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$f_1, \ldots, f_r$ is a regular sequence in $S/I$ (where $S$ is a polynomial ring in $n$ variables, and $I$ its ideal) iff $$(I, f_1, \ldots, f_{i-1}): (f_i)= (I, f_1, \ldots, f_{i-1}) \quad i \ge 2.$$

I think a sequence is regular if $f_1 + I \notin Z(S/I)$ (set of zero divisors), $f_2 + I \notin Z(S/(I, f_1))$, and so on.

Assuming $f_1 + I \notin Z(S/I)$, this means, I think, $I : (f_1) = I$, then $f_2 + I \notin Z(S/(I, f_1))$ implies $I : (f_2) = (I, f_1)$ and so on. This clearly isn't what we're asked to prove. Or have I maybe not written down the correct definitions. Can someone please help? None of this is in our text book and I couldn't find anything down to my level online.

The definition of a regular sequence in a ring $R$ that I have is:

$r_1, \dots, r_m$ is a regular sequence in $R$ if:

$r_1 \notin Z(R), r_2 \notin Z(R/(r_1)), r_3 \notin Z(R/(r_1, r_2)),$ and so on...

Also, I have it written down that $r+I \notin Z(R/I) \iff I : (r) = I.$ As you can see, I'm in desperate need of a clear picture on the topic.

Michael Hardy
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user337578
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1 Answers1

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$f_1, ..., f_r$ is a regular sequence in $S/I$ (where $S$ is a polynomial ring in $n$ variables, and $I$ its ideal) iff $(I, f_1, ..., f_{i-1}): (f_i)= (I, f_1, ..., f_{i-1}) \quad i \ge 2.$

"$\Rightarrow$" $$a\in(I, f_1, ..., f_{i-1}): (f_i)\implies af_i\in(I, f_1, ..., f_{i-1})\implies \widehat{a}\widehat{f_i}\in(\widehat{f_1},\dots,\widehat{f_{i-1}})\ (\text{in } S/I)\implies \widehat{a}\in(\widehat{f_1},\dots,\widehat{f_{i-1}})\implies a\in(I, f_1, ..., f_{i-1})$$

"$\Leftarrow$" $$\widehat{a}\widehat{f_i}\in(\widehat{f_1},\dots,\widehat{f_{i-1}})\ (\text{in } S/I)\implies af_i\in(I, f_1, ..., f_{i-1})\implies a\in(I, f_1, ..., f_{i-1}): (f_i)\implies a\in(I, f_1, ..., f_{i-1})\implies\widehat{a}\in(\widehat{f_1},\dots,\widehat{f_{i-1}})\ (\text{in } S/I)$$

user26857
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