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Prime submodules, the descent invariant, and modules of finite length. (English) Zbl 0878.13005
The authors extend the notion of prime ideals to modules over commutative rings and try to develop a (local) dimension theory for modules. As the first step, a class of particularly simple prime submodules, the so-called 0-submodules, is investigated in section 2. A submodule is defined to be a 0-submodule if the factor module is torsion-free. It is proved that over regular local rings a 0-submodule of rank 1 of a finitely generated free module is always free. This result is used in theorem 2.6 to characterize vector bundles of rank \(r-1\), generated by \(r\) elements over the punctured spectrum. The prime dimension of a module is introduced and investigated in section 3. This careful investigation leads to the notion of descent which is used to characterize reflexive modules over regular local rings by combinatorial invariants in section 4. Finitely generated modules over regular local rings satisfying Serre’s condition \(S_n\) are also discussed. In section 5 finitely generated reflexive modules over local domains are characterized by using matrices and syzygies. Torsion-free finitely generated modules over polynomial rings with two variables over algebraically closed fields are discussed in section 6.

MSC:
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13C05 Structure, classification theorems for modules and ideals in commutative rings
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[2] Evans, G.; Griffith, Ph., Syzygies, (1985), Cambridge University Press
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