Regular sequences in \({\mathbb{Z}}_ 2\)-graded commutative algebra
The paper contains a systematic study of superalgebras (or \({\mathbb{Z}}_ 2\)-graded commutative algebras) and modules over such algebras using a concept of a regular sequence. Let A be a superalgebra and let M be a \({\mathbb{Z}}_ 2\)-graded module over A. A homogeneous element a of A is said to be M-regular if aM\(\neq M\) and \(Ker(M\to^{a}M)=0\quad if\quad \deg (a)=0,\quad Ker(M\to^{a}M)=aM\quad if\quad \deg (a)=1.\) A sequence \(a_ 1,...,a_ n\) of homogeneous elements of A is said to be M-regular if \((a_ 1,...,a_ n)M\neq M\), \(a_ 1\) is M-regular and \(a_ i\) is \(M/(a_ 1,...,a_{i-1})M\)-regular for \(i=2,...,n\). If K is a commutative ring and \(A=K[x_ 1,...,x_ m]\otimes_ K\bigwedge_ K(x_{m+1},...,x_ n) \) is the tensor product of a polynomial algebra over K and an exterior algebra over K with \(\deg (x_ i)=0\quad for\quad 1\leq i\leq m\) and \(\deg (x_ i)=1\quad for\quad m+1\leq i\leq n,\) then \(x_ 1,...,x_ n\) is an A-regular sequence. The author carries over a large part of the formal machinery of regular sequences for commutative rings to superalgebras. Special attention is paid to regular sequences consisting of elements of degree \( 1.\) The classical concepts and results from commutative algebra and algebraic geometry discussed in the context of superalgebras include regular ideals, regular local rings, Cohen's structure theorem for regular local rings, conormally free ideals and flat local homomorphisms. The author introduces the notion of the odd depth of a module M over a superalgebra as the length of any maximal M-regular sequence consisting of elements of degree \( 1\) and proves that the odd depth of M is related to the growth of a minimal free resolution of M. This is the counterpart of the classical relationship between depth and projective dimension as given by the Auslander-Buchsbaum formula. [Reviewer's remark: The notion of an A-regular sequence defined in the paper under review was introduced and studied (under another name and in a different context) in the reviewer's article published in Fundam. Math. 74, 209-231 (1972; Zbl 0236.13012).
scientific article; zbMATH DE number 4110823
scientific article