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Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$.

Here $id(X)$ stands for the injective and $pd(X)$ for the projective dimension of $X$. $mod-R$ are the finitely generated $R-modules$ (or should we better take finitely presented modules?).

Define the big super global dimension as $Sgl(R):= \sup \{ pd(X)+id(X) | X \in Mod-R$ and indecomposable $\}$. (here all modules are allowed).

Has this dimension been studied before?

I looked at those dimensions only for finite dimensional algebras. Sadly I have nearly no experience with general rings so sorry in case my questions are not appropriate for mathoverflow.

Questions: 1. What is the super global dimension (small or big always) for a polynomial ring in $n$ variables, where any number of variables is allowed (possibly infinite).

  1. What is the super global dimension for the Weyl algebras?

  2. What is the super global dimension for the ring of holomorphic function?

  3. What is the super global dimension for some other interesting ring of your choice?

  4. Define the flat super global dimension as the super global dimension but with $pd(X)$ replaced by $fd(X)$. Does it coincide with the super global dimension and if not what is it in the previous questions?

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    $\begingroup$ In the definition of super global dimension, do you want to restrict to indecomposable modules X? Otherwise, it looks like $sgl(R)=2 gl.dim(R)$ trivially. $\endgroup$
    – Alex Dugas
    Commented May 10, 2018 at 21:02
  • $\begingroup$ @AlexDugas Thanks, it should be indecomposables or else it should be easy by taking the direct sum of two modules with maximal injective and projective modules. Maybe for some algebras it would also be interesting to restrict to simple modules. $\endgroup$
    – Mare
    Commented May 11, 2018 at 0:00

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