On the global dimension of quasi–hereditary algebras with triangular decomposition

König, Steffen

doi:10.1090/S0002-9939-96-03549-6

On the global dimension of quasi–hereditary algebras with triangular decomposition
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Proc. Amer. Math. Soc. 124 (1996), 1993-1999 Request permission

Abstract:

Let $A$ be a quasi–hereditary algebra with triangular decomposition ${_C}A_{C^{op}} \simeq C \otimes _S C^{op}$ such that all Verma modules are semisimple over $C^{op}$. Then we show: $gldim(A) = 2 \cdot gldim(C)$. Applying this formula to the more special class of twisted double incidence algebras of finite partially ordered sets, we get a proof of a conjecture of Deng and Xi. Another application is to the so-called dual extensions of algebras.

References

E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
B.M.Deng and C.C.Xi, Quasi–hereditary algebras which are dual extensions of algebras. Comm.Alg. 22, 4717–4736 (1994).
B.M.Deng and C.C.Xi, Quasi–hereditary algebras which are twisted double incidence algebras of posets. Contrib. Algebra and Geom. 36 (1995), 37–72.
Vlastimil Dlab and Claus Michael Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR 987824
Vlastimil Dlab and Claus Michael Ringel, The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200–224. MR 1211481
M.Dyer, Kazhdan–Lusztig–Stanley polynomials and quadratic algebras I. Preprint (1992).
S.König, Exact Borel subalgebras of quasi–hereditary algebras, I. With an appendix by L.Scott. Math. Z. 220 (1995), 399–426.
S.König, Exact Borel subalgebras of quasi–hereditary algebras, II. Comm. Alg. 23 (1995), 2331–2344. CMP 95:11
S.König, Strong exact Borel subalgebras of quasi–hereditary algebras and abstract Kazhdan–Lusztig theory. To appear in Adv.in Math.
S.König, Cartan decompositions and BGG–resolutions. Manuscr.Math. 86, 103–111 (1995).
B.Parshall and L.L.Scott, Derived categories, quasi–hereditary algebras and algebraic groups. Proc. of the Ottawa-Moosonee Workshop in Algebra 1987, Math. Lect. Note Series, Carleton University and Université d’Ottawa (1988).
Leonard L. Scott, Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 271–281. MR 933417
Chang Chang Xi, Quasi-hereditary algebras with a duality, J. Reine Angew. Math. 449 (1994), 201–215. MR 1268586
C.C.Xi, Global dimensions of dual extension algebras. Manuscripta Math. 88 (1995), 25–31.
Kunio Yamagata, A construction of algebras with large global dimensions, J. Algebra 163 (1994), no. 1, 51–67. MR 1257304, DOI 10.1006/jabr.1994.1003