{i=1}^{s}\mathcal{O}_{P}/(\pi^{g_{i}}).

(3.2)

Consequently, the center-module structure of $HH_{0}(\Lambda_{\mathcal{O}}(P))$ determines $\{\!\{g_{1},\ldots,g_{s}\}\!\}$ .

Proof.

Since the longest summand $\mathcal{O}/(\pi^{p_{s}})$ is faithful over $\mathcal{O}_{P}$ , scalar multiplication embeds $\mathcal{O}_{P}$ into the center. Conversely, let $a$ be central and write $a_{ji}=e_{j}ae_{i}$ with respect to the summands of $M_{P}$ . Commutation with every $e_{i}$ gives $a_{ji}=0$ for $i\neq j$ . Write $a_{ii}$ as multiplication by $q_{i}\in\mathcal{O}/(\pi^{p_{i}})$ . Commuting with the canonical projection $M(j)\to M(i)$ for $i<j$ gives $q_{j}\equiv q_{i}\pmod{\pi^{p_{i}}}$ . Hence all $q_{i}$ are induced by the single element $q_{s}\in\mathcal{O}_{P}$ , proving the assertion about the center.

It remains to compute the cocenter. Via scalar multiplication, identify $\End_{\mathcal{O}}(M(i))$ with $\mathcal{O}/(\pi^{p_{i}})e_{i}$ , and put $Q:=\bigoplus_{i=1}^{s}\End_{\mathcal{O}}(M(i))$ . By Lemma 3.3, the cocenter is the quotient of $Q$ by the generators in (3.1), in addition to the relations $\pi^{p_{i}}e_{i}=0$ already present in $Q$ .

Put $h_{1}:=e_{1}$ and $h_{i}:=e_{i}-e_{i-1}$ for $i\geq 2$ . Since $g_{1}=p_{1}$ , we have $\pi^{g_{1}}h_{1}=0$ . If $i\geq 2$ and $g_{i}<p_{i-1}$ , then the first relation for the pair $(i-1,i)$ gives $\pi^{g_{i}}h_{i}=0$ . If $g_{i}\geq p_{i-1}$ , then the second relation gives $\pi^{g_{i}}e_{i}=0$ , while $\pi^{g_{i}}e_{i-1}=0$ already holds in $Q$ . Thus $\pi^{g_{i}}h_{i}=0$ for every $i$ . Moreover, $e_{i}=h_{1}+\cdots+h_{i}$ , so $u_{i}\mapsto h_{i}$ defines a surjection $\bigoplus_{i=1}^{s}\mathcal{O}_{P}/(\pi^{g_{i}})u_{i}\twoheadrightarrow HH_{0}(\Lambda_{\mathcal{O}}(P))$ . Conversely, the $\mathcal{O}_{P}$ -linear map $Q\to\bigoplus_{i=1}^{s}\mathcal{O}_{P}/(\pi^{g_{i}})u_{i}$ defined by $e_{i}\mapsto u_{1}+\cdots+u_{i}$ kills all defining relations. Since $p_{i}=g_{1}+\cdots+g_{i}$ , the element $\pi^{p_{i}}$ annihilates $u_{1},\ldots,u_{i}$ . Thus the map kills $\pi^{p_{i}}e_{i}$ .

Let $i<j$ . Since $d_{ij}=g_{i+1}+\cdots+g_{j},$ the inequality $w\geq d_{ij}$ implies that $\pi^{w}$ annihilates $u_{i+1},\ldots,u_{j}$ . Hence the map kills $\pi^{w}(e_{j}-e_{i})$ in the first family. If $w\geq\max\{d_{ij},p_{i}\}$ , then $\pi^{w}$ also annihilates $u_{1},\ldots,u_{i}$ . Therefore, it annihilates the image of $e_{j}$ and kills the second family. Thus the map factors through $HH_{0}(A)$ . The two induced maps are inverse on the generators. Hence (3.2) follows.

Finally, each $\mathcal{O}_{P}/(\pi^{a})$ is indecomposable because its endomorphism ring is local. Moreover, it has Loewy length $a$ . Thus two such modules are isomorphic only if their exponents are equal. Since finite-length $\mathcal{O}_{P}$ -modules satisfy the Krull–Schmidt theorem, the module in (3.2) determines the gap multiset. ∎

The following is an application of Theorem 3.4.

Corollary 3.5.

Let $\mathcal{O}$ and $\mathcal{O}^{\prime}$ be finite-dimensional commutative local principal ideal $R$ -algebras with $\operatorname{rad}\mathcal{O}=(\pi)$ and $\operatorname{rad}\mathcal{O}^{\prime}=(\pi^{\prime})$ , and let $P$ and $Q$ be nonempty exponent sets in their respective Loewy ranges. Put $A:=\Lambda_{\mathcal{O}}(P)$ and $B:=\Lambda_{\mathcal{O}^{\prime}}(Q)$ . If $A$ and $B$ are $R$ -linearly derived equivalent, then there exist an $R$ -algebra isomorphism $\varphi\colon Z(A)\to Z(B)$ and an $R$ -linear isomorphism $\psi\colon HH_{0}(A)\to HH_{0}(B)$ such that $\psi(zm)=\varphi(z)\psi(m)$ for $z\in Z(A)$ and $m\in HH_{0}(A)$ . Consequently, one has

\mathcal{O}/(\pi^{\max P})\simeq\mathcal{O}^{\prime}/(\pi^{\prime\max Q}),\qquad H(P)=H(Q),

where the second equality is equality of multisets of positive integers. Hence $|P|=|Q|$ and $\mathrm{s}\tau\text{-}\mathrm{tilt}A\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}B$ . Thus the support $\tau$ -tilting poset is a derived invariant within this local family.

Proof.

An $R$ -linear derived equivalence preserves $HH^{0}=Z$ , $HH_{0}$ , and the cap-product action of $HH^{0}$ on $HH_{0}$ [AK17, Theorem 2.2]. Hence it identifies the algebra–module pairs $(Z(A),HH_{0}(A))$ and $(Z(B),HH_{0}(B))$ . By Theorem 3.4 and the Krull–Schmidt argument in its proof, the centers are the displayed truncated coefficient algebras and $H(P)=H(Q)$ . Thus $|P|=|Q|$ , and Corollary 3.2 yields $\mathrm{s}\tau\text{-}\mathrm{tilt}A\simeq\operatorname{Weak}(\Sigma_{|P|+1})=\operatorname{Weak}(\Sigma_{|Q|+1})\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}B$ , as required. ∎

Thus the support $\tau$ -tilting poset retains only $|P|$ , whereas adjoining the center action on $HH_{0}(\Lambda_{\mathcal{O}}(P))$ also recovers the gap multiset.

We now specialize to a primary polynomial block. Let $f\in R[x]$ be monic and irreducible, and let $P=\{p_{1}<\cdots<p_{s}\}$ be a nonempty finite set of positive integers. Put $K_{f}:=R[x]/(f)$ , $\mathcal{O}_{f}:=R[x]/(f^{p_{s}})$ , and $\pi_{f}:=f+(f^{p_{s}})$ . Then $\mathcal{O}_{f}$ is a local principal ideal $R$ -algebra with residue field $K_{f}$ , and $\mathcal{O}_{f}/(\pi_{f}^{p_{i}})\simeq R[x]/(f^{p_{i}})$ as $R[x]$ -modules for $1\leq i\leq s$ . Define $\Lambda_{f}(P):=\End_{R[x]}(\bigoplus_{i=1}^{s}R[x]/(f^{p_{i}}))$ , and let $z_{f}$ act as multiplication by $f$ on each direct summand.

Applying the preceding results gives the local weak-order classification.

Corollary 3.6.

With the above notation, $\Lambda_{f}(P)/(z_{f})\simeq B_{s}(K_{f})$ as $K_{f}$ -algebras, and

\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{f}(P)\simeq\operatorname{Weak}(\Sigma_{s+1}).

Consequently, the support $\tau$ -tilting poset depends only on $s=|P|$ and not on $f$ , $K_{f}$ , or the values of the exponents in $P$ .

Proof.

The action of $R[x]$ on every summand factors through $\mathcal{O}_{f}$ and identifies $\Lambda_{f}(P)$ with $\Lambda_{\mathcal{O}_{f}}(P)$ as $R$ -algebras. Under this identification, $z_{f}$ corresponds to $z_{\pi_{f}}$ . Thus the quotient formula follows from Theorem 3.1, and the poset formula follows from Corollary 3.2. ∎

Remark 3.7.

The weak-order formula does not determine the residue field. Indeed, if $k$ and $k^{\prime}$ are nonisomorphic $R$ -algebras, then $B_{s}(k)$ and $B_{s}(k^{\prime})$ are not Morita equivalent as $R$ -algebras, since the endomorphism division rings of their simple modules are $k$ and $k^{\prime}$ , respectively. Nevertheless, $\mathrm{s}\tau\text{-}\mathrm{tilt}B_{s}(k)\simeq\operatorname{Weak}(\Sigma_{s+1})\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}B_{s}(k^{\prime})$ .

4 Matrix centralizer algebras

Having proved the local reconstruction theorem in Section 3, we now pass from a single local block to an arbitrary matrix centralizer. Primary decomposition and Morita reduction allow us to globalize both parts of Theorem 1.1 while remaining over the ground field.

We first recover the T-type from the support $\tau$ -tilting poset. We then describe the center and interpret the gap data through its action on $HH_{0}$ , before comparing the resulting reconstruction relations and recording their homological consequences.

4.1 Primary decomposition and recovery of the T-type

This subsection proves Theorem 1.2. We first derive its product formula and then prove that the factor sizes, and hence the T-type, are intrinsic to the abstract product poset.

Let $c\in M_{n}(R)$ , and regard $V_{c}:=R^{n}$ as an $R[x]$ -module by letting $x$ act as $c$ . By the elementary-divisor decomposition, $V_{c}\simeq\bigoplus_{f\in\operatorname{Irr}(c)}X_{f}$ , where $X_{f}:=\bigoplus_{a\in P_{c}(f)}(R[x]/(f^{a}))^{m_{c}(f,a)}$ and $m_{c}(f,a)\geq 1$ .

Since $S_{n}(c,R)$ is the algebra of $R[x]$ -endomorphisms of $V_{c}$ , we have $S_{n}(c,R)\simeq\End_{R[x]}(V_{c})$ . If $f,g\in\operatorname{Irr}(c)$ are distinct, then $\Hom_{R[x]}(R[x]/(f^{a}),R[x]/(g^{b}))=0$ for $a,b\geq 1$ . Indeed, since $(f^{a},g^{b})=1$ , multiplication by $f^{a}$ is invertible on $R[x]/(g^{b})$ . If $\varphi:R[x]/(f^{a})\to R[x]/(g^{b})$ , then $f^{a}\varphi(1)=0$ , and hence $\varphi=0$ . Consequently,

S_{n}(c,R)\simeq\prod_{f\in\operatorname{Irr}(c)}\End_{R[x]}(X_{f}).

(4.1)

For each $f\in\operatorname{Irr}(c)$ , let $Y_{f}:=\bigoplus_{a\in P_{c}(f)}R[x]/(f^{a})$ . The module $Y_{f}$ contains one representative of each isomorphism class of indecomposable direct summands of $X_{f}$ . Hence $X_{f}\in\operatorname{add}Y_{f}$ and $Y_{f}\in\operatorname{add}X_{f}$ , so $\operatorname{add}X_{f}=\operatorname{add}Y_{f}$ . Therefore, the additive-generator form of Morita theory [LX25, Lemma 2.2] gives a Morita equivalence

\End_{R[x]}(X_{f})\sim_{\mathrm{Morita}}\End_{R[x]}(Y_{f})=\Lambda_{f}(P_{c}(f)).

(4.2)

Proof of the product formula in Theorem 1.2.

Support $\tau$ -tilting posets are Morita invariant. Moreover, $(A_{1}\times A_{2})\text{-}\mathrm{mod}\simeq A_{1}\text{-}\mathrm{mod}\times A_{2}\text{-}\mathrm{mod}$ , and the support $\tau$ -tilting conditions and the generation order are computed componentwise. Hence $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\prod_{f\in\operatorname{Irr}(c)}\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{f}(P_{c}(f))$ . Since $|P_{c}(f)|=\kappa_{c}(f)$ , Corollary 3.6 gives $\mathrm{s}\tau\text{-}\mathrm{tilt}\Lambda_{f}(P_{c}(f))\simeq\operatorname{Weak}(\Sigma_{\kappa_{c}(f)+1})$ . Therefore, $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\prod_{f\in\operatorname{Irr}(c)}\operatorname{Weak}(\Sigma_{\kappa_{c}(f)+1})$ . Each factor is finite, so $S_{n}(c,R)$ is $\tau$ -tilting finite. Since $|\operatorname{Weak}(\Sigma_{\kappa_{c}(f)+1})|=(\kappa_{c}(f)+1)!$ , it follows that $|\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)|=\prod_{f\in\operatorname{Irr}(c)}(\kappa_{c}(f)+1)!$ . ∎

For $s\geq 1$ , write $L_{s}:=\operatorname{Weak}(\Sigma_{s+1})$ . We now show that the integers $s$ can be recovered directly from a product of the lattices $L_{s}$ .

Let $L$ be a finite lattice with minimum element $\hat{0}$ . For $x\leq y$ in $L$ , write $[x,y]:=\{z\in L\mid x\leq z\leq y\}$ for the corresponding order interval. Recall that an atom of $L$ is an element covering $\hat{0}$ . We define the atom-interaction graph $G(L)$ to have the atoms of $L$ as its vertices, with two distinct atoms $a,b$ joined by an edge if $\bigl|[\hat{0},a\vee b]\bigr|=6.$ Since an order isomorphism between finite lattices preserves minimum elements, covers, joins, and interval cardinalities, an order isomorphism $L\simeq L^{\prime}$ induces a graph isomorphism $G(L)\simeq G(L^{\prime})$ .

The following lemma reconstructs the product factors from their atoms.

Lemma 4.1.

The graph $G(L_{s})$ is a path with $s$ vertices. More generally, if $L=\prod_{r=1}^{q}L_{s_{r}}$ , then the multiset of connected-component sizes of $G(L)$ is $\{\!\{s_{1},\ldots,s_{q}\}\!\}$ .

Proof.

The atoms of $L_{s}$ are the simple transpositions $\sigma_{1},\ldots,\sigma_{s}.$ For distinct atoms $a,b$ , the interval $[\hat{0},a\vee b]$ is the weak order on the rank-two parabolic subgroup $\langle a,b\rangle$ , which has $2m(a,b)$ elements; see [BB05, Sections 3.1–3.2]. In type $A_{s}$ , this number is six exactly for adjacent simple transpositions. Therefore, $G(L_{s})=\sigma_{1}-\sigma_{2}-\cdots-\sigma_{s}$ .

Now let $L=\prod_{r=1}^{q}L_{s_{r}}$ . Every atom of $L$ is supported in exactly one factor. If two atoms belong to distinct factors, then the interval below their join is the product of two two-element chains. Thus it has four elements, so these atoms are not adjacent in $G(L)$ .

It follows that there are no edges between atoms from distinct factors. Inside the $r$ -th factor, the preceding argument gives a path with $s_{r}$ vertices. Consequently, the connected components of $G(L)$ have sizes $s_{1},\ldots,s_{q}$ , up to permutation. ∎

Proof of the equivalences in Theorem 1.2.

Suppose first that $T_{R}(c)=T_{R}(d)$ . Then $\{\!\{\kappa_{c}(f)\mid f\in\operatorname{Irr}(c)\}\!\}=\{\!\{\kappa_{d}(g)\mid g\in\operatorname{Irr}(d)\}\!\}$ . Therefore, the product formula proved above gives $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}S_{m}(d,R)$ . Thus (1) implies (2).

Conversely, suppose that $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}S_{m}(d,R).$ By the product formula proved above, both posets are products of lattices of the form $L_{s}$ . Since an order isomorphism preserves atoms, joins and intervals, it induces an isomorphism between their atom-interaction graphs. Lemma 4.1 then gives $\{\!\{\kappa_{c}(f)\mid f\in\operatorname{Irr}(c)\}\!\}=\{\!\{\kappa_{d}(g)\mid g\in\operatorname{Irr}(d)\}\!\}$ . Hence $T_{R}(c)=T_{R}(d)$ . Thus (2) implies (1).

It remains to compare (2) and (3). By the finiteness assertion proved above, both centralizer algebras are $\tau$ -tilting finite. Therefore, every torsion class is functorially finite by [DIJ19, Theorem 3.8]. The Adachi–Iyama–Reiten correspondence consequently gives the poset isomorphisms $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\operatorname{tors}S_{n}(c,R)$ and $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{m}(d,R)\simeq\operatorname{tors}S_{m}(d,R)$ , where a support $\tau$ -tilting module $M$ corresponds to $\operatorname{Fac}M$ .

Hence the support $\tau$ -tilting posets are isomorphic if and only if the torsion lattices are isomorphic. Thus (2) and (3) are equivalent. ∎

Remark 4.2.

Let $R=\mathbb{Q}$ and let $c$ be the companion matrix of $x^{2}+1$ . Then $S_{2}(c,\mathbb{Q})\cong\mathbb{Q}(i)$ , so $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{2}(c,\mathbb{Q})$ is the two-element chain $\operatorname{Weak}(\Sigma_{2})$ . After scalar extension to $\mathbb{C}$ , the polynomial splits as $(x-i)(x+i)$ . Hence the centralizer becomes $\mathbb{C}\times\mathbb{C}$ , and its support $\tau$ -tilting poset is the four-element Boolean lattice $\operatorname{Weak}(\Sigma_{2})^{2}$ . Thus Theorem 1.2 cannot be obtained by passing formally to a splitting field and descending.

4.2 Centers and their local factors

The center of a matrix centralizer is described by the classical double-centralizer theorem for a single matrix. The remaining isomorphisms below follow from the first isomorphism theorem and the Chinese remainder theorem. Thus Proposition 4.3 is not a new description of the center. We record it because its local factors identify the blocks of the centralizer algebra and are used to define the center-enhanced invariants introduced below and the degree-zero Hochschild invariants studied in Subsection 4.3. For related structural results on matrix centralizer algebras, see [XZ21, XZ22].

For $f\in\operatorname{Irr}(c)$ , let $r_{c}(f):=\max P_{c}(f)$ and $U_{c}(f):=R[x]/(f^{r_{c}(f)})$ .

Proposition 4.3.

Let $c\in M_{n}(R)$ , and let $\mu_{c}$ be its minimal polynomial. Then

Z\bigl(S_{n}(c,R)\bigr)=R[c]\cong R[x]/(\mu_{c})\cong\prod_{f\in\operatorname{Irr}(c)}U_{c}(f).

Proof.

Let $\mathcal{C}:=S_{n}(c,R)=C_{M_{n}(R)}(c).$ Since $c\in\mathcal{C}$ , we have $C_{M_{n}(R)}(\mathcal{C})\subseteq C_{M_{n}(R)}(c)=\mathcal{C}$ . Therefore, one has

Z(\mathcal{C})=\mathcal{C}\cap C_{M_{n}(R)}(\mathcal{C})=C_{M_{n}(R)}(\mathcal{C})=C_{M_{n}(R)}\bigl(C_{M_{n}(R)}(c)\bigr).

By the classical double-centralizer theorem for a single matrix [Jac85, Chapter III], the last algebra is $R[c]$ .

Now consider the evaluation homomorphism $R[x]\to R[c]$ , $p(x)\mapsto p(c)$ . Its kernel is generated by the minimal polynomial $\mu_{c}$ . Therefore, $R[c]\cong R[x]/(\mu_{c}).$ Moreover, $\mu_{c}=\prod_{f\in\operatorname{Irr}(c)}f^{r_{c}(f)}$ . Since the factors in this product are pairwise coprime, the Chinese remainder theorem gives $R[x]/(\mu_{c})\cong\prod_{f\in\operatorname{Irr}(c)}R[x]/(f^{r_{c}(f)})$ . Combining these isomorphisms proves Proposition 4.3. ∎

Consequently, the factors $U_{c}(f)$ are intrinsic local factors of the center, even though the polynomials indexing them depend on the chosen matrix presentation.

For a finite-dimensional $R$ -algebra $B$ , write $\operatorname{LL}(B):=\min\{\ell\geq 1\mid(\operatorname{rad}B)^{\ell}=0\}$ for its Loewy length.

Corollary 4.4.

The abstract algebra $Z(S_{n}(c,R))$ determines the multiset $\{U_{c}(f)\mid f\in\operatorname{Irr}(c)\}$ up to permutation and $R$ -algebra isomorphism.

For each such $f$ , the local algebra $U_{c}(f)$ determines $r_{c}(f)=\operatorname{LL}(U_{c}(f))$ , its residue field $K_{f}:=U_{c}(f)/\operatorname{rad}U_{c}(f)\cong R[x]/(f)$ , and $\dim_{R}U_{c}(f)=r_{c}(f)\deg f$ . Consequently,

\dim_{R}Z\bigl(S_{n}(c,R)\bigr)=\deg\mu_{c}=\sum_{f\in\operatorname{Irr}(c)}r_{c}(f)\deg f.

Proof.

For each $f$ , the algebra $U_{c}(f)$ is local, with $\operatorname{rad}U_{c}(f)=(f)/(f^{r_{c}(f)})$ . Thus its Loewy length is $r_{c}(f)$ , and its residue field is $R[x]/(f)$ .

Since primitive central idempotents are unique up to permutation, every finite-dimensional commutative $R$ -algebra has a unique decomposition into local factors, up to permutation and isomorphism. Therefore, Proposition 4.3 shows that the center determines the multiset of the algebras $U_{c}(f)$ . Finally, $\dim_{R}U_{c}(f)=\dim_{R}R[x]/(f^{r_{c}(f)})=r_{c}(f)\deg f$ . Therefore all the stated invariants are determined by the corresponding local factor. ∎

Remark 4.5.

If $f$ is separable over $R$ , then we have

U_{c}(f)=R[x]/\bigl(f^{\,r_{c}(f)}\bigr)\cong K_{f}[t]/\bigl(t^{\,r_{c}(f)}\bigr),

where $U_{c}(f)$ is equipped with a suitable $K_{f}$ -algebra structure; see [LX25, Lemma 2.14]. Thus, in the separable case, the isomorphism class of $U_{c}(f)$ is determined by the pair $(K_{f},r_{c}(f))$ .

If $f$ is inseparable, then such an isomorphism need not exist. Indeed, $U_{c}(f)$ need not admit a coefficient field isomorphic to $K_{f}$ . For example, if $R=\mathbb{F}_{p}(u)$ and $f=x^{p}-u$ , then the canonical epimorphism $R[x]/(f^{2})\twoheadrightarrow R[x]/(f)$ has no $R$ -algebra section: every lift $x+fh$ of the residue class of $x$ satisfies $(x+fh)^{p}-u\equiv f\neq 0\pmod{f^{2}}$ . Therefore, over an arbitrary field, the intrinsic local factor is $U_{c}(f)$ itself, rather than a truncated polynomial algebra over its residue field. This is the point at which the arbitrary-field formulation differs from the usual separable description.

4.3 Hochschild reconstruction and derived invariance

We now apply the local Hochschild reconstruction of Section 3 to the primary blocks of a matrix centralizer. The following theorem proves the center-module formula in Theorem 1.3.

Theorem 4.6.

For every $c\in M_{n}(R)$ ,

HH_{0}(S_{n}(c,R))\simeq\bigoplus_{f\in\operatorname{Irr}(c)}\ \bigoplus_{g\in H(P_{c}(f))}U_{c}(f)/\operatorname{rad}^{g}U_{c}(f),\qquad Z(S_{n}(c,R))\simeq\prod_{f\in\operatorname{Irr}(c)}U_{c}(f).

Via the second isomorphism, the center acts on a summand indexed by $f$ through its projection to the $f$ -factor.

Proof.

Every primary block is Morita equivalent to $\Lambda_{f}(P_{c}(f))$ . Theorem 3.4 computes it as a module over its center. Hochschild homology and the center action on it are Morita invariant. They are also compatible with finite products. Hence the asserted global formula follows. ∎

The next corollary makes the blockwise gap-recovery assertion in Theorem 1.3 explicit.

Corollary 4.7.

Let $A:=S_{n}(c,R)$ and $B:=S_{m}(d,R)$ , and write their block decompositions as $A=\prod_{i\in I}A_{i}$ and $B=\prod_{j\in J}B_{j}$ . Then $c\stackrel{{\scriptstyle\mathrm{D}}}{{\sim}}d$ if and only if there is a bijection $\sigma\colon I\to J$ such that $\bigl(Z(A_{i}),HH_{0}(A_{i})\bigr)\simeq\bigl(Z(B_{\sigma(i)}),HH_{0}(B_{\sigma(i)})\bigr)$ as algebra–module pairs for every $i\in I$ .

Proof.

By Theorem 4.6, the pair attached to the $f$ -block is $\left(U_{c}(f),\bigoplus_{g\in H(P_{c}(f))}U_{c}(f)/\operatorname{rad}^{g}U_{c}(f)\right)$ . By the uniqueness of cyclic decomposition over a local principal ideal algebra, two such pairs are isomorphic if and only if their local coefficient algebras are isomorphic and their gap multisets agree. This is precisely D-equivalence. ∎

Thus the module structure recovers the individual gaps. By contrast, if $P=\{p_{1}<\cdots<p_{s}\}$ and $r=p_{s}$ , then the isomorphism class of the underlying $R$ -vector space is determined only by $\dim_{R}HH_{0}(\Lambda_{f}(P))=\deg(f)r$ , so it does not recover the individual gaps.

Corollary 4.7 yields the following result, which proves the derived-invariance assertion in Theorem 1.3.

Corollary 4.8.

If $S_{n}(c,R)$ and $S_{m}(d,R)$ are derived equivalent as $R$ -algebras, then their primary blocks can be paired so that $U_{c}(f)\simeq U_{d}(\sigma(f))$ as $R$ -algebras and $H(P_{c}(f))=H(P_{d}(\sigma(f)))$ . Thus derived equivalence preserves the local coefficient algebras and gap multisets blockwise. In particular, the center, the T-type $T_{R}$ and the support $\tau$ -tilting poset are derived invariants within this class.

Proof.

Put $A:=S_{n}(c,R)$ and $B:=S_{m}(d,R)$ , and write $A=\prod_{i}A_{i}$ and $B=\prod_{j}B_{j}$ for their block decompositions. An $R$ -linear derived equivalence gives compatible isomorphisms $Z(A)\simeq Z(B)$ and $HH_{0}(A)\simeq HH_{0}(B)$ [AK17, Theorem 2.2]. Since an algebra isomorphism preserves primitive central idempotents, it gives a bijection $e_{i}\mapsto e^{\prime}_{\sigma(i)}$ . Since $Z(A_{i})=e_{i}Z(A)$ and $HH_{0}(A_{i})=e_{i}HH_{0}(A)$ , and likewise for $B$ , compatibility with the cap-product action restricts the global isomorphisms to blockwise isomorphisms of algebra–module pairs $\bigl(Z(A_{i}),HH_{0}(A_{i})\bigr)\simeq\bigl(Z(B_{\sigma(i)}),HH_{0}(B_{\sigma(i)})\bigr)$ . Corollary 4.7 now gives these blockwise isomorphisms. They preserve the center and the numbers of gaps in each primary block. Thus they preserve the $R$ -T-type, and Theorem 1.2 gives $\mathrm{s}\tau\text{-}\mathrm{tilt}S_{n}(c,R)\simeq\mathrm{s}\tau\text{-}\mathrm{tilt}S_{m}(d,R)$ . ∎

Remark 4.9.

Conversely, suppose that the primary blocks can be paired so that the two equalities displayed in Corollary 4.8 hold. The classification theorem of Li–Xi [LX25, Theorem 1.1] then implies that $S_{n}(c,R)$ and $S_{m}(d,R)$ are derived equivalent as $R$ -algebras. Hence these blockwise isomorphisms characterize derived equivalence. This reverse implication is due to Li–Xi; our contribution in Corollary 4.8 is a direct proof of the forward implication using the center action on $HH_{0}$ .

We end this subsection with the proof of Theorem 1.3 in the introduction.

Proof of Theorem 1.3.

The center–module formula, the gap-recovery assertion, and the derived-invariance assertion in Theorem 1.3 were proved in Theorem 4.6, Corollary 4.7, and Corollary 4.8, respectively.

Taking dimensions in Theorem 4.6 gives $\dim_{R}HH_{0}(S_{n}(c,R))=\sum_{f\in\operatorname{Irr}(c)}\deg(f)r_{c}(f)=\dim_{R}Z(S_{n}(c,R))$ . This is the dimension formula in Theorem 1.3 and completes its proof. ∎

4.4 Comparison of the reconstruction relations

We now compare the support $\tau$ -tilting and center data obtained in Subsections 4.1 and 4.2 with the matrix equivalences of Li–Xi. The existing relations are recalled in our notation, while the new center-enhanced relations are defined separately. Throughout this subsection, all algebra isomorphisms are $R$ -algebra isomorphisms.

For a nonempty finite set $P=\{p_{1}<p_{2}<\cdots<p_{s}\}\subseteq\mathbb{Z}_{>0}$ , put $J(P):=\{p_{s}\}\cup\{p_{s}-p_{i}\mid 1\leq i<s\}$ . Here $H(P)$ is a multiset, while $J(P)$ is a set. These are the operations used in [LX25, Section 3.1], rewritten in increasing order. Thus $|H(P)|=|J(P)|=|P|$ , $J^{2}(P)=P$ , and $H(J(P))=H(P)$ .

The Li–Xi equivalences.

For later comparison, we recall the relations introduced in [LX25, Definition 3.1]. Let $M_{c}$ denote their set of maximal elementary divisors. Then the map $\operatorname{Irr}(c)\to M_{c}$ , $f\mapsto f^{r_{c}(f)}$ , is a bijection from the primary labels used here to the maximal elementary divisors used by Li–Xi. If $P_{c}^{\mathrm{LX}}$ denotes their power-index set, then $U_{c}(f)=R[x]/(f^{r_{c}(f)})$ and $P_{c}(f)=P_{c}^{\mathrm{LX}}(f^{r_{c}(f)})$ .

Let $c\in M_{n}(R)$ and $d\in M_{m}(R)$ . Then:

(1)

$c\stackrel{{\scriptstyle\mathrm{M}}}{{\sim}}d$ if there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that $U_{c}(f)\simeq U_{d}(\sigma(f))$ and $P_{c}(f)=P_{d}(\sigma(f))$ for every $f\in\operatorname{Irr}(c)$ .
(2)

$c\stackrel{{\scriptstyle\mathrm{AD}}}{{\sim}}d$ if there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that, for every $f\in\operatorname{Irr}(c)$ , we have $U_{c}(f)\simeq U_{d}(\sigma(f))$ and either $P_{c}(f)=P_{d}(\sigma(f))$ or $P_{c}(f)=J(P_{d}(\sigma(f)))$ .
(3)

$c\stackrel{{\scriptstyle\mathrm{D}}}{{\sim}}d$ if there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that $U_{c}(f)\simeq U_{d}(\sigma(f))$ and $H(P_{c}(f))=H(P_{d}(\sigma(f)))$ for every $f\in\operatorname{Irr}(c)$ .

These three relations correspond, respectively, to [LX25, Definition 3.1(1), (3), and (2)]; we use this order only to reflect the implications $\mathrm{M}\Rightarrow\mathrm{AD}\Rightarrow\mathrm{D}$ . Li–Xi proved that these relations characterize Morita equivalence, almost $\nu$ -stable derived equivalence, and derived equivalence, respectively [LX25, Theorem 1.1].

Consequently, Corollary 4.7 gives an intrinsic description of D-equivalence in terms of the center action on degree-zero Hochschild homology, block by block.

We now introduce the center-enhanced relations used in the comparison theorem below.

Definition 4.10.

Let $c\in M_{n}(R)$ and $d\in M_{m}(R)$ .

(1)

The matrices $c$ and $d$ are blockwise TZ-equivalent, written $c\stackrel{{\scriptstyle\mathrm{bTZ}}}{{\sim}}d$ , if there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that $U_{c}(f)\simeq U_{d}(\sigma(f))$ and $|P_{c}(f)|=|P_{d}(\sigma(f))|$ for every $f\in\operatorname{Irr}(c)$ .
(2)

They are TZ-equivalent, written $c\stackrel{{\scriptstyle\mathrm{TZ}}}{{\sim}}d$ , if their support $\tau$ -tilting posets are isomorphic and their centers are isomorphic as $R$ -algebras.

Thus TZ-equivalence records the support $\tau$ -tilting poset and the center as separate invariants, whereas bTZ-equivalence also records a blockwise correspondence between them. Recall also that $c\stackrel{{\scriptstyle\mathrm{T}}}{{\sim}}d$ means $T_{R}(c)=T_{R}(d)$ ; by Theorem 1.2, this is equivalent to an isomorphism of the corresponding support $\tau$ -tilting posets.

The next proposition gives an intrinsic interpretation of bTZ-equivalence.

Proposition 4.11.

The matrices $c$ and $d$ are bTZ-equivalent if and only if there is a bijection between the blocks of $S_{n}(c,R)$ and $S_{m}(d,R)$ such that corresponding blocks have isomorphic centers as $R$ -algebras and isomorphic support $\tau$ -tilting posets.

Proof.

By Proposition 4.3, the center of the primary factor indexed by $f$ is the local algebra $U_{c}(f)$ . Since a local algebra has no nontrivial idempotents, each primary factor has no nontrivial central idempotents. Hence the factors in (4.1) are exactly the blocks of the centralizer algebra. Moreover, Theorem 1.2 shows that the support $\tau$ -tilting poset of this block is $\operatorname{Weak}(\Sigma_{|P_{c}(f)|+1})$ and determines $|P_{c}(f)|$ . Therefore, the stated blockwise conditions are equivalent to a bijection preserving the pairs $(U_{c}(f),|P_{c}(f)|)$ . This is precisely bTZ-equivalence. ∎

Remark 4.12.

The use of $U_{c}(f)$ in Proposition 4.11 is essential: it cannot in general be replaced by the pair $(K_{f},r_{c}(f))$ . Such a replacement is valid only in the separable case of Remark 4.5.

For a nonempty finite set $P=\{p_{1}<\cdots<p_{s}\}\subseteq\mathbb{Z}_{>0}$ , let $g_{1}:=p_{1}$ and $g_{i}:=p_{i}-p_{i-1}$ for $2\leq i\leq s$ , and define $G(P):=\bigoplus_{i=1}^{s}\mathbb{Z}/g_{i}\mathbb{Z}$ . Let $C(P)$ be the Cartan matrix of the corresponding basic primary block $\Lambda_{f}(P)$ , and let $D_{\mathrm{sg}}(\Lambda_{f}(P)):=D^{\mathrm{b}}(\operatorname{mod}\Lambda_{f}(P))/K^{\mathrm{b}}(\operatorname{proj}\Lambda_{f}(P))$ be its singularity category.

Related Cartan computations for matrix centralizers appear in [DPS12]; see also [LX25, Lemma 2.18]. The next proposition gives a direct local calculation in our notation and extracts the coarser singularity invariant needed below.

Proposition 4.13.

Let $f\in R[x]$ be monic and irreducible, put $\mathcal{O}:=R[x]/(f^{p_{s}})$ , $M_{i}:=\mathcal{O}/\operatorname{rad}^{p_{i}}\mathcal{O}$ , and $\Lambda_{f}(P):=\End_{\mathcal{O}}(\bigoplus_{i=1}^{s}M_{i})$ , and let $C(P)$ be the Cartan matrix of $\Lambda_{f}(P)$ . Then

C(P)_{ij}=\min\{p_{i},p_{j}\},\qquad C(P)=Z\operatorname{diag}(g_{1},\ldots,g_{s})Z^{\mathsf{T}},

where $Z_{ij}=1$ if $j\leq i$ and $Z_{ij}=0$ otherwise. Consequently,

\det C(P)=\prod_{i=1}^{s}g_{i},\qquad K_{0}(D_{\mathrm{sg}}\Lambda_{f}(P))\cong\coker C(P)\cong\bigoplus_{i=1}^{s}\mathbb{Z}/g_{i}\mathbb{Z}=G(P).

Proof.

Write $\ell_{\mathcal{O}}$ for composition length over $\mathcal{O}$ . Let $P_{i}=\Lambda_{f}(P)e_{i}$ and $S_{j}=P_{j}/\operatorname{rad}P_{j}$ . Since $e_{j}(-)$ is exact, applying it to a composition series of $P_{i}$ gives $[P_{i}:S_{j}]=\ell_{\mathcal{O}}(e_{j}P_{i})$ . Indeed, $e_{j}S_{k}=0$ for $k\neq j$ , while $e_{j}S_{j}\simeq\mathcal{O}/\operatorname{rad}\mathcal{O}$ has $\mathcal{O}$ -length one. Since $e_{j}P_{i}=e_{j}\Lambda_{f}(P)e_{i}\simeq\Hom_{\mathcal{O}}(M_{i},M_{j})$ , it follows that $[P_{i}:S_{j}]=\ell_{\mathcal{O}}\Hom_{\mathcal{O}}(M_{i},M_{j})=\min\{p_{i},p_{j}\}$ . Since $p_{i}=g_{1}+\cdots+g_{i}$ , the stated factorization follows, and $Z$ is unimodular. Therefore $\coker C(P)\cong\bigoplus_{i}\mathbb{Z}/g_{i}\mathbb{Z}$ . Moreover, the homomorphism $K_{0}(K^{\mathrm{b}}(\proj\Lambda_{f}(P)))\to K_{0}(D^{\mathrm{b}}(\operatorname{mod}\Lambda_{f}(P)))$ is represented by $C(P)$ in the bases of indecomposable projectives and simple modules. Thus the exact sequence on Grothendieck groups for the Verdier quotient gives $K_{0}(D_{\mathrm{sg}}\Lambda_{f}(P))\cong\coker C(P)$ . ∎

Thus $G(P)$ is the Grothendieck group invariant of the singularity category. It records only $K_{0}$ and does not determine the singularity category. Chen–Xi give a complete description of singularity categories and singular equivalences for matrix centralizer algebras [CX26, Theorems 4.7 and 4.9]. Since a unit gap gives a trivial summand, $G(P)$ need not determine $|P|$ . We therefore retain the number of gaps in the next definition.

Definition 4.14.

The matrices $c$ and $d$ are gTZ-equivalent, written $c\stackrel{{\scriptstyle\mathrm{gTZ}}}{{\sim}}d$ , if there is a bijection $\sigma\colon\operatorname{Irr}(c)\to\operatorname{Irr}(d)$ such that, for every $f\in\operatorname{Irr}(c)$ , $U_{c}(f)\simeq U_{d}(\sigma(f))$ , $|P_{c}(f)|=|P_{d}(\sigma(f))|$ , and $G(P_{c}(f))\simeq G(P_{d}(\sigma(f)))$ as abelian groups.

We now compare the reconstruction relations determined by the center, the support $\tau$ -tilting poset, and the singularity Grothendieck group.

Theorem 4.15.

Over every field $R$ , there is a strict implication chain

\mathrm{D}\Rightarrow\mathrm{gTZ}\Rightarrow\mathrm{bTZ}\Rightarrow\mathrm{TZ}\Rightarrow\mathrm{T}.

Proof.

Since the gap multiset $H(P)$ determines both $|P|$ and $G(P)$ , it follows that $\mathrm{D}\Rightarrow\mathrm{gTZ}$ . The definition then gives $\mathrm{gTZ}\Rightarrow\mathrm{bTZ}$ .

If two matrices are bTZ-equivalent, then Proposition 4.11 provides a bijection between their blocks that preserves the center and the support $\tau$ -tilting poset of each block. Hence their global centers and support $\tau$ -tilting posets are isomorphic. Consequently, $\mathrm{bTZ}\Rightarrow\mathrm{TZ}$ . Finally, $\mathrm{TZ}\Rightarrow\mathrm{T}$ follows from Theorem 1.2.

We prove strictness over every field. Write $J_{a}(\lambda)$ for the Jordan block of size $a$ with eigenvalue $\lambda$ .

First, let $c_{P}:=J_{3}(0)\oplus J_{7}(0)\oplus J_{21}(0)$ and $c_{Q}:=J_{2}(0)\oplus J_{9}(0)\oplus J_{21}(0)$ . Their exponent sets $P=\{3,7,21\}$ and $Q=\{2,9,21\}$ have the same cardinality and $G(P)\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/84\mathbb{Z}\simeq G(Q)$ , whereas $H(P)=\{\!\{3,4,14\}\!\}\neq\{\!\{2,7,12\}\!\}=H(Q)$ . Thus $c_{P}$ and $c_{Q}$ are gTZ-equivalent but not D-equivalent. Second, the nilpotent matrices with exponent sets $P=\{1,4\}$ and $Q=\{2,4\}$ have the same block datum $(R[t]/(t^{4}),2)$ , whereas $G(P)\simeq\mathbb{Z}/3\mathbb{Z}$ and $G(Q)\simeq(\mathbb{Z}/2\mathbb{Z})^{2}$ . Hence bTZ-equivalence does not imply gTZ-equivalence.

For the third separation, let

	$\displaystyle c$	$\displaystyle=J_{1}(0)\oplus J_{5}(0)\oplus J_{2}(1),$
	$\displaystyle d$	$\displaystyle=J_{1}(0)\oplus J_{2}(0)\oplus J_{5}(1).$

Both posets are $\operatorname{Weak}(\Sigma_{3})\times\operatorname{Weak}(\Sigma_{2})$ , and both centers are $R[t]/(t^{5})\times R[t]/(t^{2})$ . Thus $c\stackrel{{\scriptstyle\mathrm{TZ}}}{{\sim}}d$ . However, their blockwise data are $\{\!\{(R[t]/(t^{5}),2),(R[t]/(t^{2}),1)\}\!\}$ and $\{\!\{(R[t]/(t^{2}),2),(R[t]/(t^{5}),1)\}\!\}$ , respectively, so $c\not\stackrel{{\scriptstyle\mathrm{bTZ}}}{{\sim}}d$ . Finally, the exponent sets $\{1,2\}$ and $\{1,3\}$ have the same T-type, but their centers have Loewy lengths $2$ and $3$ . Therefore T-equivalence does not imply TZ-equivalence. These examples use only $0\neq 1$ , integral exponent data, and finite abelian groups, and therefore work over every field. ∎

Combining Theorem 4.15 with the Li–Xi relations gives the strict implication chain

\mathrm{M}\Rightarrow\mathrm{AD}\Rightarrow\mathrm{D}\Rightarrow\mathrm{gTZ}\Rightarrow\mathrm{bTZ}\Rightarrow\mathrm{TZ}\Rightarrow\mathrm{T}.

The first two implications follow from [LX25, Theorem 1.1]. They are also strict. Indeed, the nilpotent matrices with exponent sets $\{1,3,5\}$ and $\{2,4,5\}$ are AD-equivalent but not M-equivalent. Moreover, let $P=\{1,3,6\}$ and $Q=\{1,4,6\}$ . Then $H(P)=H(Q)=\{\!\{1,2,3\}\!\}$ , so the corresponding nilpotent matrices are D-equivalent. However, $J(Q)=\{2,5,6\}\neq P$ , so they are not AD-equivalent. Therefore, the full chain is strict.

We conclude this section by recording the homological information detected by the preceding reconstruction data. The dimension formulas below are known; we determine which reconstruction data preserve them.

Proposition 4.16.

Let $\Lambda_{f}(P)$ be a primary block, where $r:=\max P$ . Then

\domdim\Lambda_{f}(P)=\begin{cases}\infty,&P=\{r\};\\ 2,&\text{otherwise},\end{cases}\qquad\gldim\Lambda_{f}(P)=\begin{cases}0,&P=\{1\};\\ 2,&P=\{1,\ldots,r\}\text{ and }r\geq 2;\\ \infty,&\text{otherwise}.\end{cases}

Moreover, $\gldim\Lambda_{f}(P)<\infty$ if and only if $\det C(P)=1$ . Consequently, for matrix centralizer algebras, the T-data determine dominant dimension, whereas the TZ-data determine global dimension and the T-data do not.

Proof.

The dominant-dimension formula is [LX25, Lemma 4.8]. The global-dimension formula was proved over perfect fields in [DPS12, Theorem 9.2] and over arbitrary fields in [CX26, Corollary 4.5]. By Proposition 4.13, $\det C(P)=\prod_{g\in H(P)}g$ . Hence the determinant equals $1$ if and only if $P=\{1,\ldots,r\}$ ; compare also [CX26, Lemma 6.4].

It remains to locate these formulas among the equivalence relations above. For finite direct products, dominant dimension is the minimum of the dominant dimensions of the factors, and global dimension is their maximum. Since the T-data determine the multiset of the numbers $\kappa_{c}(f)$ , they determine whether every primary block has only one exponent. Hence they determine dominant dimension. For every $f$ , we have $\kappa_{c}(f)\leq r_{c}(f)$ , with equality if and only if $P_{c}(f)=\{1,\ldots,r_{c}(f)\}$ . Therefore, $\gldim S_{n}(c,R)<\infty$ if and only if $\sum_{f\in\operatorname{Irr}(c)}\kappa_{c}(f)=\sum_{f\in\operatorname{Irr}(c)}r_{c}(f)$ . Since every difference $r_{c}(f)-\kappa_{c}(f)$ is nonnegative, equality of the two sums holds if and only if $\kappa_{c}(f)=r_{c}(f)$ for every $f$ . Thus every primary block has finite global dimension exactly when the displayed equality holds. The TZ-data determine both sums: the T-data determine the first, while the Loewy lengths of the center factors $U_{c}(f)$ determine the second. If the global dimension is finite, then it is zero if every center factor has Loewy length one. Otherwise, it is two. Hence the TZ-data determine global dimension. Finally, the exponent sets $\{1,2\}$ and $\{1,3\}$ have the same T-type, while their global dimensions are $2$ and $\infty$ , respectively. Thus the T-data do not determine global dimension. ∎

5 Rigidity and Morita reconstruction for string and gentle centralizers

Sections 3 and 4 determine the reconstruction data retained by the support $\tau$ -tilting poset, the center, and the center action on $HH_{0}$ . We now ask when these data are strong enough to recover the Morita class. Although every primary block has a gentle central quotient, gentleness need not lift to the block itself; this makes a separate rigidity analysis necessary.

We use the standard split bound-quiver conventions of [BR87, AS87]. A string algebra is a split basic algebra $RQ/I$ , where $I$ is an admissible monomial ideal, each vertex is the source and target of at most two arrows, and each arrow has at most one permitted continuation on either side. It is gentle if, in addition, $I$ is generated by paths of length two and each arrow has at most one forbidden continuation on either side. We allow $Q$ to be disconnected.

Definition 5.1.

A finite-dimensional $R$ -algebra is Morita-string if it is $R$ -linearly Morita equivalent to a split basic string $R$ -algebra. It is Morita-gentle if it is $R$ -linearly Morita equivalent to a split basic gentle $R$ -algebra.

We shall use the following elementary Morita fact to pass from the definition to an actual bound-quiver presentation.

Lemma 5.2.

Let $A$ and $B$ be finite-dimensional basic $R$ -algebras. If they are $R$ -linearly Morita equivalent, then $A\simeq B$ as $R$ -algebras. Consequently, if $B$ is split basic and has a monomial Gabriel presentation, then so does $A$ .

Proof.

An $R$ -linear equivalence $F:A\text{-}\operatorname{mod}\to B\text{-}\operatorname{mod}$ induces a bijection between the indecomposable projective modules. Since $A$ and $B$ are basic, their left regular modules contain each indecomposable projective exactly once. Hence $F({}_{A}A)\simeq{}_{B}B$ . Since $F$ is $R$ -linear and fully faithful,

A^{\mathrm{op}}=\End_{A}({}_{A}A)\simeq\End_{B}(F({}_{A}A))\simeq\End_{B}({}_{B}B)=B^{\mathrm{op}}

as $R$ -algebras. Taking opposite algebras gives $A\simeq B$ . Transporting a monomial Gabriel presentation of $B$ along this isomorphism proves the last assertion. ∎

Thus the gentle quotient $\Lambda_{\mathcal{O}}(P)/(z_{\pi})\simeq B_{s}(k)$ does not determine whether the block itself is string. Theorem 5.8 settles this lifting problem for polynomial primary blocks; the case of an arbitrary local principal $R$ -algebra $\mathcal{O}$ is not addressed here.

Over a perfect field, the Morita-string condition is strictly more restrictive than representation-finiteness. For example, if $r\geq 4$ , then $P=\{1,r\}$ gives a representation-finite block that is not Morita-string; see [LX25, Lemma 3.3] and [DM06, Theorem 2.1(i)]. Chan–Marczinzik classify the broader class of representation-finite gendo-symmetric biserial algebras over algebraically closed fields by Brauer-tree data [CM19, Theorem 3.9]. Here we give an exponent-set criterion for split string and gentle primary centralizer blocks over an arbitrary field.

5.1 The split obstruction and the local criterion

We first use the semisimple quotient to force $f$ to be linear. We then compute the Gabriel quiver and use $t$ -adic leading terms to exclude all nonmonomial cases.

The following lemma gives the linear-factor obstruction.

Lemma 5.3.

Let $f\in R[x]$ be monic and irreducible. Then $\Lambda_{f}(P)/\operatorname{rad}\Lambda_{f}(P)\simeq\prod_{i=1}^{s}K_{f}$ . Consequently, if $\Lambda_{f}(P)$ is $R$ -linearly Morita equivalent to a split basic $R$ -algebra, then $K_{f}\simeq R$ as $R$ -algebras; equivalently, $f$ is linear.

Proof.

Put $X_{i}:=R[x]/(f^{p_{i}})$ , $A:=\Lambda_{f}(P)$ , and let $e_{i}$ correspond to $X_{i}$ . Each $\End_{R[x]}(X_{i})\simeq R[x]/(f^{p_{i}})$ is local. Hence a morphism between two summands lies outside the categorical radical only if the summands are isomorphic and the morphism is invertible. Since the $X_{i}$ are pairwise nonisomorphic, it follows that

e_{j}(\operatorname{rad}A)e_{i}=\begin{cases}\operatorname{rad}\End_{R[x]}(X_{i})=(f)/(f^{p_{i}}),&i=j,\\[2.0pt] \Hom_{R[x]}(X_{i},X_{j}),&i\neq j.\end{cases}

Thus the off-diagonal corners vanish modulo $\operatorname{rad}A$ , and

A/\operatorname{rad}A\simeq\prod_{i=1}^{s}\frac{\End_{R[x]}(X_{i})}{\operatorname{rad}\End_{R[x]}(X_{i})}\simeq K_{f}^{\,s}.

Every simple $A$ -module therefore has endomorphism division ring $K_{f}$ . Since an $R$ -linear Morita equivalence preserves these endomorphism rings, whereas every simple module over a split basic $R$ -algebra has endomorphism ring $R$ , it follows that $K_{f}\simeq R$ as $R$ -algebras. Hence $\deg f=\dim_{R}K_{f}=1$ . Conversely, if $\deg f=1$ , then $f=x-\lambda$ for some $\lambda\in R$ , and hence $K_{f}\simeq R$ as $R$ -algebras. This proves the stated equivalence. ∎

If $\deg f>1$ , then Lemma 5.3 excludes an $R$ -linear Morita equivalence between $\Lambda_{f}(P)$ and any split bound-quiver $R$ -algebra. This obstruction is Morita invariant. Thus the monomial obstruction below is needed only after $f$ has been forced to be linear.

Set $R_{p}:=R[t]/(t^{p})$ and $N_{p}:=\End_{R[t]}(R_{p}\oplus R_{p+1})$ .

We first record the normal form of the two-point family that survives the string obstruction.

Lemma 5.4.

Let $Q^{(2)}$ be the two-vertex quiver

Q^{(2)}:\qquad 1\underset{b}{\overset{a}{\rightleftarrows}}2.

Then $N_{p}\simeq RQ^{(2)}/((ba)^{p})$ . Under this isomorphism, $a$ is the canonical inclusion $R_{p}\to R_{p+1}$ given by multiplication by $t$ , and $b$ is the canonical projection $R_{p+1}\to R_{p}$ . Moreover, $N_{p}$ is a connected representation-finite Nakayama string algebra, $Z(N_{p})\simeq R[z]/(z^{p+1})$ for $z:=ba+ab$ , and $Z(R_{p})=R_{p}$ .

Proof.

Write $M_{1}:=R_{p}$ and $M_{2}:=R_{p+1}$ . With right-to-left composition, let $a([g]_{t^{p}}):=[tg]_{t^{p+1}}$ and $b([g]_{t^{p+1}}):=[g]_{t^{p}}$ . Then $ba=\mu_{t}|_{M_{1}}$ and $ab=\mu_{t}|_{M_{2}}$ . Hence $(ba)^{p}=0$ , $(ab)^{p}\neq 0$ , and $(ab)^{p+1}=a(ba)^{p}b=0$ , and $a,b$ and the vertex idempotents induce a homomorphism $\Phi\colon RQ^{(2)}/((ba)^{p})\longrightarrow N_{p}.$ The nonzero paths form the basis

	$\displaystyle\mathcal{B}_{p}:={}$	$\displaystyle\{(ba)^{r}\mid 0\leq r<p\}\cup\{(ab)^{r}\mid 0\leq r\leq p\}$
		$\displaystyle}{\displaystyle\cup\{a(ba)^{r}\mid 0\leq r<p\}\cup\{(ba)^{r}b\mid 0\leq r<p\}.$

Under $\Phi$ , its four parts are the standard $t$ -power bases of $\End(M_{1})$ , $\End(M_{2})$ , $\Hom(M_{1},M_{2})$ and $\Hom(M_{2},M_{1})$ , respectively. Thus $\Phi$ is an isomorphism and $\dim_{R}N_{p}=4p+1$ . The presentation is monomial, each vertex has one incoming and one outgoing arrow, and every arrow has at most one continuation. Thus $N_{p}$ is a connected string algebra. Its indecomposable projectives are uniserial of lengths $2p$ and $2p+1$ ; therefore it is a representation-finite Nakayama algebra.

Put $z:=ba+ab$ . Since its summands lie in orthogonal corners, $z^{r}=(ba)^{r}+(ab)^{r}$ for $0\leq r<p$ , while $z^{p}=(ab)^{p}\neq 0$ and $z^{p+1}=0$ . If $c\in Z(N_{p})$ , then commutation with the vertex idempotents gives $c=\sum_{r=0}^{p-1}\lambda_{r}(ba)^{r}+\sum_{r=0}^{p}\mu_{r}(ab)^{r}$ . The equations $ca=ac$ and $cb=bc$ imply $\lambda_{r}=\mu_{r}$ for $0\leq r<p$ , while $\mu_{p}$ is free. Consequently, $Z(N_{p})=\bigoplus_{r=0}^{p}Rz^{r}\simeq R[z]/(z^{p+1})$ . Finally, $Z(R_{p})=R_{p}$ . ∎

The following elementary observation makes the obstruction independent of the choices of primitive idempotents and arrow representatives.

Lemma 5.5.

Let $A$ be a finite-dimensional split basic $R$ -algebra with Gabriel quiver $Q$ . Suppose that, for every complete set of primitive orthogonal idempotent lifts and every choice of a basis in each arrow space $e_{j}\operatorname{rad}Ae_{i}/e_{j}\operatorname{rad}^{2}Ae_{i}$ , together with representatives in $\operatorname{rad}A$ , there are distinct paths $w_{1},\dots,w_{m}$ in $Q$ such that every $w_{i}$ has nonzero image in $A$ , whereas their images are linearly dependent. Then $A$ has no monomial Gabriel presentation.

Proof.

Suppose that $\phi\colon RQ\to A$ is a Gabriel presentation with monomial kernel $I$ . Its vertex and arrow images are among the choices quantified in the statement. Since the paths not belonging to $I$ form an $R$ -basis of $RQ/I$ , distinct paths with nonzero images under $\phi$ are linearly independent. This contradicts the assumed linear dependence of their images. ∎

For the rest of this local subsection, modules are indexed by their length: for $r\geq 1$ , put $M(r):=R[t]/(t^{r})$ . If $P=\{p_{1}<\cdots<p_{s}\}$ , set $\Lambda(P):=\End_{R[t]}(\bigoplus_{i=1}^{s}M(p_{i}))$ .

We next determine the Gabriel quiver and record the valuation rigidity needed to apply the preceding observation.

Lemma 5.6.

Let $P=\{p_{1}<\cdots<p_{s}\}$ and let $Q$ be the Gabriel quiver of $\Lambda(P)$ . The arrows of $Q$ are as follows:

(1)

for consecutive $p<q$ in $P$ , the standard inclusion $M(p)\to M(q)$ and the standard projection $M(q)\to M(p)$ ;
(2)

a loop at $M(r)$ exactly when $r\geq 2$ and neither $r-1$ nor $r+1$ belongs to $P$ .

Each arrow space is one-dimensional. Moreover:

(a)

if $p<q$ are consecutive in $P$ , then a representative of the inclusion arrow $M(p)\to M(q)$ has leading order $q-p$ , a representative of the projection arrow $M(q)\to M(p)$ has leading order $0$ relative to its standard cyclic generator, and their closed two-step backtrack at $M(q)$ has leading order $q-p$ ;
(b)

a loop at $M(q)$ has order one; if $q+1\in P$ , the backtrack $M(q)\to M(q+1)\to M(q)$ has order one for $q\geq 2$ , while for $q=1$ this backtrack is zero;
(c)

if $p,p+1,p+2\in P$ , both backtracks at $M(p+1)$ have order one.

Proof.

Step 1. Set $\ord_{t}(0):=\infty$ , and for a nonzero map $\varphi\colon M(a)\to M(b)$ set $\ord_{t}(\varphi):=\ord_{t}(\varphi(1)).$ Evaluation at $1$ identifies a homomorphism with an element of $M(b)$ annihilated by $t^{a}$ . Hence

	$\displaystyle\Hom_{R[t]}(M(a),M(b))$	$\displaystyle=t^{(b-a)_{+}}R[t]/(t^{b})$
		$\displaystyle=\operatorname{span}_{R}\{t^{(b-a)_{+}},t^{(b-a)_{+}+1},\dots,t^{b-1}\},$

where $(r)_{+}:=\max\{r,0\}$ . In particular, $\dim_{R}\Hom_{R[t]}(M(a),M(b))=\min\{a,b\}$ . For $a<b$ , write $\iota_{a,b}(1):=t^{b-a}$ and $\pi_{b,a}(1):=1$ . If $M(a)\xrightarrow{\varphi}M(s)\xrightarrow{\psi}M(b)$ is any factorization, then the preceding Hom-space description gives $\ord_{t}(\psi\varphi)\geq(s-a)_{+}+(b-s)_{+}.$ For the corresponding standard inclusion–projection factorization, the composite is multiplication by $t^{(s-a)_{+}+(b-s)_{+}}.$ Thus equality holds in the preceding estimate when this power is nonzero in $M(b)$ ; if its exponent is at least $b$ , the standard composite is zero.

Step 2. Let $\mathfrak{r}$ denote the radical of the additive category generated by $\{M(p_{i})\}_{i=1}^{s}$ , and let $e_{a}$ be the idempotent of $\Lambda(P)$ corresponding to $M(a)$ . Since the modules $M(p_{i})$ are pairwise nonisomorphic and have local endomorphism rings, the argument used in Lemma 5.3 gives $e_{b}(\operatorname{rad}\Lambda(P))e_{a}=\mathfrak{r}(M(a),M(b))$ and $e_{b}(\operatorname{rad}^{2}\Lambda(P))e_{a}=\mathfrak{r}^{2}(M(a),M(b))$ . Thus the arrow space from $M(a)$ to $M(b)$ is $\mathfrak{r}(M(a),M(b))/\mathfrak{r}^{2}(M(a),M(b))$ . Its numerator is

\mathfrak{r}(M(a),M(b))=\begin{cases}\Hom_{R[t]}(M(a),M(b)),&a\neq b,\\ tR[t]/(t^{a}),&a=b.\end{cases}

Moreover, $\mathfrak{r}^{2}(M(a),M(b))=\sum_{r\in P}\mathfrak{r}(M(r),M(b))\,\mathfrak{r}(M(a),M(r))$ . If $a<s<b$ all belong to $P$ , then $\iota_{a,b}=\iota_{s,b}\iota_{a,s},\ \pi_{b,a}=\pi_{s,a}\pi_{b,s},$ so the standard maps lie in $\mathfrak{r}^{2}$ . Every higher $t$ -multiple factors through a radical endomorphism at an endpoint. Therefore there are no arrows between nonconsecutive vertices.

Now let $p<q$ be consecutive in $P$ . If $s\in P\setminus\{p,q\}$ , then $s<p$ or $s>q$ . The estimate in Step 1 shows that a factorization through $M(s)$ has order strictly larger than $q-p$ for the inclusion and strictly larger than $0$ for the projection, whereas $\ord_{t}(\iota_{p,q})=q-p$ and $\ord_{t}(\pi_{q,p})=0$ . An endpoint factorization contains a radical endomorphism and also raises the order. Hence

	$\displaystyle\frac{\mathfrak{r}(M(p),M(q))}{\mathfrak{r}^{2}(M(p),M(q))}$	$\displaystyle=R\overline{\iota}_{p,q},$
	$\displaystyle\frac{\mathfrak{r}(M(q),M(p))}{\mathfrak{r}^{2}(M(q),M(p))}$	$\displaystyle=R\overline{\pi}_{q,p}.$

These are precisely the two arrows between consecutive vertices, and both arrow spaces are one-dimensional.

Step 3. At $M(r)$ , $\operatorname{rad}\End_{R[t]}(M(r))=tR[t]/(t^{r})$ , and every power $t^{j}$ with $j\geq 2$ already factors through a radical endomorphism at $M(r)$ . Thus only multiplication by $t$ can define a loop. The standard backtrack through $M(s)$ sends $1$ to $t^{|r-s|}$ ; it is zero if this power vanishes. Therefore, $t\,\mathrm{id}_{M(r)}\in\mathfrak{r}^{2}(M(r),M(r))$ if and only if $r-1\in P$ or $r+1\in P$ . Here the equivalence holds for $r\geq 2$ . For $r=1$ , multiplication by $t$ is zero. This proves that a loop occurs exactly when $r\geq 2$ , $r-1\notin P$ , and $r+1\notin P$ , and that its arrow space is one-dimensional.

Step 4. Because every arrow space is one-dimensional, each chosen arrow representative has the form $\widetilde{\theta}_{i}=c_{i}\theta_{i}+\eta_{i}$ , where $c_{i}\in R^{\times}$ , $\eta_{i}\in\mathfrak{r}^{2}$ , and $\theta_{i}$ is the corresponding standard map. By Steps 2 and 3, $\ord_{t}(\eta_{i})>\ord_{t}(\theta_{i}).$ Consider one of the paths below and expand the composite of its chosen representatives. Since all maps are $R[t]$ -linear, every nonzero term containing some $\eta_{i}$ has larger $t$ -order than the term containing only the standard maps. The latter term has coefficient $\prod_{i}c_{i}\neq 0$ . Thus, if its predicted order is smaller than the length of the target module, it is nonzero and cannot be cancelled. Hence the chosen path has the same leading order as the standard path:

\begin{array}[]{c|c}\text{closed path}&\text{leading order}\\ \hline\cr M(q)\to M(p)\to M(q),\ p<q\text{ consecutive}&q-p\\ \text{a loop at }M(q)&1\\ M(q)\to M(q+1)\to M(q),\ q\geq 2&1\\ \text{either backtrack at }M(p+1),\ p,p+1,p+2\in P&1.\end{array}

For $q=1$ , the third path is multiplication by $t$ on $M(1)$ and is zero. This proves (a)–(c) for arbitrary choices of arrow representatives. ∎

We can now isolate the two obstructions to a monomial Gabriel presentation.

Proposition 5.7.

Let $P=\{p_{1}<\cdots<p_{s}\}$ with $s\geq 2$ .

(1)

If consecutive $p<q$ in $P$ satisfy $q-p>1$ , then $\Lambda(P)$ has no monomial Gabriel presentation.
(2)

If $p,p+1,p+2\in P$ , then $\Lambda(P)$ has no monomial Gabriel presentation.

Proof.

Let $\{e_{i}\}$ be the standard primitive idempotents and let $\{e_{i}^{\prime}\}$ be any other complete set of primitive orthogonal idempotents. After reindexing, $e_{i}^{\prime}-e_{i}\in\operatorname{rad}\Lambda(P)$ for every $i$ . Put $u:=\sum_{i}e_{i}^{\prime}e_{i}$ . Then $u\equiv 1\pmod{\operatorname{rad}\Lambda(P)}$ , so $u$ is a unit, and $e_{i}^{\prime}u=ue_{i}$ . Thus $e_{i}^{\prime}=ue_{i}u^{-1}$ . Since conjugation preserves nonvanishing and linear dependence, it suffices to use the standard idempotents. Fix arbitrary arrow representatives and apply Lemma 5.6.

(1) Write $d:=q-p>1$ . Let $\rho$ be the image in $\End_{R[t]}(M(q))$ of the backtrack at $M(q)$ through $M(p)$ . By Lemma 5.6(a), $\rho=t^{d}u(t)$ with $u(0)\neq 0$ . Let $\omega$ be the loop at $M(q)$ if $q+1\notin P$ , and otherwise the backtrack at $M(q)$ through $M(q+1)$ . Since $p<q$ are consecutive and $q-p>1$ , we have $q-1\notin P$ and $q\geq 3$ . Thus the loop exists in the first case. By Lemma 5.6(b), $\omega=tv(t)$ with $v(0)\neq 0$ . For $1\leq j<q$ , write $\omega^{j}=t^{j}v(t)^{j}=t^{j}(v(0)^{j}+th_{j}(t))$ for a polynomial $h_{j}(t)$ . Thus the change-of-basis matrix from $\{t,t^{2},\ldots,t^{q-1}\}$ to $\{\omega,\omega^{2},\ldots,\omega^{q-1}\}$ is triangular with nonzero diagonal entries $v(0)^{j}$ . Consequently the powers of $\omega$ form a basis of the maximal ideal and $\omega^{q}=0$ . Hence

\rho=c_{d}\omega^{d}+c_{d+1}\omega^{d+1}+\cdots+c_{q-1}\omega^{q-1}

with $c_{d}\neq 0$ because $\ord_{t}\rho=d$ . The elements $\rho,\omega^{d},\omega^{d+1},\dots,\omega^{q-1}$ are images of distinct paths in the Gabriel quiver: $\rho$ uses the two arrows between $M(p)$ and $M(q)$ , while the powers of $\omega$ use only the loop at $M(q)$ or the two arrows between $M(q)$ and $M(q+1)$ . In $\End_{R[t]}(M(q))=R[t]/(t^{q})$ , an element of order less than $q$ is nonzero. Hence every path just described has nonzero image, and Lemma 5.5 applies.

(2) Let $\rho_{-}$ and $\rho_{+}$ be the images in $\End_{R[t]}(M(p+1))$ of the backtracks at $M(p+1)$ through $M(p)$ and $M(p+2)$ , respectively. By Lemma 5.6(c), $\rho_{-}=tu_{-}(t)$ and $\rho_{+}=tu_{+}(t)$ , with $u_{-}(0),u_{+}(0)\neq 0$ . Since $\rho_{+}$ is a uniformizer of $R[t]/(t^{p+1})$ and $\rho_{+}^{p+1}=0$ ,

\rho_{-}=c_{1}\rho_{+}+c_{2}\rho_{+}^{2}+\cdots+c_{p}\rho_{+}^{p}

with $c_{1}\neq 0$ . The two-step path representing $\rho_{-}$ has middle vertex $M(p)$ , whereas every positive power of the path representing $\rho_{+}$ alternates through $M(p+2)$ ; paths with different powers also have different lengths. Thus all the paths just described are distinct. Each has order at most $p<p+1$ in $\End_{R[t]}(M(p+1))=R[t]/(t^{p+1})$ , and hence has nonzero image. Lemma 5.5 again applies. ∎

Combining the split and monomial obstructions gives the following local classification, which proves parts (1) and (2) of Theorem 1.4.

Theorem 5.8.

Let $f\in R[x]$ be monic and irreducible, and let $P$ be a nonempty finite set of positive integers. Then $\Lambda_{f}(P)$ is Morita-string if and only if

\deg f=1\quad\text{and}\quad P=\{p\}\ \text{or}\ P=\{p,p+1\}\quad(p\geq 1).

In these cases, $\Lambda_{f}(P)$ is isomorphic to $R_{p}$ or $N_{p}$ , respectively, and is already split basic. Hence it is Morita-gentle if and only if it is gentle, which occurs precisely for

R_{1}=R,\qquad R_{2}=R[\varepsilon]/(\varepsilon^{2}),\qquad N_{1}=\End_{R[\varepsilon]/(\varepsilon^{2})}(R\oplus R[\varepsilon]/(\varepsilon^{2})).

Proof.

Suppose that $\Lambda_{f}(P)$ is Morita-string. By Lemma 5.3, $\deg f=1$ . Thus $f=x-\lambda$ for some $\lambda\in R$ , and the change of variable $t:=x-\lambda$ gives $\Lambda_{f}(P)\simeq\End_{R[t]}(\bigoplus_{r\in P}M(r))=\Lambda(P)$ . Since $\Lambda(P)/\operatorname{rad}\Lambda(P)\simeq R^{|P|},$ the algebra $\Lambda(P)$ is split basic. Since split basic representatives of an $R$ -linear Morita class are isomorphic as $R$ -algebras by Lemma 5.2, the string algebra in the Morita class transports its monomial Gabriel presentation to $\Lambda(P)$ .

Write $P=\{p_{1}<\cdots<p_{s}\}$ . Proposition 5.7 then gives

p_{i+1}-p_{i}=1\quad(1\leq i<s),\qquad\{p,p+1,p+2\}\nsubseteq P\quad(p\geq 1).

If $s\geq 3$ , then the first condition yields $\{p_{1},p_{1}+1,p_{1}+2\}\subseteq P$ , a contradiction. Hence $P=\{p\}$ or $P=\{p,p+1\}$ . Conversely, $\Lambda(\{p\})=\End_{R[t]}(M(p))\simeq R_{p}$ and $\Lambda(\{p,p+1\})=N_{p}$ . The algebras $R_{p}$ are monomial string algebras, and the algebras $N_{p}$ are string by Lemma 5.4. This proves the string criterion.

It remains to determine the gentle cases. If $P=\{p\}$ , then $R_{1}=R$ and $R_{p}\simeq R[\gamma]/(\gamma^{p})$ for $p\geq 2$ , where $\gamma$ is the unique loop in the Gabriel quiver. Any representative $u$ of this loop is a uniformizer. Hence $1,u,\ldots,u^{p-1}$ is an $R$ -basis of $R_{p}$ , while $u^{p}=0$ . Therefore every Gabriel presentation has kernel $(\gamma^{p})$ , which is generated by quadratic paths if and only if $p=2$ . Hence $R_{p}$ is gentle precisely for $p=1,2$ .

Now let $P=\{p,p+1\}$ . Lemma 5.4 gives $N_{p}\simeq RQ^{(2)}/((ba)^{p})$ and $\dim_{R}N_{p}=4p+1$ . If $p=1$ , this is a gentle presentation. Conversely, suppose that $p\geq 2$ and that $N_{p}$ is gentle. Since $N_{p}$ is split basic, every gentle presentation has the same Gabriel quiver $Q^{(2)}$ . Finite dimensionality forces at least one of the quadratic paths $ba$ and $ab$ to be a relation; otherwise their alternating powers give nonzero paths of arbitrary length. Consequently, the only possible nonzero paths are the two trivial paths, the arrows $a,b$ , and at most one of $ba,ab$ . Thus the quotient has dimension at most $5$ , contradicting $\dim_{R}N_{p}=4p+1\geq 9$ . Therefore $N_{p}$ is gentle if and only if $p=1$ .

Finally, suppose that one of the algebras in the string classification is Morita-gentle. It is split basic, and it is Morita equivalent to a split basic gentle algebra. Lemma 5.2 shows that the two algebras are isomorphic. Hence it is gentle itself. The converse is immediate. Therefore Morita-gentle and gentle coincide within this classified family. ∎

5.2 Global criteria and Morita reconstruction

We now apply Theorem 5.8 to the primary decomposition. For $p\geq 1$ , define $u_{p}(c):=\#\{f\in\operatorname{Irr}(c)\mid\deg f=1,\ P_{c}(f)=\{p\}\}$ and $v_{p}(c):=\#\{f\in\operatorname{Irr}(c)\mid\deg f=1,\ P_{c}(f)=\{p,p+1\}\}$ .

Corollary 5.9.

Let $c\in M_{n}(R)$ . Then $S_{n}(c,R)$ is Morita-string if and only if every $f\in\operatorname{Irr}(c)$ is linear and every $P_{c}(f)$ is a singleton or a consecutive pair. In this case,

S_{n}(c,R)_{b}\simeq\prod_{\begin{subarray}{c}p\geq 1\\ u_{p}(c)>0\end{subarray}}R_{p}^{\,u_{p}(c)}\times\prod_{\begin{subarray}{c}p\geq 1\\ v_{p}(c)>0\end{subarray}}N_{p}^{\,v_{p}(c)}.

Both products are finite. Moreover, $S_{n}(c,R)$ is Morita-gentle if and only if every $f$ is linear and $P_{c}(f)\in\{\{1\},\{2\},\{1,2\}\}$ . Equivalently, its basic algebra has the form

S_{n}(c,R)_{b}\simeq R^{u_{1}(c)}\times R_{2}^{u_{2}(c)}\times N_{1}^{v_{1}(c)}.

Proof.

Set $B_{c}:=\prod_{f\in\operatorname{Irr}(c)}\Lambda_{f}(P_{c}(f))$ . By the primary decomposition and the additive-generator reduction, $B_{c}$ is a basic algebra in the Morita class of $S_{n}(c,R)$ . If $S_{n}(c,R)$ is Morita-string, respectively Morita-gentle, then $B_{c}$ is $R$ -linearly Morita equivalent to a split basic string, respectively gentle, algebra. By Lemma 5.2, $B_{c}$ is isomorphic to that algebra. Its bound quiver decomposes along the factors of $B_{c}$ , so every $\Lambda_{f}(P_{c}(f))$ has the corresponding property. Conversely, a finite product of split basic string, respectively gentle, algebras has the same property, using the disjoint union of their bound quivers. Thus $S_{n}(c,R)$ has either Morita property if and only if every primary factor does. Theorem 5.8 now gives both criteria and the displayed decomposition. ∎

We next prove the comparison statement in Theorem 1.4(3).

Theorem 5.10.

(1)

If two matrix centralizers are Morita-string, then

$\mathrm{M}\Longleftrightarrow\mathrm{AD}\Longleftrightarrow\mathrm{D}\Longleftrightarrow\mathrm{gTZ}\Longleftrightarrow\mathrm{bTZ}.$

Within this class, both $\mathrm{bTZ}\Rightarrow\mathrm{TZ}$ and $\mathrm{TZ}\Rightarrow\mathrm{T}$ are strict.
(2)

Within the Morita-gentle class, TZ is equivalent to Morita equivalence.

Proof.

(1) The forward implications follow from Theorem 4.15 and [LX25, Theorem 1.1]. Suppose that $c\stackrel{{\scriptstyle\mathrm{bTZ}}}{{\sim}}d$ , and pair their primary blocks via the bijection in the definition of bTZ-equivalence. For a paired block, put $r:=\operatorname{LL}(U_{c}(f))=\operatorname{LL}(U_{d}(\sigma(f)))$ and $s:=|P_{c}(f)|=|P_{d}(\sigma(f))|$ . By Corollary 5.9, both $f$ and $\sigma(f)$ are linear, $s\in\{1,2\}$ , and each exponent set is a singleton or a consecutive pair. Hence $U_{c}(f)\simeq R[t]/(t^{r})\simeq U_{d}(\sigma(f))$ , where $r$ is their common Loewy length and the largest exponent in each set. Consequently,

P_{c}(f)=P_{d}(\sigma(f))=\begin{cases}\{r\},&s=1,\\ \{r-1,r\},&s=2.\end{cases}

Thus $c\stackrel{{\scriptstyle\mathrm{M}}}{{\sim}}d$ , and the five relations coincide.

To separate bTZ from TZ, take $c_{1}:=J_{1}(0)\oplus J_{2}(0)\oplus J_{3}(1)$ and $c_{2}:=J_{2}(0)\oplus J_{2}(1)\oplus J_{3}(1)$ . Their basic centralizers are $N_{1}\times R_{3}$ and $R_{2}\times N_{2}$ . Both have center $R_{2}\times R_{3}$ and support $\tau$ -tilting poset $\operatorname{Weak}(\Sigma_{3})\times\operatorname{Weak}(\Sigma_{2})$ , whereas their blockwise data are $\{\!\{(R_{2},2),(R_{3},1)\}\!\}$ and $\{\!\{(R_{2},1),(R_{3},2)\}\!\}$ . Hence $c_{1}\stackrel{{\scriptstyle\mathrm{TZ}}}{{\sim}}c_{2}$ but $c_{1}\not\stackrel{{\scriptstyle\mathrm{bTZ}}}{{\sim}}c_{2}$ .

To separate TZ from T, choose distinct $\lambda_{1},\lambda_{2}\in R$ and let $d_{1},d_{2}$ have elementary-divisor sets $\{x-\lambda_{1},(x-\lambda_{2})^{3}\}$ and $\{(x-\lambda_{1})^{2},(x-\lambda_{2})^{2}\}$ , respectively. Both have T-type $\{\!\{1,1\}\!\}$ , but their centers are $R\times R_{3}$ and $R_{2}\times R_{2}$ . Thus $d_{1}\stackrel{{\scriptstyle\mathrm{T}}}{{\sim}}d_{2}$ and $d_{1}\not\stackrel{{\scriptstyle\mathrm{TZ}}}{{\sim}}d_{2}$ .

(2) Let $A_{c}:=S_{n}(c,R)$ and $A_{d}:=S_{m}(d,R)$ be Morita-gentle, and suppose $c\stackrel{{\scriptstyle\mathrm{TZ}}}{{\sim}}d$ . By Corollary 5.9, write their basic algebras as

(A_{e})_{b}\simeq R^{u_{1}(e)}\times R_{2}^{u_{2}(e)}\times N_{1}^{v_{1}(e)}\qquad(e\in\{c,d\}).

The T-data determine $v_{1}(e)$ and $u_{1}(e)+u_{2}(e)$ . Moreover, $Z(A_{e})\simeq R^{u_{1}(e)}\times R_{2}^{u_{2}(e)+v_{1}(e)}$ . Since the centers are isomorphic, uniqueness of the decomposition into local factors gives equality of $u_{1},u_{2},v_{1}$ . Hence the displayed basic algebras are isomorphic, and therefore the centralizers are Morita equivalent. Conversely, Morita equivalence preserves centers and support $\tau$ -tilting posets. Thus it implies TZ-equivalence. ∎

Remark 5.11.

Theorem 5.10(1) shows that the first five relations in the strict implication chain coincide within the Morita-string class, while Theorem 5.10(2) shows that the separate center and support $\tau$ -tilting data determine the Morita class within the Morita-gentle class. However, the reduction (4.2) removes the elementary-divisor multiplicities $m_{c}(f,a)$ . Hence neither conclusion implies matrix similarity or algebra isomorphism.

Acknowledgments

Jiangsheng Hu was supported by the National Natural Science Foundation of China (Grant No. 12571035). Yu-Zhe Liu was supported by the National Natural Science Foundation of China (Grant Nos. 12401042 and 12561008), the Science and Technology Foundation of the Guizhou S&T Department (Grant Nos. VZD[2026]001, ZD[2025]085, and ZK[2024]YiBan066), and the Scientific Research Foundation of Guizhou University (Grant No. [2023]16). Tiwei Zhao was supported by the National Natural Science Foundation of China (Grant No. 12471036) and the Hubei Provincial Natural Science Foundation of China (Grant No. 2026AFA094). The authors thank Professor Changchang Xi for valuable comments.

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[XZ22] C. C. Xi, J. B. Zhang, Centralizer matrix algebras and symmetric polynomials of partitions, J. Algebra 609 (2022), 688–717.

Support τ\tau-tilting posets and Hochschild reconstruction for matrix centralizer algebras

1 Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

2 Central reduction for support τ\tau-tilting posets

2.1 Support τ\tau-tilting posets

2.2 Central reduction

Proposition 2.1.

Proof.

3 Reconstruction over local principal ideal algebras

Theorem 3.1.

Proof.

Corollary 3.2.

Proof.

Lemma 3.3.

Proof.

Theorem 3.4.

Proof.

Corollary 3.5.

Proof.

Corollary 3.6.

Proof.

Remark 3.7.

4 Matrix centralizer algebras

4.1 Primary decomposition and recovery of the T-type

Proof of the product formula in Theorem 1.2.

Lemma 4.1.

Proof.

Proof of the equivalences in Theorem 1.2.

Remark 4.2.

4.2 Centers and their local factors

Proposition 4.3.

Proof.

Corollary 4.4.

Proof.

Remark 4.5.

4.3 Hochschild reconstruction and derived invariance

Theorem 4.6.

Proof.

Corollary 4.7.

Proof.

Corollary 4.8.

Proof.

Remark 4.9.

Proof of Theorem 1.3.

4.4 Comparison of the reconstruction relations

The Li–Xi equivalences.

Definition 4.10.

Proposition 4.11.

Proof.

Remark 4.12.

Proposition 4.13.

Proof.

Definition 4.14.

Theorem 4.15.

Proof.

Proposition 4.16.

Proof.

5 Rigidity and Morita reconstruction for string and gentle centralizers

Definition 5.1.

Lemma 5.2.

Proof.

5.1 The split obstruction and the local criterion

Lemma 5.3.

Proof.

Lemma 5.4.

Proof.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Proposition 5.7.

Proof.

Theorem 5.8.

Proof.

5.2 Global criteria and Morita reconstruction

Corollary 5.9.

Proof.

Support $\tau$ -tilting posets and Hochschild reconstruction for matrix centralizer algebras

2 Central reduction for support $\tau$ -tilting posets

2.1 Support $\tau$ -tilting posets