×

zbMATH — the first resource for mathematics

Cohomology of algebras of semidihedral type. VII: Local algebras. (English. Russian original) Zbl 1183.16009
J. Math. Sci., New York 161, No. 4, 530-536 (2009); translation from Zap. Nauchn. Semin. POMI 365, 130-142 (2009).
Summary: The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information” is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration. It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras.
For part V of this series cf. Zap. Nauchn. Semin. POMI 330, 131-154 (2006); translation in J. Math. Sci., New York 140, No. 5, 676-689 (2007; Zbl 1105.16016).

MSC:
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16G20 Representations of quivers and partially ordered sets
16E05 Syzygies, resolutions, complexes in associative algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. I. Generalov, ”Cohomology of algebras of dihedral type. I,”Zap. Nauchn. Semin. POMI, 265, 139–162 (1999). · Zbl 1053.16005
[2] O. I. Balashov and A. I. Generalov, ”Yoneda algebras of a class of dihedral algebras,” Vestn. St.Peterbary. Univ., Ser. 1, No. 15, 3-10 (1999). · Zbl 1043.16500
[3] O. I. Balashov and A. I. Generalov, ”Cohomology of algebras of dihedral type. II,” Algebra Analiz, 13, No. 1, 3–25 (2001). · Zbl 1053.16004
[4] A. I. Generalov, ”Cohomology of algebras of semidihedral type. I,” Algebra Analiz, 13, No. 4, 54–85 (2001).
[5] N. V. Kosmatov, ”Cohomology of algebras of dihedral type: automatic calculation,” in: International Algebraic Conference Dedicated to the Memory Z. I. Barevich, St.Petersburg (2002), pp. 115–116.
[6] M. A. Antipov and A. I. Generalov, ”Cohomology of algebras of semidihedral type. II,” Zap. Nauchn. Semin. POMI, 289, 9–36 (2002). · Zbl 1070.16009
[7] A. I. Generalov, ”Cohomology of algebras of dihedral type. IV: the family D(2B),” Zap. Nauchn. Semin. POMI, 289, 76-89 (2002). · Zbl 1070.16007
[8] A. I. Generalov and E. A. Osiuk, ”Cohomology of algebras of dihedral type. lll: the family D(2A),” Zap. Nauchn. Semin. POMI, 289, 113–133 (2002).
[9] A. I. Generalov, ”Cohomology of algebras of semidihedral type. III: the family SD(3K),” Zap. Nauchn. Semin. POMI, 305, 84-100 (2003).
[10] A. I. Generalcv and N. V. Kosmatov, ”Computation of the Yoneda algebras of dihedral type,” Zap. Nauchn. Semin. POMI, 305, 101-120 (2003).
[11] A. I. Generalov, ”Cohomology of algebras of semidihedral type. IV,” Zap. Nauchn. Semin. POMI, 319, 81-116 (2003). · Zbl 1079.16005
[12] A. I. Generalov and N. V. Kusmatov, ”Projective resolutions and Yoneda algebras for algebras of dihedral type: the family D(3Q),” Funclam. Prikl. Matem., 10, No. 4, 65-89 (2004). · Zbl 1069.18003
[13] A. I. Generalov, ”Cohomology of algebras of semidihedral type. V,” Zap. Nauchn. Semin. POMI, 330, 131-154 (2006). · Zbl 1105.16016
[14] A. Generaluv and N. Kusmatov, ”Projective resolutions and Yoneda algebras for algebras of dihedral type,” Algebras Repr. Theory, 10, No. 3, 241-256 (2007). · Zbl 1144.16009 · doi:10.1007/s10468-006-9012-7
[15] A. I. Generalcv, ”Cohomology of algebras of semidihedral type. VI,” Zap. Nauchn. Semin. POMI, 343, 183-198 (2007).
[16] K. Erdniann, ”Blocks 0f tame representation type and related algebras,” Lect. Notes Math., 1428, Berlin, Heidelberg (1990).
[17] A. I. Generalov, ”Hochschild cohomology of algebras of dihedral type. I: the family D(3K) in characteristic 2,” Algebra Analiz, 16, No. 6, 53-122 (2004).
[18] A. I. Generalov, ”Hochschild cohomology of algebras of quaternion type. I: generalized quaternion groups,” Algebra Analiz, 18, No. 1, 55-107 (2006).
[19] A. I. Generalov and N. Yu. Kussuvskaya, ”Hochschild cohumulogy of Liu-Schulz algebras,” Algebra Analiz, 18, No. 4, 39-82 (2006).
[20] A. I. Generalcv, A. A. Ivanov, and S. O. Ivanov, ”Hochschild cohomology of algebras of quaternion type. II. The family Q(2B)1 in characteristic 2,” Zap. Naachn. Semin. POMI, 349, 53-134 (2007).
[21] A. I. Generalov, ”Hochschild cohomology of algebras of quaternion type. III. Algebras with a small parameter,” Zap. Nauchn. Semin. POMI, 356, 46-84 (2008).
[22] A. I. Generalcv, ”Hochschild cohomology of algebras of semidihedral type. I. Group algebras of semidihedral groups,” Algebra Analiz, 21 (2009) (to appear).
[23] H. Sasaki, ”The mod 2 cohomology algebras of finite groups with semidihedral Sylow 2–subgroups,” Commun. Algebra, 22, 4123-4156 (1994). · Zbl 0837.20064 · doi:10.1080/00927879408825071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.