Homogeneous ideal and its system of generators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:57:18Z http://mathoverflow.net/feeds/question/108973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108973/homogeneous-ideal-and-its-system-of-generators Homogeneous ideal and its system of generators Knot 2012-10-06T02:47:50Z 2013-03-13T20:45:03Z <p>Let $I$ be a homogeneous ideal in a graded commutative ring $R$, $S$ be its minimal system of generators.</p> <p>What is the conclusion that we can say about the element in $S$ ? Is the cardinality of $S$ uniquely determined by $I$ ? </p> <p>In the book Commutative ring theory of Matsumura, theorem 2.3, page 8 there is a theorem for local ring which we can deduce that the number of generator is unique. So, is there a version of that theorem for the graded ring ?</p> <p>What about the degree of generator in $S$ ? </p> http://mathoverflow.net/questions/108973/homogeneous-ideal-and-its-system-of-generators/109108#109108 Answer by Youngsu for Homogeneous ideal and its system of generators Youngsu 2012-10-08T02:06:09Z 2012-10-08T02:06:09Z <p>Also, Proposition 1.5.15 in Cohen-Macaulay rings by Bruns and Herzog might help you. This is a bit more general than what Karl mentioned above. </p> http://mathoverflow.net/questions/108973/homogeneous-ideal-and-its-system-of-generators/109994#109994 Answer by YACP for Homogeneous ideal and its system of generators YACP 2012-10-18T09:25:14Z 2012-11-03T20:09:55Z <p>I gave a short answer to this question here: <a href="http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators" rel="nofollow">http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators</a>.</p> <p>A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then </p> <p>$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$ </p> <p>where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.</p>