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Sequences for complexes

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  • Mathematics


CHRISTENSEN MATH. SCAND. 89 (2001), 161–180 SEQUENCES FOR COMPLEXES LARS WINTHER CHRISTENSEN Introduction Let R be a commutative Noetherian ring and let M �= 0 be a finite (that is, finitely generated) R-module. The concept of M-sequences is central for the study of R-modules by methods of homological algebra. Largely, the useful- ness of these sequences is based on the following properties: 1◦ When � is an ideal in R and M/�M �= 0, the number inf{� ∈ Z | Ext�R(R/�,M) �= 0}, the so-called �-depth of M , is the maximal length of an M-sequence in �, and any maximal M-sequence in � is of this finite length. 2◦ If x1, . . . , xn is an M-sequence contained in � ∈ SuppR M , then the sequence of fractions x1/1, . . . , xn/1, in the maximal ideal of R�, is an M�-sequence. In commutative algebra, a wave of work dealing with complexes of modules was started by A. Grothendieck, see [9]. The underlying idea is the following: Complexes (that is, complexes of modules) are tacitly involved whenever ho- mological methods are applied, and since hyperhomological algebra, that is, homological algebra for complexes, is a very powerful tool, it is better to work consistently with complexes. Modules are also complexes, concentrated in degree zero, so results for complexes yield results for modules as special cases. Like most concepts for modules that of M-sequences can be extended to complexes in several non-equivalent ways; this short paper explores two such possible extensions: (ordinary) sequences and strong sequences for com- plexes. Ordinary sequences have a property corresponding to 1◦, at least over local rings where they coincide with the regular sequences suggested by H. -B. Foxby in [8, Sec. 12]. But ordinary sequences may fail to localize properly, Received June 22, 1998; in revised form December 7, 1998. 162 lars winther christensen whereas strong sequences not only enjoy the correspondent property of 2◦, but also that of 1◦ in the special case where R is local and � the m

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