# An identity for Ext for rings

Let $$A$$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every non-zero module has finite grade). For simplicity we can also assume first that $$A$$ is a finite dimensional algebra and modules are finitely-generated (but non-finitely generated examples or non-Gorenstein examples are also welcome but Im mainly interested in finite dimensional algebras).

For a module $$M$$, define the grade of $$M$$ as: $$g_M:= \inf \{ i \geq 0 | Ext_A^i(M,A) \neq 0 \}$$. Define the Ext-dual of $$M$$ to be $$U(M):=Ext_A^{g_M}(M,A)$$ and the double Ext-dual of $$M$$ as $$G(M):=U(U(M))=Ext_{A^{op}}^{g_{Ext_A^{g_M}(M,A)}}(Ext_A^{g_M}(M,A),A)$$. Note that $$G(M)$$ is always non-zero.

Question: Do we have always that $$G^l(M) \cong G^{l-1}(M)$$ (at least in the stable category of $$A$$) for some $$l \geq 1$$ and indecomposable modules $$M$$, so that the sequence of the $$G^l(M)$$ becomse stationary?

This is true for $$A$$ selfinjective or hereditary (in those cases we have $$G(M) \cong M$$ for all $$M$$). In all examples this was even true for $$l \leq 2$$ so I wonder whether we have $$G^2(M) \cong G(M)$$ (at least in the stable category).

It seems that any module $$M$$ whose double $$A$$-dual $$M^{**}$$ is not reflexive gives a counterexample. In this case $$G(M) = M^{**}$$ is a summand of $$G^2(M) = (M^{**})^{**}$$ with non-trivial complement, and $$G^2(M)$$ can't be reflexive since this property is inherited by summands. Repeating this argument with $$G(M)$$ instead of $$M$$, we see that $$G^l(M)$$ is a summand of $$G^{l+1}(M)$$ with non-trivial complement for each $$l \geq 0$$.

Here's an example over a non-Gorenstein ring:

Take $$S$$ to be the unique simple module over $$A = k[x,y]/(x,y)^2$$. Then $$g_S = 0$$ and its $$A$$-dual is $$S^* \cong S^{\oplus 2}$$ with $$g_{S^*} = 0$$, so that $$G(S) = S^{**} \cong S^{\oplus 4}$$ and in general $$G^{l}(S) = S^{\oplus 4l}$$.

• Thanks, do you also know an example where $add(G^l(M))$ does not become stationary?
– Mare
Sep 22 '20 at 18:54
• I don't. That sort of stability would be very interesting if it did occur. Sep 22 '20 at 19:00
• I will do some more experiments, especially with local algebras to see whether this might be true.
– Mare
Sep 22 '20 at 19:02
• If you're intent on Gorenstein rings, then I'd suggest taking a look at Auslander-Gorenstein rings (in particular any commutative Gorenstein ring). Over such rings the grade of $Ext^n(M, A)$ is always $n$, and there are various sequences relating $M$ to its double Ext-dual, for example in the work of Levasseur (math.univ-brest.fr/perso/thierry.levasseur/files/…). My guess is modules over these rings might satisfy the stability $G(M) = G^2(M) = G^3(M) = \cdots$ that you wanted. Sep 22 '20 at 20:32
• (Small correction: The grade of $Ext^n(M, A)$ is always at least $n$, and if $n = g_M$ then the grade of $Ext^{g_M}(M, A)$ is also $g_M$.) Sep 22 '20 at 20:39