On the existence of non-free totally reflexive modules

Totally reflexive modules over connected sums with m^3=0

“…Assume that R has a minimal totally acyclic complexes. Then the conclusion follows immediately from Theorem (3.5) and Lemma (4).…”

“…It was shown in [4] , Theorem 5.1 that there is a countable intersection Similar open conditions imposed on the transpose matrices A t and B t ensure the acyclicity of the dual complex.…”

“…Inspecting the defining equations of the Stanly-Reisner ring, we see that no such relation exists other than f g = 0. Proposition 3.9 in [4] shows that rings obtained from the constructin described above do not have exact zero divisors. On the other hand, Example 4.1 in [4] is an example of such a ring that has non-free totally reflexive modules.…”

“…Proposition 3.9 in [4] shows that rings obtained from the constructin described above do not have exact zero divisors. On the other hand, Example 4.1 in [4] is an example of such a ring that has non-free totally reflexive modules. The rings studied in this paper can be viewed as generalizations of this example.…”

“…Even for rings with m 3 R = 0, there are no known necessary and sufficient conditions for G-regularity. In [4], it was shown that rings obtained from Stanley-Reisner rings of graphs after modding out by a linear system of parameters satisfy m 3 R = 0, and some conditions for G-regularity of such rings were studied. Example (4.1) in [4] prompted us to consider the class of rings studied in this paper.…”