(Regular sequence)
Let \(M\) be an \(A\)-module and \(x_1,\ldots,x_r \in A\). This is a regular sequence for \(M\) if
for each \(i\), \(x_i\) is not a zero divisor on \(M/(x_1,\ldots,x_{i-1})M\), and in particular \(x_1\) is not a zero divisor on \(M\);
the inclusion \((x_1,\ldots,x_r)M \subsetneq M\) is proper. For \(M=A\), this is called a regular sequence in \(A\); see the regular-sequence definition.
(Permutation over a local ring)
Let \((A,\mathfrak m)\) be a noetherian local ring and let \(M\) be a finite \(A\)-module. Every permutation of an \(M\)-regular sequence is again \(M\)-regular; see the permutation lemma. The entries of such a sequence necessarily lie in \(\mathfrak m\), since an entry that is a unit would make the final quotient zero.
(Order can matter outside the local setting)
In \(A=k[x,y,z]\), the sequence \(x,y(1-x),z(1-x)\) is regular: after quotienting by \(x\), its remaining entries are \(y,z\) in \(k[y,z]\), and the final quotient is \(k\). The reordering \(y(1-x),z(1-x),x\) is not regular. Modulo \(y(1-x)\), the second entry annihilates the nonzero class of \(y\). That class is nonzero because \(y=y(1-x)h\) in the polynomial domain would imply \(1=(1-x)h\), impossible for a polynomial \(h\).
(Depth along an ideal)
For a finite \(A\)-module \(M\) and an ideal \(I\), define \(\depth_I M\) as the supremum of the lengths of \(M\)-regular sequences in \(I\) when \(IM\ne M\), and set \(\depth_I M=\infty\) when \(IM=M\). This includes \(\depth_I0=\infty\) and \(\depth_A M=\infty\); see the depth convention. For a noetherian ring and \(IM\ne M\), the supremum is a finite maximum. For a local ring \((A,\mathfrak m)\), write \(\depth M=\depth_{\mathfrak m}M\).
(Dimension bound)
For a nonzero finite module \(M\) over a noetherian local ring, \[ \depth M\le\dim\operatorname{Supp}(M). \] Equality defines a Cohen–Macaulay module; see the depth dimension bound. For \(M=A\), this recovers the Cohen--Macaulay ring condition.
(Koszul coefficients)
If \(x_1,\ldots,x_r\) is \(M\)-regular, the Koszul complex \(K_\bullet(x_1,\ldots,x_r;M)\) has zero homology in positive degrees and degree-zero homology \(M/(x_1,\ldots,x_r)M\). Its terms are finite direct sums of \(M\). In particular, an \(A\)-regular sequence gives a finite free resolution \(K_\bullet(x_1,\ldots,x_r;A)\) of \(A/(x_1,\ldots,x_r)\) [@Har10a, Proposition III.7.10A]. For the module assertion, tensor the one-element Koszul complexes successively with \(M\). At each step the mapping-cone homology sequence reduces positive homology to the kernel of the next element on the preceding quotient, which is zero by regularity; degree-zero homology is the next quotient.
(Module regularity is not ring regularity)
Let \(A=k[x,y]/(xy)\) and \(M=A/(x)\cong k[y]\). Multiplication by \(y\) on \(M\) is injective and \(M/yM\cong k\ne0\), so \(y\) is \(M\)-regular. But multiplication by \(y\) on \(A\) kills the nonzero class of \(x\). Thus \(K_\bullet(y;A)\) has nonzero first homology and is not a free resolution of \(A/(y)\), whereas \(K_\bullet(y;M)\) resolves \(M/yM\).
Dependencies
D-DEFNOETHNoetherian rings and modulesuses
Backlinks
D-SCHCMGORCohen--Macaulay, Gorenstein and local complete intersection schemesuses