ABSTRACT
The “Homological Conjectures” in local algebra date back to Serre’s beautiful 1957-58 course at the Colle`ge de France [19]. How to count the multiplicities of components in an intersection of two algebraic varieties was reduced to questions in local algebra involving the celebrated Tor-formula. Several important results were proven, other questions remained. These led to related but also different conjectures by other mathematicians, notably M. Auslander, Bass [2], Vasconcelos [26], Peskine-Szpiro [15], and Hochster [10], [12]. Some of these were proved for all noetherian local rings, others up to a certain dimension or in equal characteristic; others, again, remain virtually untouched. A masterly survey of
results and of the many interconnections between these Homological Conjectures was presented in [11]. This was updated in [16]. Progress depending on De Jong’s theory of alterations is discussed in [3] and [17]. Finally, a substantial part of these conjectures is treated more leisurely in the monographs [24] and [5].
