Flat modules over commutative noetherian rings

Vasconcelos, Wolmer V.

doi:10.1090/S0002-9947-1970-0265341-5

Flat modules over commutative noetherian rings
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Trans. Amer. Math. Soc. 152 (1970), 137-143 Request permission

Abstract:

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

References

Tomoharu Akiba, Remarks on generalized rings of quotients, Proc. Japan Acad. 40 (1964), 801–806. MR 180573
H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
Claude Chevalley, Fundamental concepts of algebra, Academic Press, Inc., New York, 1956. MR 0082459
Harley Flanders, On free exterior powers, Trans. Amer. Math. Soc. 145 (1969), 357–367. MR 249417, DOI 10.1090/S0002-9947-1969-0249417-6
Daniel Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128 (French). MR 254100
Fred Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794–799. MR 181653, DOI 10.1090/S0002-9939-1965-0181653-1
Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI 10.1090/S0002-9947-1969-0238839-5
Wolmer V. Vasconcelos, On projective modules of finite rank, Proc. Amer. Math. Soc. 22 (1969), 430–433. MR 242807, DOI 10.1090/S0002-9939-1969-0242807-2
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581