A generalized Koszul complex. III. A remark on generic acyclicity

A Remark on Koszul Complexes

j j mj m b mj mi for all m mi M When M is a nite free R module then the vanishing of the homology of K is determined by the grade g of the ideal Im one hasHj K for j n g n and Hn g K Giving a linear form on a nite free R module of rank n is equivalent to specifying a sequence a an of elements of R If M is not a free R module then the picture changes drastically In this note we investigate the s...

A Generalized Koszul Complex. I

Introduction. In [1], the Koszul complex was used to study the relationship between codimension and multiplicity. It also helped us investigate Macaulay modules and rings, and provided a context in which to prove the Cohen-Macaulay Theorem concerning the unmixedness of complete intersections. Now there is a generalization of the Cohen-Macaulay Theorem (known, we believe, as the generalized Cohe...

Confluence Algebras and Acyclicity of the Koszul Complex

The N -Koszul algebras are N -homogeneous algebras satisfying a homological property. These algebras are characterised by their Koszul complex: an N -homogeneous algebra is N -Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove N -Koszulness: an algebra admitting a side-confluent presentation is N -Koszul if and only if the extra-con...

A Remark on the Generalized Smashing Conjecture

Using one of Wodzicki's examples of H-unital algebras [14] we exhibit a ring whose derived category contains a smashing subcategory which is not generated by small objects. This disproves the generalization to arbitrary triangulated categories of a conjecture due 1. Statement of the conjecture We refer to [7] for a nicely written analysis of the following setup: Let S be a triangulated category...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem