References¶
The curated, verified bibliography that quiverlab cites, rendered from the single packaged source src/quiverlab/citations/references.bib.
Algorithms¶
[bardzell] Bardzell, Michael J. (1997). The alternating syzygy behavior of monomial algebras. Journal of Algebra 188, 69-89. The minimal projective bimodule resolution for monomial algebras (quiverlab's Bardzell engine and the truncated/radical-square-zero families). [chouhy_solotar] Chouhy, Sergio; Solotar, Andrea (2015). Projective resolutions of associative algebras and ambiguities. Journal of Algebra 432, 22-61. The general kQ/I bimodule resolution from a reduction system -- quiverlab's Chouhy-Solotar engine for non-monomial algebras. [bracket_liftings] Negron, Cris; Witherspoon, Sarah (2016). An alternate approach to the Lie bracket on Hochschild cohomology. Homology, Homotopy and Applications 18, 265-285. The Gerstenhaber bracket computed directly on a non-bar resolution (with Volkov2019), transported onto bar representatives. [bracket_liftings_volkov] Volkov, Yury (2019). Gerstenhaber bracket on the Hochschild cohomology via an arbitrary resolution. Proceedings of the Edinburgh Mathematical Society. Series II 62, 817-836. A bracket formula valid on any projective bimodule resolution (companion to Negron-Witherspoon). [oke_koszul] Oke, Femi Emmanuel (2021). Bracket structure on Hochschild cohomology of Koszul quiver algebras using homotopy liftings. Communications in Algebra. The homotopy-lifting bracket formulation quiverlab's native CS bracket implements (Plan 51); source of the section-7 Koszul-quiver worked tables. [minimal_resolution] Green, Edward L.; Solberg, Oyvind; Zacharia, Dan (2001). Minimal projective resolutions. Transactions of the American Mathematical Society 353, 2915-2939. The Green-Solberg-Zacharia minimal module resolution algorithm (quiverlab's minimal engine and module Ext). [module_ext] Green, Edward L.; Solberg, Oyvind; Zacharia, Dan (2001). Minimal projective resolutions. Transactions of the American Mathematical Society 353, 2915-2939. Module-level Ext^n over minimal resolutions (Plan 05 module engine). [hodge] Gerstenhaber, Murray; Schack, Samuel D. (1987). A Hodge-type decomposition for commutative algebra cohomology. Journal of Pure and Applied Algebra 48, 229-247. The eigenspace splitting HH^n = (+) HH^{n,(i)} of commutative/tensor and incidence-algebra pieces. [cyclic] Connes, Alain (1985). Non-commutative differential geometry. Publications Math'ematiques de l'IH'ES 62, 41-144. Connes' B-operator and the SBI sequence -- quiverlab's cyclic homology. [barakat_homalg] Barakat, Mohamed; Lange-Hegermann, Markus (2011). An axiomatic setup for algorithmic homological algebra and an alternative approach to localization. Journal of Algebra and Its Applications 10, 269-293. The Grothendieck spectral sequence over general module categories (the homalg framing) -- quiverlab's Grothendieck / Cartan-Eilenberg change-of-rings preset. [cup] Gerstenhaber, Murray (1963). The cohomology structure of an associative ring. Annals of Mathematics. Second Series 78, 267-288. The associative cup product on HH^ (Gerstenhaber-algebra structure). [bracket] Gerstenhaber, Murray (1963). The cohomology structure of an associative ring. Annals of Mathematics. Second Series 78, 267-288. The graded Lie bracket making HH^ a Gerstenhaber algebra. [bv_tradler] Tradler, Thomas (2008). The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Annales de l'Institut Fourier 58, 2351-2379. The symmetric-algebra BV operator Delta on HH^ induced by a symmetric, invariant, nondegenerate inner product -- quiverlab's symmetric (nu-inner) BV route (Plan 54). [bv_lzz] Lambre, Thierry; Zhou, Guodong; Zimmermann, Alexander (2016). The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra. Journal of Algebra 446, 103-131. The Frobenius-algebra BV operator under the semisimple-nu hypothesis -- quiverlab's semisimple-nu / twisted BV route (Plan 54). [bv_volkov] Volkov, Yury (2016). BV-differential on Hochschild cohomology of Frobenius algebras. The BV differential under ord(nu) coprime to char k (over GF(p): the squarefree-minpoly / p does not divide ord(nu) phrasing recorded in quiverlab's provenance) -- Plan 54. [bv_biklz] Bian, Xiaojun; Itagaki, Tomohiro; Kou, Cuili; Lyu, Junjun; Zhou, Guodong (2026). The Hochschild cohomology ring of a self-injective Nakayama algebra is a Batalin-Vilkovisky algebra. The self-injective Nakayama BV structure; Sec 3.2 (e=1) gives the explicit k[x]/(x^N) Delta values that back quiverlab's char-sensitivity + Delta-rank literature oracle (Plan 54). [priddy] Priddy, Stewart B. (1970). Koszul resolutions. Transactions of the American Mathematical Society 152, 39-60. The Priddy PBW / G-quadratic certifier: a quadratic Gröbner basis (all reduction tips length 2) proves the algebra is Koszul (Plan 27). [froberg_koszul] Fr"oberg, Ralf (1999). Koszul algebras. Advances in Commutative Ring Theory (Fez, 1997) 205, 337-350. The Hilbert-series Koszulity criterion P(t)C_A(-t) = I -- quiverlab's Fröberg numeric Koszulity falsifier (Plan 27). [green_hartman_marcos_solberg] Green, Edward L.; Hartman, Gregory; Marcos, Eduardo N.; Solberg, Oyvind (2005). Resolutions over Koszul algebras. Arch. Math. (Basel) 85, 118-127. Green-Hartman-Marcos-Solberg: the minimal graded A^e-resolution P_n = A (x)_S K_n (x)_S A of a Koszul algebra with the comultiplicative differential; K_n = the intersection of V^i (x) R (x) V^j, and dim K_n is the n-th Koszul-dual Hilbert coefficient. The GHMS fast-HH engine (Plan 75). [clms_bounded_extensions] Cibils, Claude; Lanzilotta, Marcelo; Marcos, Eduardo N.; Solotar, Andrea (2022). Han's conjecture for bounded extensions. Journal of Algebra. CLMS: B subset A left/right bounded (A/B tensor-nilpotent, finite pd over B^e, one-sided B-projective) implies B satisfies Han iff A does (Thm 4.6). Examples 5.3 (bounded) / 5.5 (not bounded) are the P73 oracles. [igusa_todorov] Igusa, Kiyoshi; Todorov, Gordana (2005). On the finitistic global dimension conjecture for Artin algebras. Representations of algebras and related topics 45, 201-204. The Igusa-Todorov functions phi/psi on the finite K0 + syzygy operator (phi = pd for finite projective dimension) -- quiverlab's Plan-40 IT engine. [butler_ringel] Butler, M. C. R.; Ringel, Claus Michael (). Auslander-Reiten sequences with few middle terms and applications to string algebras. Communications in Algebra. Butler-Ringel: the string/band module classification and the hook/cohook description of the AR translate -- the ground truth for the string subsystem. [suarez_alvarez] Su'arez-'Alvarez, Mariano (). A Simple Homological Characterization of String Algebras of Finite Representation Type. Algebras and Representation Theory 26}, number = {5}, pages = {1759--1772}, year = {2023. Suarez-Alvarez: among rep-finite algebras, string <=> the middle term of EVERY extension of indecomposables has <= 2 summands (all Ext^1 classes, not just AR sequences) -- the homological string test. [avella_geiss] Avella-Alaminos, Diana; Geiss, Christof (). Combinatorial derived invariants for gentle algebras. Journal of Pure and Applied Algebra. The AG-invariant: a multiset of (n,m) pairs from permitted/forbidden threads; a DERIVED invariant, provably NOT complete. [amiot_skew_gentle] Amiot, Claire (). Indecomposable objects in the derived category of a skew-gentle algebra using orbifolds. arXiv preprint (ICRA 2020 proceedings). Amiot: skew-gentle as Z2-skew-group of a gentle algebra; the orbifold/double-cover geometric model backing the geometric-vs-engine tau-tilting cross-check. [garcia_lavoue] Garcia, Monica; Lavou'e, L'ea (). Brick-finite skew-gentle algebras are representation-finite. arXiv preprint. Garcia-Lavoue Thm 3.1 (char != 2): brick-finite <=> rep-finite for skew-gentle -- composed with DIJ (brick-finite <=> tau-tilting-finite) gives the rep-type certificate. [buan_marsh_tau_exceptional] Buan, Aslak Bakke; Marsh, Robert J. (2021). \(tau\)-exceptional sequences. J. Algebra 585, 36-68. Buan-Marsh: signed tau-exceptional sequences via the Jasso tau-perpendicular reduction; bijection with ordered support tau-tilting modules (#signed = n!*#sTt) -- the Plan-65 / R27 count oracle. [degraaf_lie] de Graaf, Willem A. (2000). Lie Algebras: Theory and Algorithms. Elsevier 56. de Graaf: the algorithms behind the char-0 classification -- solvable radical rad = [L,L]^perp, Levi-Malcev decomposition, and direct-sum-of-simple-ideals type of a semisimple Lie algebra.
Families¶
[quantum_ci] Buchweitz, Ragnar-Olaf; Green, Edward L.; Madsen, Dag; Solberg, Oyvind (2005). Finite Hochschild cohomology without finite global dimension. Mathematical Research Letters 12, 805-816.
The algebra k
Finite fields¶
[conway] L"ubeck, Frank (). Conway polynomials for finite fields. Lubeck's Conway-polynomial tables fixing canonical generators of GF(p^n). [finite_fields] L"ubeck, Frank (). Conway polynomials for finite fields. Deterministic cross-compatible GF(q) arithmetic via Conway polynomials.
Foundations¶
[witherspoon_gsm204] Witherspoon, Sarah J. (2019). Hochschild Cohomology for Algebras. American Mathematical Society 204.
The textbook derivation of the homotopy-lifting Gerstenhaber bracket (the expository anchor for quiverlab's native CS bracket).
[bar] Hochschild, Gerhard (1945). On the cohomology groups of an associative algebra. Annals of Mathematics. Second Series 46, 58-67.
Hochschild's original definition of the cohomology of an associative algebra; the normalized bar complex is quiverlab's HH^/HH_ oracle in any characteristic.
[happel_question] Happel, Dieter (1989). Hochschild cohomology of finite-dimensional algebras. S'eminaire d'Alg`ebre Paul Dubreil et Marie-Paul Malliavin (Paris, 1987/1988) 1404, 108-126.
Whether finite global dimension is equivalent to eventual vanishing of HH^n -- the motivating question for the hereditary and truncated families.
[weibel_homological] Weibel, Charles A. (1994). An Introduction to Homological Algebra. Cambridge University Press 38.
The section-5.4 spectral-sequence page formulas and the strong-convergence theorem -- quiverlab's spectral-sequence engine (Plan 42).
[cartan_eilenberg] Cartan, Henri; Eilenberg, Samuel (1956). Homological Algebra. Princeton University Press 19.
The change-of-rings spectral sequence -- quiverlab's Cartan-Eilenberg preset (Plan 42).
[gerstenhaber] Gerstenhaber, Murray (1963). The cohomology structure of an associative ring. Annals of Mathematics. Second Series 78, 267-288.
The definitional source of the cup product and Gerstenhaber bracket.
[assem_book] Assem, Ibrahim; Simson, Daniel; Skowro'nski, Andrzej (2006). Elements of the Representation Theory of Associative Algebras. Volume 1: Techniques of Representation Theory. Cambridge University Press 65.
The standard reference for bound quivers and the representation theory quiverlab implements.
[ars_book] Auslander, Maurice; Reiten, Idun; Smalo, Sverre O. (1995). Representation Theory of Artin Algebras. Cambridge University Press 36.
The Auslander-Reiten theory reference: almost-split sequences, irreducible maps, the AR quiver, and the Nakayama functor -- the ground truth for Plan 41.
[liu_degrees] Liu, Shiping (1992). Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. Journal of the London Mathematical Society. Second Series 45, 32-54.
Liu's left/right degrees of irreducible morphisms and their control of the AR-quiver shape -- the R21 degree theory Plan 57 computes on the knit.
[liu_semistable] Liu, Shiping (1993). Semi-stable components of an Auslander-Reiten quiver. Journal of the London Mathematical Society. Second Series 47, 405-416.
Liu's component classification (sectional paths, semistable/directed components) underlying Plan 57's partition and rep-directed recognizer.
[chaio_liu_radical] Chaio, Claudia; Liu, Shiping (2013). A note on the radical of a module category. Communications in Algebra 41, 4419-4424.
Chaio-Liu: rep-finiteness through the infinite radical; the nilpotency of rad(mod A) in the rep-finite case is the maximal depth of composites -- the theorem behind Plan 57's nilpotency index and rad^inf=0 gate.
[cmms_radsq] Coelho, Fl'avio U.; Marcos, Eduardo N.; Merklen, H'ector A.; Skowro'nski, Andrzej (1994). Module categories with infinite radical square zero are of finite type. Communications in Algebra 22, 4511-4517.
CMMS: (rad^inf)^2 = 0 implies representation-finite -- the class oracle justifying Plan 57's finite-nilpotency-index rep-finiteness certificate (documented, not a per-instance decider off the rep-finite domain).
[ringel_tame] Ringel, Claus Michael (1984). Tame Algebras and Integral Quadratic Forms. Springer 1099.
Ringel LNM 1099: directing modules, the postprojective/regular/preinjective trichotomy -- the R21 component-invariant reference.
[han_conjecture] Han, Yang (2006). Hochschild (co)homology dimension. Journal of the London Mathematical Society. Second Series 73, 657-668.
Finite global dimension iff finite Hochschild homology dimension -- the conjecture the zoo scans probe.
[polishchuk_positselski] Polishchuk, Alexander; Positselski, Leonid (2005). Quadratic Algebras. American Mathematical Society 37.
The quadratic-dual conventions (A^! = kQ^op/R^perp) behind quiverlab's Koszul dual and the E(A) = (A^!)^op cross-check (Plan 27).
[amiot_cluster_category] Amiot, Claire (2009). Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l'Institut Fourier 59, 2525-2590.
The generalized cluster category C_(Q,W) of a quiver with potential: when (Q,W) is Jacobi-FINITE it carries a cluster-tilting object whose endomorphism algebra IS the Jacobian algebra Jac(Q,W). quiverlab CITES this identification -- it computes the Jacobian side and never forms Hom_C (no dg engine) -- and certifies the algebra-level statement (Plan 79).
[bmrrt_cluster] Buan, Aslak Bakke; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana (2006). Tilting theory and cluster combinatorics. Advances in Mathematics 204, 572-618.
The cluster category C_Q = D^b(kQ)/tau^-1[1] and its FUNDAMENTAL DOMAIN ind(mod kQ) || {P_v[1]} -- the finite model quiverlab computes on, giving #indec(C_Q) = #ind(mod kQ) + n (the almost-positive roots) and the reduction Ext^1_C(X,Y) = Ext^1_A(X,Y) (+) D Ext^1_A(Y,X) behind the 2-CY certificate (Plan 79).
[bmr_cluster_tilted] Buan, Aslak Bakke; Marsh, Robert; Reiten, Idun (2007). Cluster-tilted algebras. Transactions of the American Mathematical Society 359, 323-332.
End{C_Q}(T) for a cluster-tilting object T is the cluster-tilted algebra; with Amiot it is the Jacobian algebra of the cluster-tilted quiver-with-potential. quiverlab certifies the QUIVER step by Fomin-Zelevinsky matrix mutation and verifies the Jacobian algebra's presentation (Plan 79).
[keller_reiten_gorenstein] Keller, Bernhard; Reiten, Idun (2007). Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Advances in Mathematics 211, 123-151.
Cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary iff of finite global dimension -- quiverlab's self-certificate on every cluster-tilted algebra it builds (Plan 79).
[ginzburg_cy] Ginzburg, Victor (2006). Calabi-Yau algebras.
The Ginzburg dg algebra Gamma(Q,W) underlying the generalized cluster category. OUT-OF-SCOPE machinery for quiverlab -- cited only to name what a general non-acyclic C_(Q,W) would need (Plan 79 honest scope).
[keller_yang_mutation] Keller, Bernhard; Yang, Dong (2011). Derived equivalences from mutations of quivers with potential. Advances in Mathematics 226, 2118-2168.
The 'Keller' half of Amiot-Keller: derived equivalences from QP mutation. OUT-OF-SCOPE machinery (no dg engine ships) -- cited at the scope boundary (Plan 79).
[berger_nonquadratic] Berger, Roland (2001). Koszulity for nonquadratic algebras. Journal of Algebra 239, 705-734.
Berger's N-Koszul property for N-homogeneous algebras: the minimal resolution of the trivial module is PURE, generated in the single internal degree delta(n) = (N/2)n (n even) / (N/2)(n-1)+1 (n odd) -- the 2-N alternation quiverlab's n_koszul_certificate checks. For N >= 3 the property is equivalent to the Yoneda algebra being generated in degrees 0, 1, 2 (Plan 77).
[cassidy_shelton] Cassidy, Thomas; Shelton, Brad (2008). Generalizing the notion of Koszul algebra. Mathematische Zeitschrift 260, 93-114.
The K2 property: E(A) = Ext(k,k) is generated as an algebra in cohomological degrees 1 and 2. K2 generalizes BOTH Koszul and N-Koszul and allows relations in several degrees -- quiverlab's k2_certificate, decided through an explicit certified window (Plan 77).
[brenner_butler_king] Brenner, Sheila; Butler, Michael C. R.; King, Alastair D. (2002). Periodic algebras which are almost Koszul. Algebras and Representation Theory 5, 331-368.
(p,q)-almost-Koszul: A is concentrated in degrees 0..p and a linear complex of projectives resolves the simple up to an error in internal degree p+q. The Dynkin preprojective algebras are (h-2, 2)-Koszul (h = Coxeter number) and periodic of period 2(h-1) -- quiverlab's almost_koszul_certificate reproduces the (p,q) label via p = top degree, q = e - p (Plan 77).
[herscovich_multikoszul] Herscovich, Estanislao (2013). On the multi-Koszul property for connected algebras.
Multi-Koszul for locally finite-dimensional nonnegatively graded CONNECTED (A_0 = k) algebras, generalizing N-Koszul to relations in several degrees. Prop. 3.30: a finitely generated multi-Koszul algebra with a finite-dimensional relation space has Yoneda algebra generated in degrees 1 and 2, i.e. is K2 -- the transfer quiverlab reports for multi-vertex kQ/I, whose A_0 = k^{Q_0} is semisimple, not connected (Plan 77).
[herscovich_ainfty_ext] Herscovich, Estanislao (2019). Applications of one-point extensions to compute the \(A_infty\)-(co)module structure of several Ext (resp., Tor) groups. Journal of Pure and Applied Algebra 223, 1054-1072.
The A-infinity structure on the Yoneda algebra of a multi-Koszul algebra. CONTEXT for Plan 77's multi-Koszul scope -- quiverlab computes no A-infinity structure.
[chouhy_degenerations] Chouhy, Sergio (2019). On geometric degenerations and Gerstenhaber formal deformations. Bulletin of the London Mathematical Society 51, 787-797.
For finite-dimensional associative algebras the N-Koszul property is preserved under the degeneration relation, for every N >= 2. CONTEXT for the robustness of Plan 77's N-Koszul verdict -- quiverlab computes no degenerations.
[green_marcos_martinezvilla_zhang] Green, Edward L.; Marcos, Eduardo N.; Mart'inez-Villa, Roberto; Zhang, Pu (2004). D-Koszul algebras. Journal of Pure and Applied Algebra 193, 141-162.
N-Koszul (delta-Koszul) theory over a SEMISIMPLE base A_0 = k^{Q_0} -- the several-vertex foundation that legitimizes Plan 77's N-Koszul recognizer on multi-vertex kQ/I, where Berger's connected-graded setting does not apply verbatim.
[happel_trace] Happel, Dieter (1997). The trace of the Coxeter matrix and Hochschild cohomology. Linear Algebra and its Applications 258, 169-177.
Happel's trace identity tr(Coxeter) = sum (-1)^i dim HH^i for finite global dimension -- the Hochschild/Coxeter cross-invariant consistency oracle (Plan 29).
[keller_cyclic_invariance] Keller, Bernhard (1998). Invariance and localization for cyclic homology of DG algebras. Journal of Pure and Applied Algebra 123, 223-273.
Derived invariance of cyclic homology (with HH^/HH_): reflection-equivalent orientations of one graph share HH^/HH_/HC_ -- the derived-invariance oracle scheme.
[rickard_derived] Rickard, Jeremy (1989). Morita theory for derived categories. Journal of the London Mathematical Society. Second Series 39, 436-456.
Derived-equivalent algebras (e.g. a Brauer tree and its Brauer star) share Hochschild and cyclic homology -- the derived-invariance oracle.
[lenzing_delapena_spectral] Lenzing, Helmut; de la Pe na, Jos'e Antonio (2008). Spectral analysis of finite dimensional algebras and singularities. Trends in Representation Theory of Algebras and Related Topics (ICRA XII), 541-588.
The Dynkin / extended-Dynkin / canonical Coxeter-polynomial tables and Happel's trace identity -- the spectral-invariant oracle.
[delapena_mahler] de la Pe na, Jos'e Antonio (2014). On the Mahler measure of the Coxeter polynomial of an algebra. Advances in Mathematics.
The wild star [2,3,7] realizes Lehmer's polynomial -- the spectral_radius / mahler_measure oracle (Plan 29).
[dlPena2014mahler] de la Pe na, Jos'e Antonio (2014). On the Mahler measure of the Coxeter polynomial of an algebra. Advances in Mathematics.
de la Pena's class-conditional Mahler-measure dichotomy for accessible algebras (M = 1 or M >= mu_0, mu_0 = Lehmer's number) -- the Plan-58 Lehmer-class documentation note (shares the dlPena2014mahler eprint with the Plan-29 delapena_mahler oracle key).
[dlPena2013cyclotomic] de la Pe na, Jos'e Antonio (2013). Algebras whose Coxeter polynomials are products of cyclotomic polynomials.
de la Pena's separation of periodic (finite-order) Coxeter transformations from merely cyclotomic-type (quasi-unipotent) ones -- the Plan-58 quasi-unipotent / finite-order verdict.
[dlPenaTakane1990spectral] de la Pe na, Jos'e Antonio; Takane, Martha (1990). Spectral properties of Coxeter transformations and applications. Archiv der Mathematik 55, 120-134.
de la Pena-Takane: reality of the dominant Coxeter eigenvalue on the wild-hereditary locus -- the Plan-58 real-dominant spectral-radius foundation (Arch. Math. 55 (1990) 120-134).
[gerstenhaber_schack_1983] Gerstenhaber, Murray; Schack, Samuel D. (1983). Simplicial cohomology is Hochschild cohomology. J. Pure Appl. Algebra 30, 143-156.
Gerstenhaber-Schack: HH^(kP) is isomorphic to the simplicial cohomology of the order complex of P AS A RING (cup product). The ring source for the incidence-vs-nerve oracle (with Cibils 1989 / Redondo 2008). Proves the CUP iso for the FACE POSET of a simplicial complex; Cibils 1989 extends it to an ARBITRARY finite poset. NB: makes no bracket-vanishing claim.
[wang_singular_hh] Wang, Zhengfang (2015). Singular Hochschild cohomology and Gerstenhaber algebra structure.
Wang: the DEFINITIONAL origin of singular (= Tate) Hochschild cohomology HH_sg^i(A,A) = Hom_{D_sg(A (x) A^op)}(A, A[i]) for every i in Z, with its Gerstenhaber (and, for symmetric A, BV) structure. The object Plan 76 computes.
[keller_singular_hh] Keller, Bernhard (2018). Singular Hochschild cohomology via the singularity category. C. R. Math. Acad. Sci. Paris 356, 1106-1111.
Keller: singular Hochschild cohomology is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category. The singularity-category identification (building on Wang) -- context for Plan 76, not its computational recipe.
[usui_tate_periodic] Usui, Satoshi (2021). Tate-Hochschild cohomology rings for eventually periodic Gorenstein algebras.
Usui: for a GORENSTEIN algebra, eventual periodicity is equivalent to the existence of an INVERTIBLE homogeneous element of the Tate-Hochschild cohomology ring; also that eventually periodic algebras need NOT be Gorenstein (so such an algebra has no complete resolution and no Tate ring -- the honest refusal). The source of Plan 76's periodicity certificate.
[bergh_jorgensen_tate] Bergh, Petter Andreas; Jorgensen, David A. (2013). Tate-Hochschild homology and cohomology of Frobenius algebras. J. Noncommut. Geom. 7, 907-937.
Bergh-Jorgensen: the computational reference for Plan 76 -- the definition via a complete resolution over A^e; the THRESHOLD (Gorenstein dimension d of the enveloping algebra => HHhat^n = Ext^n_{A^e}(A,B) for n >= d+1, so n >= 1 when A is self-injective); the Frobenius duality dim HHhat^n(L,L) = dim HHhat^{-(n+1)}(L, {}{nu^2}L_1), symmetric when nu^2 = id; and the explicit quantum-complete-intersection computation (1,2,1 in degrees 0,1,2 and 0 elsewhere, q not a root of unity).
[cmrs_split] Cibils, Claude; Marcos, Eduardo; Redondo, Mar'ia Julia; Solotar, Andrea (2003). Cohomology of split algebras and of trivial extensions. Glasgow Mathematical Journal 45, 21-40.
HH^1(T(A)) is never zero -- Z(A) is always a summand -- for every finite-dimensional A; the trivial-extension HH^1 oracle.
[crs_trivial_ext_hh1] Cibils, Claude; Redondo, Mar'ia Julia; Saor'in, Manuel (2004). The first cohomology group of the trivial extension of a monomial algebra. Journal of Algebra and its Applications 3, 143-159.
The HH^1(T(A)) decomposition and Example 2.20 (the Z_5 cycle) -- the trivial-extension first-cohomology oracle.
[clms_arrow_removal] Cibils, Claude; Lanzilotta, Marcelo; Marcos, Eduardo N.; Solotar, Andrea (2020). Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology. Proceedings of the American Mathematical Society 148, 2421-2432.
Inert-arrow deletion (Def. 3.1) gives a clean HH_n isomorphism for n >= 2 (Thm 3.2) and a cohomology Ext-correction (Thm 4.2); arrow addition = the tensor algebra T_B(N), finite iff no relative cycle (Thm 3.5/3.6) -- the Plan-72 certified arrow-removal reduction and the P73 Han-conjecture seam.
[clms_jacobi_zariski] Cibils, Claude; Lanzilotta, Marcelo; Marcos, Eduardo N.; Solotar, Andrea (2022). Jacobi-Zariski long nearly exact sequences for associative algebras. Bulletin of the London Mathematical Society 54.
CLMS: the Jacobi-Zariski long nearly exact sequence relating HH(A), HH_(B), HH_(A|B) ('exact twice in three') -- the computational tool + self-cert gate for the P73 Han transport.
[kaygun_jacobi_zariski] Kaygun, Atabey (2012). Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology. Homology, Homotopy and Applications 14, 65-78.
Kaygun: the classical noncommutative Jacobi-Zariski sequence (B subset A with A/B flat) -- the origin CLMS credit for the noncommutative case; CLMS 2009.05017 is the 'long nearly exact / exact twice in three' refinement P73 computes with.
[clms_split_bounded] Cibils, Claude; Lanzilotta, Marcelo; Marcos, Eduardo N.; Solotar, Andrea (2020). Split bounded extension algebras and Han's conjecture. Pacific Journal of Mathematics 307, 63-77.
CLMS: the split-case predecessor of the bounded-extension theory (context).
[wang_recollement_han] Wang, Ren; Xu, Xiaoxiao; Zhang, Jinbi; Zhou, Guodong (2024). A recollement approach to Han's conjecture.
Wang-Xu-Zhang-Zhou: an independent recollement/derived reduction of Han's conjecture (also proves Han for skew-gentle algebras -- ties to P68). Authorship verified.
[chaparro_schroll_solotar] Chaparro, Cristian; Schroll, Sibylle; Solotar, Andrea (2020). On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras. Journal of Algebra 558, 293-326.
Determines HH^1(A, M) with different coefficients for gentle algebras via ribbon-graph combinatorics -- the Plan-52 gentle HH^1-with-coefficients provenance (numeric pin BLOCKED-until-transcribed).
[lindell_rubio_relative] Lindell, Jonathan; Rubio y Degrassi, Lleonard (2024). On the first relative Hochschild cohomology and the contracted fundamental group. arXiv preprint.
The Lie-algebra structure of the first RELATIVE (vertex-relative, E = kQ_0) Hochschild cohomology, with radical-square-zero computations -- the Plan-52 relative HH provenance (numeric pin BLOCKED-until-transcribed).
[xhj_truncated] Xu, Yunge; Han, Yang; Jiang, Wenfeng (2007). Hochschild cohomology of truncated quiver algebras. Science in China Series A: Mathematics 50, 727-736.
For a truncated algebra kQ/R^N, dim HH^ is finite iff Q is acyclic -- the truncated finiteness boolean oracle.
[cibils_acyclic] Cibils, Claude (1986). Hochschild homology of an algebra whose quiver has no oriented cycles. Representation Theory I (Ottawa, 1984) 1177, 55-59.
An acyclic quiver has HH_n = 0 for n >= 1 -- the acyclic Hochschild-homology vanishing oracle.
[skowronski_yamagata] Skowro'nski, Andrzej; Yamagata, Kunio (2011). Frobenius Algebras I: Basic Representation Theory. European Mathematical Society.
Symmetric and Frobenius representation theory, incl. the symmetric Nakayama criterion n | (L-1) -- the is_symmetric regression oracle.
[happel_trivial_extension] Happel, Dieter (1988). Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press 119.
The trivial extension T(A) = A |x D(A) is symmetric for every finite-dimensional A; the repetitive-algebra framework -- the anchor for the certified double-quiver TrivialExtension presentation (Plan 31).
[happel_triangulated] Happel, Dieter (1988). Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press 119.
The derived-category reference: the Serre functor / AR triangles of D^b(mod A) exist iff gl.dim < infinity, tau_{D^b} = nu[-1] -- the ground truth for the Plan-43 derived surface.
[schremmer_wpl] Schremmer, Felix (2025). Weighted projective lines and Hochschild cohomology.
HH^ of the canonical algebras (dim HH^2 = t-3, after Happel LNM 1404) -- the canonical-algebra Hochschild oracle.
[gelinas_delooping] G'elinas, Vincent (2022). The depth, the delooping level and the finitistic dimension. Advances in Mathematics 394, 108052.
The delooping level dell(A) as an upper bound for the finitistic dimension -- the deferred Plan-40 Task-F invariant (honest-scope note).
[fernandes_lanzilotta_mendoza] Fernandes, Sonia; Lanzilotta, Marcelo; Mendoza, Octavio (2015). The \(Phi\)-dimension: a new homological measure. Algebras and Representation Theory 18, 463-476.
phidim(A) = sup phi(M) as an algebra invariant + derived-equivalence invariance of its finiteness -- the Plan-53 phidim/psidim ground truth.
[bravo_lanzilotta_mendoza_vivero] Bravo, Diego; Lanzilotta, Marcelo; Mendoza, Octavio; Vivero, Jos'e (2021). Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras. Journal of Algebra 580, 63-83.
The LIT algebras + the proof-carrying finitistic bound psi_D(V) + n + 1 -- the Plan-53 LIT finitistic certificate.
[erdmann_skowronski_scy] Erdmann, Karin; Skowro'nski, Andrzej (2006). The stable Calabi-Yau dimension of tame symmetric algebras. Journal of the Mathematical Society of Japan 58, 97-128.
Introduced the stable Calabi-Yau dimension of a self-injective algebra as the weak CY dimension of mod-bar A -- the Plan-53 fractional-CY foundation.
[wald_waschbusch] Wald, Burkhard; Waschb"usch, Josef (). Tame biserial algebras. Journal of Algebra.
Biserial / special-biserial structure underlying string and Brauer graph algebras.
[geiss_delapena] Geiss, Christof; de la Pe na, Jos'e Antonio (). Auslander-Reiten components for clans. Bolet'in de la Sociedad Matem'atica Mexicana. Tercera Serie.
Geiss-de la Pena: the original skew-gentle / clan definition (char != 2).
[crawley_boevey_clans] Crawley-Boevey, William (). Functorial filtrations II: clans and the Gelfand problem. Journal of the London Mathematical Society.
Crawley-Boevey: the classification of indecomposables for clans, underlying the special-string re-gluing (the +/- forms).
[bongartz_tilting] Bongartz, Klaus (1981). Tilted algebras. Representations of Algebras (Puebla, 1980) 903, 26-38.
Bongartz's count criterion for tilting modules (# non-iso indecomposable summands = # vertices, given pd<=1 and self-Ext vanishing) and the Bongartz completion of a partial tilting module (Plan 44 / C7).
[fomin_shapiro_thurston] Fomin, Sergey; Shapiro, Michael; Thurston, Dylan (2008). Cluster algebras and triangulated surfaces. Part I: Cluster complexes. Acta Mathematica 201, 83-146.
The arc/triangulation combinatorics: the ideal-arc count n=6g-6+3(b+p)+Sum k_i, the admissibility exclusion list, and flip<->mutation -- the ground truth for the Plan-48 surface subsystem.
[fomin_zelevinsky_ca1] Fomin, Sergey; Zelevinsky, Andrei (2002). Cluster algebras I: Foundations. Journal of the American Mathematical Society 15, 497-529.
The skew-symmetric matrix mutation mu_k that surface flip is certified against (Plan 48).
[kac_canonical] Kac, Victor G. (1980). Infinite root systems, representations of graphs and invariant theory. Inventiones Mathematicae 56, 57-92.
Kac's roots, Schur roots, and the canonical decomposition of a dimension vector -- the ground truth for Plan 49's canonical_decomposition.
[schofield_general_reps] Schofield, Aidan (1992). General representations of quivers. Proceedings of the London Mathematical Society. Third Series 65, 46-64.
Schofield's generic hom/ext and the general-representation identities hom - ext =
Raw BibTeX¶
@article{Hochschild1945,
author = {Hochschild, Gerhard},
title = {On the cohomology groups of an associative algebra},
journal = {Annals of Mathematics. Second Series},
volume = {46},
number = {1},
year = {1945},
pages = {58--67},
doi = {10.2307/1969145},
}
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author = {Bardzell, Michael J.},
title = {The alternating syzygy behavior of monomial algebras},
journal = {Journal of Algebra},
volume = {188},
number = {1},
year = {1997},
pages = {69--89},
doi = {10.1006/jabr.1996.6813},
}
@article{ChouhySolotar2015,
author = {Chouhy, Sergio and Solotar, Andrea},
title = {Projective resolutions of associative algebras and ambiguities},
journal = {Journal of Algebra},
volume = {432},
year = {2015},
pages = {22--61},
doi = {10.1016/j.jalgebra.2015.02.019},
note = {arXiv:1406.2300},
}
@article{NegronWitherspoon2016,
author = {Negron, Cris and Witherspoon, Sarah},
title = {An alternate approach to the {L}ie bracket on {H}ochschild cohomology},
journal = {Homology, Homotopy and Applications},
volume = {18},
number = {1},
year = {2016},
pages = {265--285},
doi = {10.4310/HHA.2016.v18.n1.a14},
note = {arXiv:1406.0036},
}
@article{Volkov2019,
author = {Volkov, Yury},
title = {Gerstenhaber bracket on the {H}ochschild cohomology via an arbitrary resolution},
journal = {Proceedings of the Edinburgh Mathematical Society. Series II},
volume = {62},
number = {3},
year = {2019},
pages = {817--836},
doi = {10.1017/S0013091518000901},
note = {arXiv:1610.05741},
}
@article{Oke2021,
author = {Oke, Femi Emmanuel},
title = {Bracket structure on {H}ochschild cohomology of {K}oszul quiver algebras using homotopy liftings},
journal = {Communications in Algebra},
year = {2021},
note = {arXiv:2103.12331},
}
@book{WitherspoonGSM204,
author = {Witherspoon, Sarah J.},
title = {Hochschild Cohomology for Algebras},
series = {Graduate Studies in Mathematics},
volume = {204},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2019},
isbn = {9781470449315},
}
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author = {Green, Edward L. and Solberg, {\O}yvind and Zacharia, Dan},
title = {Minimal projective resolutions},
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number = {7},
year = {2001},
pages = {2915--2939},
doi = {10.1090/S0002-9947-01-02687-3},
}
@incollection{Happel1989,
author = {Happel, Dieter},
title = {Hochschild cohomology of finite-dimensional algebras},
booktitle = {S\'eminaire d'Alg\`ebre Paul Dubreil et Marie-Paul Malliavin (Paris, 1987/1988)},
series = {Lecture Notes in Mathematics},
volume = {1404},
publisher = {Springer},
address = {Berlin},
year = {1989},
pages = {108--126},
doi = {10.1007/BFb0084073},
}
@book{Happel1988,
author = {Happel, Dieter},
title = {Triangulated Categories in the Representation Theory of Finite Dimensional Algebras},
series = {London Mathematical Society Lecture Note Series},
volume = {119},
publisher = {Cambridge University Press},
year = {1988},
doi = {10.1017/CBO9780511629228},
}
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author = {Buchweitz, Ragnar-Olaf and Green, Edward L. and Madsen, Dag and Solberg, {\O}yvind},
title = {Finite {H}ochschild cohomology without finite global dimension},
journal = {Mathematical Research Letters},
volume = {12},
number = {6},
year = {2005},
pages = {805--816},
doi = {10.4310/MRL.2005.v12.n6.a2},
note = {arXiv:math/0407108},
}
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author = {Bergh, Petter Andreas and Erdmann, Karin},
title = {Homology and cohomology of quantum complete intersections},
journal = {Algebra \& Number Theory},
volume = {2},
number = {5},
year = {2008},
pages = {501--522},
doi = {10.2140/ant.2008.2.501},
note = {arXiv:0709.3029},
}
@article{GerstenhaberSchack1987,
author = {Gerstenhaber, Murray and Schack, Samuel D.},
title = {A {H}odge-type decomposition for commutative algebra cohomology},
journal = {Journal of Pure and Applied Algebra},
volume = {48},
number = {3},
year = {1987},
pages = {229--247},
doi = {10.1016/0022-4049(87)90112-5},
}
@book{CartanEilenberg1956,
author = {Cartan, Henri and Eilenberg, Samuel},
title = {Homological Algebra},
series = {Princeton Mathematical Series},
volume = {19},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1956},
isbn = {9780691049915},
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@book{Weibel1994,
author = {Weibel, Charles A.},
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publisher = {Cambridge University Press},
address = {Cambridge},
year = {1994},
doi = {10.1017/CBO9781139644136},
isbn = {9780521559874},
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author = {Barakat, Mohamed and Lange-Hegermann, Markus},
title = {An axiomatic setup for algorithmic homological algebra and an alternative approach to localization},
journal = {Journal of Algebra and Its Applications},
volume = {10},
number = {2},
year = {2011},
pages = {269--293},
doi = {10.1142/S0219498811004562},
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author = {Connes, Alain},
title = {Non-commutative differential geometry},
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year = {1985},
pages = {41--144},
doi = {10.1007/BF02698807},
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@article{Gerstenhaber1963,
author = {Gerstenhaber, Murray},
title = {The cohomology structure of an associative ring},
journal = {Annals of Mathematics. Second Series},
volume = {78},
number = {2},
year = {1963},
pages = {267--288},
doi = {10.2307/1970343},
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@article{Tradler2008,
author = {Tradler, Thomas},
title = {The {B}atalin--{V}ilkovisky algebra on {H}ochschild cohomology induced by infinity inner products},
journal = {Annales de l'Institut Fourier},
volume = {58},
number = {7},
year = {2008},
pages = {2351--2379},
doi = {10.5802/aif.2417},
}
@article{LambreZhouZimmermann2016,
author = {Lambre, Thierry and Zhou, Guodong and Zimmermann, Alexander},
title = {The {H}ochschild cohomology ring of a {F}robenius algebra with semisimple {N}akayama automorphism is a {B}atalin--{V}ilkovisky algebra},
journal = {Journal of Algebra},
volume = {446},
year = {2016},
pages = {103--131},
doi = {10.1016/j.jalgebra.2015.09.018},
eprint = {1405.5325},
archivePrefix = {arXiv},
}
@misc{Volkov2016bv,
author = {Volkov, Yury},
title = {{BV}-differential on {H}ochschild cohomology of {F}robenius algebras},
year = {2016},
eprint = {1405.5155},
archivePrefix = {arXiv},
primaryClass = {math.KT},
}
@misc{BianItagakiKouLyuZhou2026,
author = {Bian, Xiaojun and Itagaki, Tomohiro and Kou, Cuili and Lyu, Junjun and Zhou, Guodong},
title = {The {H}ochschild cohomology ring of a self-injective {N}akayama algebra is a {B}atalin--{V}ilkovisky algebra},
year = {2026},
eprint = {2603.04834},
archivePrefix = {arXiv},
primaryClass = {math.RT},
}
@misc{Luebeck_ConwayPolynomials,
author = {L{\"u}beck, Frank},
title = {Conway polynomials for finite fields},
howpublished = {\url{https://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html}},
note = {Online database; accessed 2026-07-18},
}
@book{ASS2006,
author = {Assem, Ibrahim and Simson, Daniel and Skowro\'nski, Andrzej},
title = {Elements of the Representation Theory of Associative Algebras.
Volume 1: Techniques of Representation Theory},
series = {London Mathematical Society Student Texts},
volume = {65},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2006},
isbn = {9780521586313},
doi = {10.1017/CBO9780511614309},
}
@book{ARS1995,
author = {Auslander, Maurice and Reiten, Idun and Smal{\o}, Sverre O.},
title = {Representation Theory of {A}rtin {A}lgebras},
series = {Cambridge Studies in Advanced Mathematics},
volume = {36},
publisher = {Cambridge University Press},
year = {1995},
}
@article{Liu1992degrees,
author = {Liu, Shiping},
title = {Degrees of irreducible maps and the shapes of {A}uslander-{R}eiten quivers},
journal = {Journal of the London Mathematical Society. Second Series},
volume = {45},
number = {1},
pages = {32--54},
year = {1992},
}
@article{Liu1993semistable,
author = {Liu, Shiping},
title = {Semi-stable components of an {A}uslander-{R}eiten quiver},
journal = {Journal of the London Mathematical Society. Second Series},
volume = {47},
number = {3},
pages = {405--416},
year = {1993},
}
@article{ChaioLiu2013,
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author = {Ringel, Claus Michael},
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year = {1984},
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year = {2006},
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author = {Fr{\"o}berg, Ralf},
title = {Koszul algebras},
booktitle = {Advances in Commutative Ring Theory (Fez, 1997)},
series = {Lecture Notes in Pure and Applied Mathematics},
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address = {New York},
year = {1999},
pages = {337--350},
isbn = {9780824771478},
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publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2005},
isbn = {9780821838341},
doi = {10.1090/ulect/037},
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@incollection{cibils1998radsq,
author = {Cibils, Claude},
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@article{cibilsredondosaorin2004,
author = {Cibils, Claude and Redondo, Mar\'ia Julia and Saor\'in, Manuel},
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@misc{qpa,
author = {Green, Edward L. and Solberg, {\O}yvind},
title = {{QPA} -- {Quivers}, path algebras and representations, a {GAP} package, Version 1.37},
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key = {GAP}, organization = {The GAP Group},
title = {{GAP -- Groups, Algorithms, and Programming}, Version 4.16.0}, year = {2026},
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@misc{sagemath,
author = {{The Sage Developers}}, title = {{SageMath}, the {Sage} {Mathematics} {Software} {System}},
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author = {Armenta, Marco}, title = {quiverlab: exact Hochschild theory for quivers with relations},
year = {2026}, version = {0.1.0}, url = {https://github.com/MarcoArmenta/quiverlab}
}
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