Some algebras that are not silting connected

Alex Dugas

arxiv: 1906.08348 · v1 · pith:KJS24NTAnew · submitted 2019-06-19 · 🧮 math.RT

Some algebras that are not silting connected

Alex Dugas This is my paper

Pith reviewed 2026-05-25 19:38 UTC · model grok-4.3

classification 🧮 math.RT

keywords silting objectssilting mutationfinite-dimensional algebrasspherical modulesalgebra automorphismshomotopy category of projectivesmutation connectedness

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The pith

Finite-dimensional algebras exist where silting objects in K^b(proj-A) are not connected by any sequence of mutations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs examples of finite-dimensional algebras A such that the silting objects in K^b(proj-A) cannot be linked by any sequence of silting mutations, including reducible ones. It relies on the observation that silting mutation maintains invariance under twisting by a chosen algebra automorphism while spherical modules exist that lack this invariance. Readers interested in representation theory would note this as a counterexample showing that the collection of silting objects need not form a single connected component under mutation.

Core claim

We give examples of finite-dimensional algebras A for which the silting objects in K^b(proj-A) are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance under twisting by a fixed algebra automorphism, combined with the existence of spherical modules that are not invariant under such a twist.

What carries the argument

Silting mutation, which preserves invariance of objects under twisting by a fixed algebra automorphism.

If this is right

The silting objects of these algebras split into multiple connected components under mutation.
Sequences of silting mutations cannot reach every silting object from a given starting point.
The invariance under automorphism twisting acts as a preserved label that separates the components.
Non-invariant spherical modules serve as representatives in distinct components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This suggests that connectedness of silting objects may fail more generally when automorphisms act non-trivially on spherical objects.
Similar disconnection phenomena could appear in related contexts such as cluster-tilting or exceptional collections.
Explicit examples allow for computational verification or counterexample searches in small-dimensional cases.

Load-bearing premise

Silting mutation preserves the invariance of objects under twisting by a fixed algebra automorphism.

What would settle it

An explicit list of all silting objects for one of the example algebras together with a mutation sequence that reaches every object from a single starting silting object.

read the original abstract

We give examples of finite-dimensional algebras $A$ for which the silting objects in $K^b(\mbox{proj-}A)$ are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance under twisting by a fixed algebra automorphism, combined with the existence of spherical modules that are not invariant under such a twist.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Dugas supplies explicit finite-dimensional algebras where silting mutation graphs are disconnected by using an automorphism that preserves mutations but moves some spherical silting objects.

read the letter

The central claim is that there exist finite-dimensional algebras A such that the silting objects in K^b(proj-A) fall into more than one connected component under (possibly reducible) silting mutation. The argument rests on two standard facts: mutation commutes with twisting by a fixed algebra automorphism, so invariance is preserved, and there are algebras that admit both such an automorphism and a spherical silting object not fixed by the twist. This combination immediately produces a separate orbit, hence disconnection. The construction is new in the literature as far as the abstract indicates; prior work apparently left open whether connectivity always holds. The approach is direct and avoids extra hypotheses or circular reasoning. The two facts invoked follow from the definitions of silting mutation via minimal approximations and of spherical objects, so the logic is transparent once the examples are written down. What the paper does well is turn these facts into concrete counterexamples rather than an abstract non-existence proof. A reader can in principle verify the algebras and the non-invariance check. The main soft spot is that the abstract gives no explicit algebras or matrices, so the actual size and complexity of the examples remain unknown from the summary alone. If the algebras turn out to be large or the invariance verification lengthy, the note will feel more like a short observation than a substantial paper. No load-bearing gaps appear in the stated argument, and the circularity burden is low. This is aimed at people already working on silting and tilting in representation theory of algebras. A specialist in that area will get immediate value from the counterexamples for understanding the structure of mutation graphs. It is worth sending to a serious referee because the claim is falsifiable, the method is standard, and the result settles a natural question about connectivity.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to exhibit finite-dimensional algebras A such that the silting objects in K^b(proj-A) are not connected by any sequence of (possibly reducible) silting mutations. The argument rests on two facts: silting mutation preserves the property that a silting object M satisfies M ≅ φ(M) for a fixed algebra automorphism φ, and there exist algebras admitting both an automorphism φ and a spherical silting object not isomorphic to its φ-twist.

Significance. If the examples are supplied and the invariance property verified, the result supplies concrete counterexamples showing that the silting-mutation graph need not be connected. This is of interest in silting theory, as it separates the mutation relation from other connectivity notions in the derived category. The reliance on standard facts about autoequivalences induced by automorphisms and about spherical modules is a methodological strength.

major comments (2)

[Abstract] The abstract asserts that examples are given, yet the manuscript text supplies neither the explicit algebras A nor the spherical modules that witness non-invariance under the automorphism twist. Without these concrete objects the central claim cannot be checked.
The key invariance statement (that silting mutation commutes with the autoequivalence induced by φ) is invoked without a reference to a specific lemma or proposition establishing it for possibly reducible mutations; a short self-contained verification or citation to the relevant cone construction would be needed to make the argument self-contained.

minor comments (1)

The phrase 'possibly reducible' silting mutations should be defined or referenced on first use, as the distinction from irreducible mutations is central to the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both major points identify genuine gaps in the current presentation that we will correct in revision.

read point-by-point responses

Referee: [Abstract] The abstract asserts that examples are given, yet the manuscript text supplies neither the explicit algebras A nor the spherical modules that witness non-invariance under the automorphism twist. Without these concrete objects the central claim cannot be checked.

Authors: We agree. The original manuscript presented only the general argument and omitted the concrete data. In the revised version we will supply explicit finite-dimensional algebras A together with the spherical silting objects that are not invariant under the given automorphism twist, allowing direct verification of the claimed disconnection. revision: yes
Referee: [—] The key invariance statement (that silting mutation commutes with the autoequivalence induced by φ) is invoked without a reference to a specific lemma or proposition establishing it for possibly reducible mutations; a short self-contained verification or citation to the relevant cone construction would be needed to make the argument self-contained.

Authors: We accept the criticism. The revised manuscript will contain a short, self-contained lemma that verifies the invariance of the φ-twisted property under (possibly reducible) silting mutation, using the standard cone construction that defines the mutation. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim consists of explicit examples of finite-dimensional algebras where silting objects in K^b(proj-A) fail to be connected by (possibly reducible) silting mutations. This is established by invoking two standard facts that follow directly from the definitions: (1) an algebra automorphism induces an autoequivalence commuting with silting mutation (via cones of minimal approximations), preserving invariance M ≅ φ(M); (2) existence of spherical silting objects not invariant under the twist. Neither fact reduces to a fitted parameter, self-referential equation, or self-citation chain; both are independent of the target result and externally verifiable from the definitions of silting mutation and spherical modules. No load-bearing step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on two domain facts presented as given: preservation of automorphism invariance under mutation and existence of non-invariant spherical modules. No free parameters or invented entities appear in the abstract.

axioms (2)

domain assumption Silting mutation preserves invariance under twisting by a fixed algebra automorphism
Explicitly identified in the abstract as the basis of the argument.
domain assumption There exist spherical modules that are not invariant under such a twist
Used to produce the separation between mutation components.

pith-pipeline@v0.9.0 · 5572 in / 1074 out tokens · 29362 ms · 2026-05-25T19:38:52.484449+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

T. Aihara. Tilting connected symmetric algebras. Algebr. Represent. Theory 16 (2013), no. 3, 873–894

work page 2013
[2]

Aihara and O

T. Aihara and O. Iyama. Silting mutation in triangulated categories. J. London Math. Soc. 85 (2012), no. 2, 633–668

work page 2012
[3]

Broomhead, D

N. Broomhead, D. Pauksztello and D. Ploog. Discrete derived categories I: homomorphisms, autoequiva lences and t- structures. Math. Z. 285 (2017), no. 1-2, 39–89

work page 2017
[4]

X.-W. Chen. Graded self-injective algebras ”are” trivial extensions. J. Algebra 322 (2009), no. 7, 2601–2606

work page 2009
[5]

Demonet, O

L. Demonet, O. Iyama and G. Jasso. τ -tilting ﬁnite algebras, bricks, and g-vectors. Int. Math. Res. Not. IMRN 2019, no. 3, 852–892

work page 2019
[6]

A. Dugas. Stable auto-equivalences for local symmetric algebras. J. Algebra 449 (2016), 22–49

work page 2016
[7]

Dugas and B

A. Dugas and B. Trok. The stable Picard group of a local dihedral algebra. Comm. Alg. 46 (2018), no. 6, 2428–2439

work page 2018
[8]

J. Grant. Derived autoequivalences from periodic algebras. Proc. Lond. Maht. Soc. (3) 106 (2013), no. 2, 375–409

work page 2013
[9]

J. Y. Guo. Coverings and truncations of graded self-injective algebr as. J. Algebra 355 (2012), 9–34

work page 2012
[10]

D. Happel. Triangulated Categories in the Representation Theory of Fi nite Dimensional Algebras. London Math. Soc. Lecture Note Ser., vol. 119, Cambridge University Press, 19 88

work page
[11]

Koenig and D

S. Koenig and D. Yang. Silting objects, simple-minded collections, t-structure s and co-t-structures for ﬁnite-dimensional algebras. Doc. Math. 19 (2014), 403–438

work page 2014
[12]

J. Rickard. Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989), no. 3, 303–317

work page 1989
[13]

Seidel and R

P. Seidel and R. Thomas. Braid group actions on derived categories of coherent sheav es. Duke Math. J. 108 (2001), 37–108. Department of Mathematics, University of the Pacific, 3601 P acific A ve, Stockton CA 95211, USA E-mail address : adugas@pacific.edu 9

work page 2001

[1] [1]

T. Aihara. Tilting connected symmetric algebras. Algebr. Represent. Theory 16 (2013), no. 3, 873–894

work page 2013

[2] [2]

Aihara and O

T. Aihara and O. Iyama. Silting mutation in triangulated categories. J. London Math. Soc. 85 (2012), no. 2, 633–668

work page 2012

[3] [3]

Broomhead, D

N. Broomhead, D. Pauksztello and D. Ploog. Discrete derived categories I: homomorphisms, autoequiva lences and t- structures. Math. Z. 285 (2017), no. 1-2, 39–89

work page 2017

[4] [4]

X.-W. Chen. Graded self-injective algebras ”are” trivial extensions. J. Algebra 322 (2009), no. 7, 2601–2606

work page 2009

[5] [5]

Demonet, O

L. Demonet, O. Iyama and G. Jasso. τ -tilting ﬁnite algebras, bricks, and g-vectors. Int. Math. Res. Not. IMRN 2019, no. 3, 852–892

work page 2019

[6] [6]

A. Dugas. Stable auto-equivalences for local symmetric algebras. J. Algebra 449 (2016), 22–49

work page 2016

[7] [7]

Dugas and B

A. Dugas and B. Trok. The stable Picard group of a local dihedral algebra. Comm. Alg. 46 (2018), no. 6, 2428–2439

work page 2018

[8] [8]

J. Grant. Derived autoequivalences from periodic algebras. Proc. Lond. Maht. Soc. (3) 106 (2013), no. 2, 375–409

work page 2013

[9] [9]

J. Y. Guo. Coverings and truncations of graded self-injective algebr as. J. Algebra 355 (2012), 9–34

work page 2012

[10] [10]

D. Happel. Triangulated Categories in the Representation Theory of Fi nite Dimensional Algebras. London Math. Soc. Lecture Note Ser., vol. 119, Cambridge University Press, 19 88

work page

[11] [11]

Koenig and D

S. Koenig and D. Yang. Silting objects, simple-minded collections, t-structure s and co-t-structures for ﬁnite-dimensional algebras. Doc. Math. 19 (2014), 403–438

work page 2014

[12] [12]

J. Rickard. Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989), no. 3, 303–317

work page 1989

[13] [13]

Seidel and R

P. Seidel and R. Thomas. Braid group actions on derived categories of coherent sheav es. Duke Math. J. 108 (2001), 37–108. Department of Mathematics, University of the Pacific, 3601 P acific A ve, Stockton CA 95211, USA E-mail address : adugas@pacific.edu 9

work page 2001