## Topics in the Homological Theory of Modules Over Commutative Rings, Issue 24This volume contains expository lectures by Melvin Hochster from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. The lectures deal mainly with recent developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings. A good deal of attention is given to the role ``big'' Cohen-Macaulay modules play in clearing up some of the open questions. A modest knowledge of commutative rings and familarity with (the long exact sequences for) Tor and Ext should suffice as a background for the reader. |

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### Contents

Introduction | 1 |

Preliminaries | 3 |

Some Open Questions | 8 |

Consequences of the Existence of CohenMacaulay Modules | 15 |

Modifications and the Existence of Big CohenMacaulay Modules in Characteristic p 0 | 20 |

Henselian Rings M Artins Approximation Theorem and Big CohenMacaulay Modules over Fields of Characteristic 0 | 27 |

DepthSensilivity Theorems and Exactness Criteria | 34 |

Modules of Projective Dimension 2 Generic Modules of Finite Projective Dimension and the BuchsbaumEisenbud Structure Theorems | 44 |

Linear Algebraic Groups | 52 |

Applications of Homological Methods and Some More Open Questions | 61 |

Supplement | 70 |

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### Common terms and phrases

algebraically closed field Artin's approximation theorem big Cohen-Macaulay modules biggest integer commutative ring contains a field denotes depth-sensitivity direct summand discrete valuation ring elements entries equivalent exact sequence existence of big fact field of characteristic finite length finite projective dimension finite type fraction field free acyclic complexes free modules free resolution GL(n hence Henselian homomorphism injective integral closure intersection conjecture isomorphism K-algebra Koszul complex Lemma Let G linear algebraic group linearly reductive M-sequence matrix of indeterminates maximal ideal minimal prime minors module of syzygies module-finite Moreover multiplicities Noetherian ring pdRM perfect polynomial ring projective resolution proof Proposition prove R-module of finite rank refer the reader regular local ring result ring and let ring of invariants solution Spec(R structure theorems suppose system of parameters Theorem 8.4 theory trivial type and finite unique vector space zerodivisor