Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-20T02:54:36.918Z Has data issue: false hasContentIssue false

3 - The canonical module. Gorenstein rings

Published online by Cambridge University Press:  04 December 2009

Get access

Summary

The concept of a canonical module is of fundamental importance in the study of Cohen–Macaulay local rings. The purpose of this chapter is to introduce the canonical module and derive its basic properties. By definition it is a maximal Cohen–Macaulay module of type 1 and of finite injective dimension.

In the first two sections we investigate the injective dimension of a module, and prove Matlis duality which plays a central role in Grothendieck's local duality theorem. Actually the canonical module has its origin in this theory. Here the canonical module is introduced independently of local cohomology which is an important notion in itself and will be treated later in this chapter.

A ring which is its own canonical module is called a Gorenstein ring. Next to regular rings and complete intersections, Gorenstein rings are in many ways the ‘nicest’ rings. Distinguished by the fact that they are of finite injective dimension, they have various symmetry properties, as reflected in their free resolution, their Koszul homology, and their Hilbert function. The last aspect will be discussed in the next chapter.

Gorenstein rings of embedding dimension at most two are complete intersections. The first non-trivial Gorenstein rings occur in embedding dimension three, and they are classified by the Buchsbaum–Eisenbud structure theorem.

In the final section the canonical module of a graded ring is introduced.

Finite modules of finite injective dimension

In this section we study injective resolutions of finite modules. We shall see that the injective dimension of a finite module M over a Noetherian local ring R either is infinite or equals the depth of R, and is bounded below by the dimension of M.

Type
Chapter
Information
Cohen-Macaulay Rings , pp. 88 - 146
Publisher: Cambridge University Press
Print publication year: 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×