NTNU

Summer School 2001:
Homological conjectures for finite dimensional algebras

The program of the first part

This is the list of the topics for the lectures in the first part of the Summer School. All lectures are maximum 50 minutes long.
  • The conjectures
    1. Origins and statements, [H1], [HZ], [HZ2]
  • Resolutions and syzygies
    1. Resolutions and finitistic dimension for monomial algebras, [AG, sections 1 and 2], [GKK, Corollary 2.5]
    2. Finite finitistic dimension for radical cubed 0 and generalisations, [W]
    3. Resolutions in general, [GSZ, sections 1 and 2], [BK, section 7]
    4. Gröbner bases, [E], [G1, (G2)]
  • Homologically finite subcategories
    1. Basic definitions and examples, [AR1, section 1], [AR2, Corollary 1.8], [AS, Theorem 4.5, Proposition 4.6]
    2. Contravariant finiteness for the modules of finite projective dimension, [AR1, Corollary 3.10], [IST]
    3. Torsion theories and tilting modules, [A, section 1]
    4. Correspondence between (co)tilting modules and special homologically finite subcategories, [A, Theorem 3.2], [AR1, Theorem 5.5]
  • Representation dimension
    1. Representation dimension of artin algebras, [Au, Chapter III], [X1, X2]
  • Geometric aspects
    1. Varieties of algebras and modules, [MHS, section 1], [G, Lemma 3.2]
    2. Bounds for global and finistic dimensions, [MHS, section 2, 3]
  • Commutative theory
    1. The Auslander-Buchsbaum formula, [BH, Theorem 1.3.3]
    2. Characterisations of regular local rings, [BH, Theorem 2.2.7]
    3. Characterisations of complete intersections, [BH, section 2.3]
    4. Finitistic dimension equals Krull dimension, [GR, section 3.2] and [B, Corollary 5.5]
  • Infinitely generated modules
    1. The difference between the little and the big finitistic dimension, [S]
    2. Criteria for equality of the little and the big finitistic dimension, [HZS], [Kr]
    3. Contravariant finiteness for the modules of projective dimension less than n, [AEJO], [KS]
  • Derived categories
    1. Introduction to derived categories and tilting, [H3], [K1], [K2], [K3]
    2. Reduction techniques for homological conjectures, [H2]
References for the papers listed above.

Distribution of the lectures in the first part.

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