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If we have a regular sequence $a_1,\dots, a_r$ in a ring $A$, I think it means the subschemes $A/(a_1,\dots,a_i)$ cut out step by step are all equi-dimensional. (when $A$ is affine coordinate ring, by the krull principal ideal theorem).

But what does regular sequence mean for a module? How to understand the concept of depth, which can be defined to be the length of a maximal regular sequence? How to understand the concept of Cohen-Macauley where the depth and dimension coincide?

And an counterexample to C.M. is the union of two planes meeting at one point in $k^4$, how to see this?

  • http://mathoverflow.net/questions/6704/how-to-think-about-cm-rings –  Jun 01 '14 at 14:35
  • http://mathoverflow.net/questions/54876/geometric-meaning-of-cohen-macaulay-schemes –  Jun 01 '14 at 14:35

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Regular sequences $(a_1, \dots, a_r)$ are defined for modules $M$ in the following way:

$a_1$ is a nonzerodivisor on $M$ and for $i>1$, $a_i$ is a nonzerodivisor on the module $M/(a_1,...,a_{i-1})M$.

For your last question, one consequence of a variety being Cohen-Macaulay is that it is "connected in codimension 1", which means that if you remove a subvariety of codimension at least 2, then the resulting topological space is still connected. For this statement, see Theorem 18.12 of Eisenbud's book, "Commutative Algebra".

Steven Sam
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