What can I say for free about a module with dominant dimension 2 (other than the double centralizer property)?

Question

Let's say I have my favorite finite dimensional algebra $A$, and favorite module $T$. Now assume that the reason $T$ is my favorite module is that it has a cool property:

1. there is an injection $A\to T$ which is a $\mathrm{add}(T)$-approximation (that is, any map of $A$ to a summand of $T$ factors through this map).
2. better yet, this map extends to a copresentation $0\to A\to T\to T^m$.

Konig, Slungard and Xi define such a module to have dominant dimension 2. This has one cool consequence, the double centralizer property $A=\mathrm{End}_{\mathrm{End}_A(T)}(T)$.

Does this condition have any other interesting consequences? Has anything other than the KSX paper been written on this? (Looking on Google Scholar at papers referencing theirs didn't turn up much).

Answer 1

I don't think the idea of dominant dimension originated in KSX 2001. I don't know of any particularly interesting results, but probably a reasonable seed to start your search from is:

Auslander, M.; Solberg, Ø. "Gorenstein algebras and algebras with dominant dimension at least 2". Comm. Algebra 21 (1993), no. 11, 3897-3934. MR1238133 DOI:10.1080/00927879308824773

There were too many papers referencing these to check for interesting results, but some of them were even just reference books, so I think it should not be too hard to find interesting properties.

Answer 2

The first reference I know for dominant dimension is Nakayama's 1958 paper, where he (among other things) conjectured [something equivalent to] a finite-dimensional algebra of infinite dominant dimension is quasi-Frobenius. See Auslander and Reiten's generalized Nakayama conjecture paper from 1975.

Auslander's Queen Mary notes (not widely available online, but here's a Google books link to a later, similar, Auslander paper) are the canonical place for the phrase "dominant dimension two", I think. He characterized the Artin algebras $\Gamma$ of global dimension at most two and dominant dimension at least two: they are obtained by taking an Artin algebra $\Lambda$ of finite representation type, with representation generator $M$, and setting $\Gamma = \mathrm{End}_\Lambda(M)$. (Now called the Auslander algebra of $\Lambda$.) This gives a bijection on Morita equivalence classes of finite-representation type algebras and such algebras $\Gamma$.

Answer 3

When the algebra has dominant dimension at least two, then a module has dominant dimension at least two iff it is reflexive iff it is projecctive or a 2. syzygy, see http://www.tandfonline.com/doi/abs/10.1080/00927879208824528 an for example http://arxiv.org/abs/1608.04212 for references for this and more on dominant dimensions of modules.