Unsolved problems concerning Artinian Rings and Artinian Modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:43:43Z http://mathoverflow.net/feeds/question/60060 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60060/unsolved-problems-concerning-artinian-rings-and-artinian-modules Unsolved problems concerning Artinian Rings and Artinian Modules To be cont'd 2011-03-30T12:13:28Z 2011-04-09T05:24:17Z <p>I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to find and understand the statement of <a href="http://en.wikipedia.org/wiki/Monomial_conjecture" rel="nofollow">the Monomial Conjecture</a>. Upon googling, I did not find problems that would serve my need as they were well over my head.</p> <p><strong>Question</strong>: Are there more well-known and simply-stated problems?</p> <p><strong>P.S.</strong>: By simply-stated, I mean understandable by a student comfortable with some theory of modules and rings, chain conditions on rings and modules, radicals(nil and Jacobson's), Nakayama's lemma and the Krull dimension of a ring plus some special topics. You can assume this to be my 'bailiwick' while suggesting problems.</p> <p>Your assistance is greatly appreciated.</p> http://mathoverflow.net/questions/60060/unsolved-problems-concerning-artinian-rings-and-artinian-modules/61115#61115 Answer by Hailong Dao for Unsolved problems concerning Artinian Rings and Artinian Modules Hailong Dao 2011-04-09T05:24:17Z 2011-04-09T05:24:17Z <p>There are many open problems which are fairly easy to state, also one might need some basic definitions, for example derived functors. I will provide mostly pointers to the ones I know, you can google for more details (one can easily fill many projects with each topic below). Interestingly, most people do not consider the Monomial conjecture as a problem about Artinian rings (the word Artinian only appear as in the definition of a system of parameters).</p> <p><strong>Representation theory of Artin algebras</strong>: you can start with the list at page 411 of <a href="http://books.google.com/books?id=aiRmz22Ou3wC&amp;dq=isbn:0521599237&amp;ei=D-mfTZjYG4XeNd_L5acG" rel="nofollow">this book</a>. </p> <p><strong>Numerics of Betti numbers, Hilbert functions</strong>: perhaps the most famous one is the Buchsbaum-Eisenbud-Horrocks Conjecture; </p> <blockquote> <p>If $R$ is a regular local ring of dimension $n$ and $M$ is an Artinian $R$-module, then the rank of the $i$-th module in the minimal free resolution of $M$ is at least $n \choose i$. (weaker version, but as open: the sum of the ranks is at least $2^n$, graded version is also open). </p> </blockquote> <p>An important open problem is to characterize the Hilbert functions of Gorenstein graded artinian algebras. See the surveys by Irena Peeva or Valla for some other questions. </p> <p><strong>Homological problems</strong>: Here the big one, and also one of the simplest looking, is the Auslander-Reiten Conjecture </p> <blockquote> <p>If $R$ is commutative Artinian ring and $M$ is a finitely generated $R$-module such that $\text{Ext}^i(M, M\oplus R) = 0$ for all $i>0$, then $M$ is projective. </p> </blockquote> <p><strong>Mathoverflow</strong>: see for example <a href="http://mathoverflow.net/questions/36396/length-of-i-i2-versus-anni-anni2-in-artinian-rings" rel="nofollow">here</a> or <a href="http://mathoverflow.net/questions/26013/selforthogonal-modules-over-artinian-gorenstein-rings" rel="nofollow">here</a>. </p>