# Some places I can't understand in the paper “Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra”

I am reading the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", the link is here: https://arxiv.org/pdf/1608.04212.pdf

There are some places I can't understand:

1. On page 10, proposition 3.11 says that we can apply Mueller's theorem to calculate the dominant dimension of $C$, then $dom.dim(C)=inf \{ i \geq 1 | Ext^i_B(M,M) \not =0 \}+1$(It says Theorem 2.2 (4) is refered as Mueller's theorem)
2. In the proof of Corollary 3.15 on page 11, $1 \Longrightarrow 2$, we get that $Dom_d=Gp(A)$, then it says by proposition 3.14, we get $A$ is Gorenstein with Gorenstein dimension d. But to use 3.14, we need that $dom.dim(A)=d$ and I can only get $dom.dim(A) \geq d$ by $Dom_d=proj$. So how to get $dom.dim(A)=d$ ? Another question is that by $Gp(A)=proj$, then $A$ is CM-free. Since a Gorenstein algebra is CM-free iff it has finite global dimension. But why it says the global dimension equal to the Gorenstein dimensin d?

Sorry for that I have not explain the concrete meanings for those symbols in this question since it needs quite much word, these can be found in page 3 and 5.

• I do not understand the question in 1. To 2.:I think it is assumed in 3.15. that d is the dominant dimension of A as in 3.14. (but should have been stated better) It is a general fact that the global dimension coincides with the Gorenstein dimension in case the global dimension is finite. – Mare Mar 28 '17 at 9:03
• @Mare My first question is how to use theorem 2.2(4) get that $dom.dim(C)= inf \{ i \geq 1 | Ext^i_B(M,M) \not = 0 \} +1$? – Xiaosong Peng Mar 28 '17 at 9:07