Independence of the Total Reflexivity Conditions for Modules

By David A. Jorgensen, Liana Sega. 2004

62 citations recorded by the source corpus.

Abstract We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive R-module M such that ExtRi(M,R)=0 for all i>0, but ExtRi(M*,R)≠0 for all i>0.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb2090b1b6b4776d418fc0e65f1e023cb163b652

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

On algebras of finite Cohen-Macaulay type (2011, background): “In fact they constructed a commutative Noetherian local (graded Koszul) ring and a finitely generated stable module which is not torsion-free, see [47] for details.”
Free resolutions over short local rings (2007, background): “Patterns that had been conjectured not to exist at one time have subsequently been discovered over rings with m3 = 0: not finitely generated algebras ExtR(k, k) (Roos [22]); transcendental Poincaré series PRk (t) (Anick [1]); modules with constant Betti numbers and aperiodic minimal free resolution”
On Gorenstein projective, injective and flat dimensions—A functorial description with applications (2004, background): “Indeed, recent work of Jorgensen and ¸Sega [34] shows that for a reﬂexive module M the vanishing of Ext > 0 R (M,R) and Ext > 0 R ( Hom R (M,R),R) cannot be inferred from vanishing of any ﬁnite number of these cohomology modules.”
Beyond totally reflexive modules and back (2008, background): “In general, though, one can not infer from such vanishing thatM is totally reflexive—explicit examples to this effect are constructed by Jorgensen and Şega in [77]—and this has motivated a search for alternative criteria for finiteness of G-dimension.”