Need some suggestion for an introductory talk on 'Local Cohomology'?

Question

Next week i am to give a talk on 'Local Cohomology' and i am writing to request suggestions for some basic interesting results for the talk.The relevant information is as follows:

(1) The audience for the talk consists of first year graduate students who have some idea(at least definition) about Local Cohomology modules.

(2) The talk is to be $90$ minutes in length.

(3) I want to start the talk with the computation of Local Cohomology modules for the polynomial ring.

What are some other interesting results (with reference, may be related with resolution and regularity or something else) that can be presented in this talk?

Thank you in advance for all the suggestions!

I've also computed some local cohomology modules on this site, in particular of some ideals in polynomial rings. For instance, see here. Furthermore, one can relate the local cohomology with some important numaerical invariants like ara, cd, reg (Castelnuovo-Mumford regularity), and so on. — user26857, Apr 13 '16 at 11:03
@user26857 Any reference for these relations of local cohomology and numerical invariants cd,reg etc?Sorry for troubling you again. — Arpit Kansal, Apr 13 '16 at 11:43

score 1 · Answer 1 · edited Apr 13 '17 at 12:58

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Te book "Local Cohomology(An Algebraic Introduction with Geometric Applications)-2nd edition" by "BRODMANN-SHARP" has topics in the relationship of Local Cohomology with depth , dim (Chapter 6), Arithmetic rank (3.3), Castelnuovo-Mumford regularity (Chapter 16),... . As an application of the fact that Local Cohomology can measure depth and dim, in some cases Local Cohomology help us detect Cohen-Macaulay-ness. For computation of Local Cohomology there are examples in the book. (For some computation with software see here).

Introduction of the book "Twenty-Four Hours of Local Cohomology" can be a help. In (Lecture 7) there are some examples. Maybe Exercise 7.17 is near to what you need. Being Gorenstein can be characterized by Local Cohomology: Theorem 11.26 (Bruns-Herzog's book also has this). Lecture 16 is about "Polyhedral Applications"

edited Apr 13 '17 at 12:58

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answered Apr 13 '16 at 12:48

user 1

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Thank you for your suggestions,i'll surely have a look! – Arpit Kansal Apr 13 '16 at 13:12
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@Arpit Kansal here, Karl Schwede says "I could give you a long list of properties of rings commonly measured by local cohomology " but he didnt develope this comment to an answer. if he do so it would be very helpful to you and me . – user 1 Apr 13 '16 at 13:15
you are welcome. – user 1 Apr 13 '16 at 13:15

Need some suggestion for an introductory talk on 'Local Cohomology'?

1 Answers1