Auslander regular algebras and Coxeter matrices

Hugh Thomas; Rene Marczinzik; Vikt\'oria Kl\'asz

REVIEW 4 major objections 5 minor 3 cited by

Auslander regular algebras and Coxeter matrices

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Pith's one-line read The paper shows that the grade bijection of an Auslander regular algebra coincides with the permutation matrix in the Bruhat decomposition of its Coxeter matrix, and derives consequences for permanents, distributive lattices, and category…

desk verdict A genuinely new and mostly clean paper identifying the grade bijection with the Bruhat permutation of the Coxeter matrix, with one load-bearing citation to a proof-part of Iyama that should be promoted to a lemma. read the letter →

arxiv 2501.09447 v2 pith:ENSAKFIM submitted 2025-01-16 math.RT math.CO

Viktória Klász , Rene Marczinzik , Hugh Thomas This is my paper · ORCID

classification math.RTmath.CO MSC 16G1016E1006A11

keywords PartiallyorderedsetsDistributivelatticesAuslanderregularalgebrasCoxetermatrixRowmotionbijectiongradeBruhatdecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that two apparently different ways of associating a permutation to an Auslander regular algebra—one homological (the grade bijection on simple modules), one linear algebraic (the permutation part of a Bruhat factorization of the Coxeter matrix)—always give the same answer. This makes the grade bijection computable by Gaussian elimination instead of by injective resolutions. As consequences, the permanent of any Auslander regular Coxeter matrix is 1 or −1, giving a quick numerical way to disprove Auslander regularity; and a finite lattice is distributive exactly when the first nonzero entry of each row of its Coxeter matrix lies in a distinct column. The paper also shows that the grade bijection agrees with the classical injective-to-projective syzygy bijection, and applies the results to blocks of category O.

What carries the argument

The load-bearing identity is the Bruhat factorization $C = P U_2$. The proof identifies the Coxeter matrix $C = -\omega^T \omega^{-1}$ with the matrix $M$ recording, with signs, which indecomposable injectives appear in the minimal injective coresolutions of indecomposable projectives; the row/column dimension-vector relation then gives $M = -C$. The admissible ordering of simples (grades nonincreasing along the order) makes the matrix $\varphi^{-1}M$ upper triangular, using the key property that the grade bijection $\varphi$ sends grade to cograde. The permutation matrix $P$ in the Bruhat decomposition is thus forced to be $\varphi$.

What would settle it

Compute the permanent of the Coxeter matrix of any finite-dimensional algebra of finite global dimension; if an Auslander regular algebra has a Coxeter matrix with permanent other than 1 or −1, the central theorem fails, and since the permanent is independent of the ordering of simples, a single such example suffices.

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Extended reading notes

Core claim

The central claim is that for a finite-dimensional Auslander regular algebra, the grade bijection—the map sending a simple module to the top of the socle of its first nonzero Ext against the algebra—is encoded verbatim by linear algebra. When the simple modules are ordered so that grades increase with index, the Coxeter matrix factorizes as $C = P U$, where $P$ is the permutation matrix of the grade bijection and $U$ is upper triangular with diagonal entries 1 or −1. Equivalently, the grade bijection is the unique permutation part of the Bruhat decomposition of the Coxeter matrix. Along the way the paper proves that the grade bijection coincides with the classical bijection between indecomposable injective and projective modules given by syzygies, and uses the factorization to show that the permanent of any Auslander regular Coxeter matrix is 1 or −1, and to characterize distributive lattices by the pivot pattern of their Coxeter matrix.

Load-bearing premise

The argument assumes a known structural property of the grade bijection, cited from the literature rather than proved here: in the module $\mathrm{Ext}^r_A(S,A)$, every simple composition factor other than the socle has grade greater than $r$, and dually every top except one has cograde greater than $r$.

Editorial extensions

If this is right

The permanent of the Coxeter matrix of any Auslander regular algebra is either 1 or −1, providing a fast necessary test: a permanent outside this set shows the algebra is not Auslander regular.
For a finite lattice ordered by a linear extension, distributivity is equivalent to the condition that the first nonzero entry of each row of the Coxeter matrix lies in a different column; in that case the Coxeter permutation is the rowmotion bijection.
For blocks of category O, the set of grades of modules equals twice the values of Lusztig's a-function.
For an Auslander regular algebra with a simple-preserving duality, the Cartan matrix is symmetric and the grade bijection is the identity.
The grade bijection of an Auslander-Gorenstein algebra equals the classical injective-to-projective syzygy bijection, so the two historical definitions describe the same map.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because a Bruhat permutation is computed by pivoting, the result should make grade bijections computable in polynomial time for algebras where injective resolutions are slow; an implementation on quiver algebras would be a direct test.
Beyond the paper: the ±1 permanent constraint is a necessary condition that may be much easier to verify than Auslander regularity; applying it to families of algebras of finite global dimension with unknown homological dimension could identify new candidates for Auslander regular algebras.
Beyond the paper: the lattice theorem suggests a purely matrix-theoretic route to rowmotion on distributive lattices; one could test whether the entries of $U_2$ in the factorization carry additional order information about the lattice.

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Formalized claims in Lean

Claim #1: The central claim is that for a finite-dimensional Auslander regular algebra, the grade bijection—the map sending a simple module to the top of the socle of its first nonzero Ext against the algebra—is encoded verbatim by linear algebra. When the simple modules are ordered so that grades increase with index, the Coxeter matrix factorizes as $C = P U$, where $P$ is the permutation matrix of the gra

/-- @claim 1 The central claim is that for a finite-dimensional Auslander regular algebra, the grade bijection—the map sending a simple module to the top of the socle of its first nonzero Ext against the algebra—is encoded verbatim by linear algebra. When the simple modules are ordered so that grades increase with index, the Coxeter matrix factorizes as $C = P U$, where $P$ is the permutation matrix of the gra -/
def central_claim : Prop :=

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one before sending it back. The headline claim is real: for Auslander regular algebras, the permutation part of the Bruhat decomposition of the Coxeter matrix is exactly Iyama's grade bijection, and this also matches the Auslander-Reiten bijection. That is a new and useful way to compute a homological invariant from a matrix factorization.

The paper does a lot right. Theorem 3.3, the characterization of Auslander-Gorenstein by grade S = pdim I(S), is clean and self-contained. The proof of Theorem 3.13 is short: once you accept Proposition 3.9, the upper-triangularity argument via admissible orderings is elegant. The permanent corollary is a genuinely fast numerical obstruction, and the example with permanent -1501 is nice. The distributive lattice section gives a new characterization (U1 = id in a Bruhat decomposition) and connects to rowmotion, building on [IM]. I also like the category O application.

The soft spot is exactly the one the stress-test note highlights. Proposition 3.9 cites 'part (ii) of the proof of [I, Theorem 2.10]' for two structural facts: all composition factors of Ext^r(S,A) except the socle have grade > r, and the socle occurs with multiplicity one. The second fact is essential for the ±1 diagonal of U2; without it you could get other diagonal entries. The first is essential for upper triangularity. These facts may well be true and proved by Iyama, but citing a step inside a proof rather than a stated theorem is fragile, and the authors should either state and prove them as a lemma or point to a theorem number that covers them. This is addressable, not fatal.

Section 4 is more compressed. Lemma 4.5's formula for the Coxeter matrix of a lattice with all proper upper intervals distributive is plausible but only sketched; Lemma 4.8's induction is quick. The rowmotion identification in Theorem 4.7 is dense, and I'd want a referee to check the edge cases around 'min Y^c' and the inverse rowmotion convention. Also, the relation to the authors' earlier [MTY] is only a one-line remark; they should say a bit more about what is genuinely new beyond that paper. (The reader's note that [MTY] is never cited is wrong; it appears in Section 2.3.4.)

All in all, this paper deserves a serious referee. It is a genuine contribution, the main arguments are sound on a first pass, and the issues are fixable with a revision. I would cite it and would bring it to a reading group that cares about homological algebra or lattice combinatorics.

Referee Report

4 major / 5 minor

Summary. The paper establishes that for Auslander-Gorenstein algebras, Iyama's grade bijection coincides with the Auslander-Reiten bijection, and for Auslander regular algebras with an admissible ordering of simples, the Coxeter matrix admits a Bruhat decomposition C = P U2 in which P is the permutation matrix of the grade bijection and U2 is upper triangular with diagonal entries ±1. From this it derives that the permanent of the Coxeter matrix is ±1. In a second part, it gives a characterisation of finite distributive lattices in terms of the Bruhat decomposition of their incidence algebra's Coxeter matrix, and applies the results to blocks of category O.

Significance. The linear-algebraic reinterpretation of the grade bijection is elegant and has immediate computational value, and the permanent corollary gives a fast numerical necessary condition for Auslander regularity. The lattice characterisation is striking and connects rowmotion to Coxeter matrices. The proof strategy is mostly clean: Theorem 3.3's equivalence and the AR/grades bijection proof are well structured, and the matrix argument for Theorem 3.13 is transparent once its hypotheses are granted. The main caveat is that the proof of Theorem 3.13 delegates a structurally essential point to a fact quoted from inside the proof of Iyama's Theorem 2.10 rather than to a stated theorem.

major comments (4)

[Proposition 3.9 / Theorem 3.13] The proof of Theorem 3.13 rests on Proposition 3.9, which invokes "part (ii) of the proof of Theorem 2.10 in [I]" for two facts: all composition factors of Ext^r_A(S,A) other than the socle Dφ(S) have grade > r, and Dφ(S) occurs with multiplicity one. These facts are not stated in [I, Theorem 2.10] (as cited) and not proved here. The upper triangularity of φ^{-1}M in the proof of Theorem 3.13 depends on the first fact, and the assertion that the diagonal entries of U2 are ±1 depends on the second: if the multiplicity of Dφ(S) were m > 1, the corresponding diagonal entry would be ±m. Please provide a self-contained proof or a direct reference to a stated theorem containing these properties.
[Lemma 4.1] The dot-product rule in the proof of Lemma 4.1 reads "C_{x·} · [P(y)] is −1 if y ≥ x and 0 otherwise," but no convention for the dimension vectors of projectives and injectives in the incidence algebra is specified. In particular, with Cartan entries e_{i,j} = dim e_j A e_i, the identity C·dim P(y)=-dim I(y) determines the direction of the inequality, and the subsequent equivalence "for y ∈ I, this is the same as −1 if y ≥ x ∨ z" depends on that direction. Since Corollary 4.2 and Theorem 4.9 rely on Lemma 4.1, please state the convention explicitly and verify the proof of Lemma 4.1 under that convention.
[Lemma 4.5] The verification of the Coxeter matrix formula is given as a few dot-product assertions ("the dot product is therefore zero, as desired"). This is load-bearing for Theorems 4.7 and 4.9, which identify leftmost nonzero entries of rows via Δ(Y) and χ_red(Δ(Y)). Please expand the proof to show explicitly how the entries follow from the definition C = -ω^T ω^{-1} and the Möbius inversion for the incidence algebra, and state the precise support conventions for projective and injective dimension vectors.
[Theorem 4.7] The statement "the leftmost non-zero entry in row Y is in column ⟨min Y^c⟩ = row^{-1}(Y)" needs clarification: the standard rowmotion formula row(I) = ⟨min(P \ I)⟩ gives ⟨min Y^c⟩ = row(Y), not row^{-1}(Y). The inverse appears because Y is a set of meet-irreducibles and the correspondence with R reverses order, but this is not spelled out. Please state the convention explicitly and define the rowmotion used on R and on order ideals of meet-irreducibles, so that the conclusion that the Coxeter permutation is the grade bijection/rowmotion is verifiable.

minor comments (5)

[Title page] The running title contains a typo: "ma trices" should be "matrices".
[Section 2.3.4] The phrase "where is acts as ν" should read "where it acts as ν".
[Proof of Theorem 3.13] The matrix M is introduced as containing "multiplicity (with sign)"; please specify the sign convention, since the diagonal entries of U2 depend on it.
[Example 3.15] The poset example would be easier to verify if a Hasse diagram were included.
[Section 2.1] The notation "idim" is used without definition; it should be defined where first used, parallel to "pdim".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main Bruhat-factorization theorem is derived, not assumed, and self-citations are background only.

full rationale

The central result, Theorem 3.13, is not circular. The matrix M is defined directly from the multiplicities of indecomposable injectives in the minimal injective coresolutions of projectives; the identity M = -C follows from the Cartan/Coxeter definitions, and the argument that phi^{-1}M is upper triangular uses only the admissible ordering and the external grade-bijection property cograde(phi(S)) = grade(S). No parameter is fitted, no prediction is renamed as an input, and the conclusion C = P U2 is not assumed. Theorem 3.3 is a genuine characterization proved from Lemma 3.2 and standard facts, not from the target equality. Proposition 3.9 does cite part (ii) of the proof of Iyama's Theorem 2.10 for two structural facts, including the multiplicity-one property of D phi(S); this is an external published result rather than a self-citation, and while it is a load-bearing dependency for the +/-1 diagonal, depending on external theorems is not circularity. The self-citations [IM] and [MTY] are used for background and for the rowmotion remark, but Theorem 4.7 independently proves that the Coxeter permutation for a distributive lattice is the rowmotion bijection using Lemma 4.5, so the self-citation is not needed for the main claim. No equation in the paper reduces to its own input by construction, and no fitted quantity is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. Its results are theorems that depend on standard theorems from representation theory, especially Iyama's published properties of the grade bijection and Auslander-Reiten's bijection.

assumptions (6)

standard math Iyama's grade bijection properties (cograde(phi(S)) = grade(S); composition series structure of Ext^r(S,A) in [I, Theorem 2.10])
Used in Proposition 3.9 and Lemma 3.10 to locate the unique projective of injective dimension r and to prove upper triangularity in Theorem 3.13.
standard math Auslander-Reiten bijection results, especially [AR, Prop 5.4(b)] giving the bijection Omega^{-r} between projectives of injective dimension r and injectives of projective dimension r in r-Gorenstein algebras
Used in Theorem 3.12 to identify the final terms of resolutions.
standard math Fundamental theorem of finite distributive lattices and the structure lemma for lattices with all proper upper intervals distributive (Lemma 4.3)
Underlies the Coxeter matrix description in Lemma 4.5 and the distributivity characterization.
standard math Bruhat decomposition existence and uniqueness of the permutation matrix for invertible matrices (Theorem 2.6)
Basis for defining the Coxeter permutation and for the factorization C = P U2.
standard math Morita equivalence reduction to quiver algebras over algebraically closed fields
Stated at the start of the paper as no loss of generality; all notions are Morita invariant.
standard math Auslander-Gorenstein algebras are Iwanaga-Gorenstein and the property is left-right symmetric
Used in Lemma 2.2, Lemma 3.10 and Theorem 3.3.

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Pith. "Pith review of Auslander regular algebras and Coxeter matrices." pith.science (2026). https://pith.science/paper/ENSAKFIM

@misc{pith2026250109447,
  author       = {Pith},
  title        = {Pith review of: Auslander regular algebras and Coxeter matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENSAKFIM}},
  note         = {Machine review of arXiv:2501.09447}
}

abstract

We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category $\mathcal{O}$ of semisimple Lie algebras.

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Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 3 Pith papers

[1]

Classics in Mathematics

Aigner, M.: Combinatorial theory. Classics in Mathematics. Berlin: Springer. x, 484 p. (1997)

work page 1997
[2]

Assem, I.; Simson, D.; Skowronski, A.: Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory. London Mathematical Society Student Texts 65. Cambridge: Cambridge University Press, 458 p. (2006)

work page 2006
[3]

O.: Representation theory of Artin algebras

Auslander, M.; Reiten, I.; Smalo, S. O.: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, Volume 36, Cambridge University Press, 1997

work page 1997
[4]

Auslander, M., Reiten, I.: k-Gorenstein algebras and syzygy modules. J. Pure Appl. Algebra, 92, 1-27 (1994)

work page 1994
[5]

Proceedings of the American Mathematical Society Vol

Auslander, M., Reiten, I.: On a Generalized Version of the Nakayama Conjecture. Proceedings of the American Mathematical Society Vol. 52, No. 1 (1975), pp. 69-74

work page 1975
[6]

Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8-28 (1963)

work page 1963
[7]

Beilinson, A.; Ginzburg, V.; Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473-527

work page 1996
[8]

Cambridge Studies in Advanced Mathematics, Volume 30, Cambridge University Press, 1991

Benson, D.: Representations and cohomology I: Basic representation theory of finite groups and associative algebras. Cambridge Studies in Advanced Mathematics, Volume 30, Cambridge University Press, 1991

work page 1991

Show all 38 references

[9]

Graduate Texts in Mathematics 225

Bump, D.: Lie groups. Graduate Texts in Mathematics 225. New York, NY: Springer. (2013)

2013
[10]

https://arxiv.org/abs/2012.11927

Chan, A.; Darp\"o, E.; Iyama, O.; Marczinzik, R.: Periodic trivial extension algebras and fractionally Calabi-Yau algebras. https://arxiv.org/abs/2012.11927

2012 arXiv
[11]

https://arxiv.org/abs/1712.04587

Chen, X.: Gorenstein Homological Algebra of Artin Algebras. https://arxiv.org/abs/1712.04587

arXiv
[12]

Eilenberg, S.: Algebras of cohomologically finite dimension. Comment. Math. Helv. 28 (1954), no.1, 310--319

1954
[13]

Fuller, K.; Zimmermann-Huisgen, B.: On the generalized Nakayama conjecture and the Cartan determinant problem. Trans. Am. Math. Soc. 294, 679-691 (1986)

1986
[14]

Grant, J.: The Nakayama automorphism of a self-injective preprojective algebra. Bull. Lond. Math. Soc. 52, No. 1, 137-152 (2020)

2020
[15]

Humphreys, J. E. Representations of semisimple Lie algebras in the BGG category O . Graduate Studies in Mathematics 94. Providence, RI: American Mathematical Society (AMS) 289 p. (2008)

2008
[16]

2, 528-535

Iyama, O.: Symmetry and duality on n -Gorenstein rings , Journal of Algebra, 269 (2003), no. 2, 528-535

2003
[17]

Iyama, O.: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210, No. 1, 22-50 (2007)

2007
[18]

Iyama, O.: The relationship between homological properties and representation theoretic realization of artin algebras. Trans. Amer. Math. Soc. 357 (2005), no. 2, 709-734

2005
[19]

Advances in Mathematics Volume 398, 2022

Iyama, O.; Marczinzik, R.: Distributive lattices and Auslander regular algebras. Advances in Mathematics Volume 398, 2022

2022
[20]

In progress

Kl\'asz, V.: The Auslander-Gorenstein condition for monomial algebras. In progress
[21]

Ko, H.; Mazorchuk, V.; Mrden, R.: Some homological properties of category O . VI. Doc. Math. 26, 1237-1269 (2021)

2021
[22]

A.: Spectral analysis of finite dimensional algebras and singularities

Lenzing, H.; de la Pena, J. A.: Spectral analysis of finite dimensional algebras and singularities. Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop. EMS Series of Congr...

2008
[23]

Fields Institute Communications

Madsen, D.: Projective dimensions and Nakayama algebras. Fields Institute Communications. 45. Amer. Math. Soc., Providence, RI, 2005. 247-265

2005
[24]

Marczinzik, R.; Thomas, H.; Yildirim, E.: On the interaction of the Coxeter transformation and the rowmotion bijection. J. Combinatorial Algebra 8 (2024) no. 3--4, 359--374

2024
[25]

Mazorchuk, V.: Some homological properties of the category O . II. Represent. Theory 14 (2010), 249-263

2010
[26]

Encyclopedia of Mathematics and its Applications 6

Minc, H.: Permanents. Encyclopedia of Mathematics and its Applications 6. Cambridge: Cambridge University Press. 205 p. (2011)

2011
[27]

H.; Olesky, D

Odeh, O. H.; Olesky, D. D.; van den Driessche, P.: Bruhat decomposition and numerical stability. SIAM J. Matrix Anal. Appl. 19, No. 1, 89-98 (1998)

1998
[28]

J.; Shakiban, C.: Applied linear algebra

Olver, P. J.; Shakiban, C.: Applied linear algebra. Undergraduate Texts in Mathematics. Springer. xxv, 679 p. (2018)

2018
[29]

The QPA-team, QPA - Quivers, path algebras and representations - a GAP package, Version 1.33; 2022 https: //folk.ntnu.no/oyvinso/QPA/

2022
[30]

M.: The finitistic dimension of a Nakayama algebra

Ringel, C. M.: The finitistic dimension of a Nakayama algebra. Journal of Algebra, Volume 576, 2021, Pages 95-145, ISSN 0021-8693

2021
[31]

Stein et al., Sage Mathematics Software, The Sage Development Team, 2022, http: //www.sagemath.org

A. Stein et al., Sage Mathematics Software, The Sage Development Team, 2022, http: //www.sagemath.org

2022
[32]

Gordon and Breach Science Publishers, Algebra, Logic and Applications, Volume 4, 1992

Simson, D.: Linear representations of partially ordered sets and vector space categories. Gordon and Breach Science Publishers, Algebra, Logic and Applications, Volume 4, 1992

1992
[33]

I: Basic representation theory

Skowronski, A.; Yamagata, K.: Frobenius algebras. I: Basic representation theory. EMS Textbooks in Mathematics. Z\"urich: European Mathematical Society (2011)

2011
[34]

Notices Amer

Striker, J.: Dynamical algebraic combinatorics: promotion, rowmotion, and resonance. Notices Amer. Math. Soc. 64 (2017), no. 6, 543-549

2017
[35]

Thomas, H.; Williams, N.: Rowmotion in slow motion. Proc. Lond. Math. Soc. (3) 119 (2019), no. 5, 1149-1178

2019
[36]

N.; Bau, D

Trefethen, L. N.; Bau, D. III: Numerical linear algebra. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xii, 361 p. (1997)

1997
[37]

Pure and Applied Mathematics 232

Van Oystaeyen, F.: Algebraic geometry for associative algebras. Pure and Applied Mathematics 232. New York, NY: Marcel Dekker. vi, 287 p. (2000)

2000
[38]

Journal of Algebra Volume 82, Issue 2, June 1983, Pages 353-357

Zacharia, D.: On the Cartan matrix of an artin algebra of global dimension two. Journal of Algebra Volume 82, Issue 2, June 1983, Pages 353-357

1983

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Classics in Mathematics

Aigner, M.: Combinatorial theory. Classics in Mathematics. Berlin: Springer. x, 484 p. (1997)

work page 1997

[2] [2]

Assem, I.; Simson, D.; Skowronski, A.: Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory. London Mathematical Society Student Texts 65. Cambridge: Cambridge University Press, 458 p. (2006)

work page 2006

[3] [3]

O.: Representation theory of Artin algebras

Auslander, M.; Reiten, I.; Smalo, S. O.: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, Volume 36, Cambridge University Press, 1997

work page 1997

[4] [4]

Auslander, M., Reiten, I.: k-Gorenstein algebras and syzygy modules. J. Pure Appl. Algebra, 92, 1-27 (1994)

work page 1994

[5] [5]

Proceedings of the American Mathematical Society Vol

Auslander, M., Reiten, I.: On a Generalized Version of the Nakayama Conjecture. Proceedings of the American Mathematical Society Vol. 52, No. 1 (1975), pp. 69-74

work page 1975

[6] [6]

Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8-28 (1963)

work page 1963

[7] [7]

Beilinson, A.; Ginzburg, V.; Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473-527

work page 1996

[8] [8]

Cambridge Studies in Advanced Mathematics, Volume 30, Cambridge University Press, 1991

Benson, D.: Representations and cohomology I: Basic representation theory of finite groups and associative algebras. Cambridge Studies in Advanced Mathematics, Volume 30, Cambridge University Press, 1991

work page 1991