MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow I\longrightarrow 0$ be a finite presentation of $I.$ What is the relation between symmetric algebra of $I,$ i.e. $Sym(I)$ and the syzygies matrix?

share|cite|improve this question

Quite in general, let $M$ be an $R$-module represented as the cockernel of a linear map $f: G\to F$ of free R-moduels $F$ and $G$ of rank $n$ and $m$. Then $\text{Sym}(M)$ is isomorphic (as an R-algebra) to $\text{Sym}(F)/J$ where J is the ideal of $\text{Sym}(F)$ generated by the image of $f$ (the syzygies of $M$). Choosing basis $\text{Sym}(F)$ is isomorphic to $R[x_1,\dots,x_n]$ and $J$ is generated by $m$ elements of degree 1 that correspond to the syzygies of $M$.

These are well known properties of the symmetric algebra, see for example Bourbaki Algebra I: Chapters 1-3.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.