Proof explanation of a theorem in the paper "On the finitistic global dimension conjecture for Artin algebras"

Question

I am reading the paper "On the finitistic global dimension conjecture for Artin algebras", the link is here:https://www.researchgate.net/publication/240152799_On_the_finitistic_global_dimension_conjecture_for_Artin_algebras.

On page 3, there is a theorem 0.4:

I am trying to understand the proof. But there are some places I can't understand:

1. What is the meaning of $\Omega^nA\approx \Omega^nB$?
2. Why $n \leq pdC$ and $n \leq \phi(A \oplus B)$?
3. By Horseshoe lemma, I can get the exact sequence $0 \rightarrow X \oplus P \rightarrow X \oplus Q \rightarrow \Omega^nC \rightarrow 0$ where I think $X= \Omega^n A$. But I don't know why $X=Y \oplus Z$ and how to get $0 \rightarrow Z \oplus P \rightarrow Z \oplus Q \rightarrow \Omega^nC \rightarrow 0$. Thank you for any help.

Answer 1

Ad 1: I guess it means projective equivalence, i.e. $M\approx N$ if and only if there exist projective modules $P,Q$ such that $M\oplus P\cong N\oplus Q$ - the operation of taking syzygies is well-defined up to this notion of equivalence. Then the claim follows from the fact that for any short exact sequence $0\to A\to B\to C\to 0$ and any $n\geq 0$, by the Horseshoe Lemma there is a choice of $n$-th syzygies such that there is a short exact sequence $0\to \Omega^n A\to \Omega^n B\to \Omega^n C\to 0$. If $n\geq \text{pdim}(C)$, then $\Omega^n C$ is projective, so $\Omega^n A\approx\Omega^n B$.

Ad 2: If $\Omega^n A\approx\Omega^n B$ then also $[\Omega^n A]=[\Omega^n B]$ in $K_0$, and we have just seen that $n\geq\text{pdim}(C)$ suffices. Concerning $n\leq\phi(A\oplus B)$: By definition of $\phi$, stably equal elements of $L^{\phi(A\oplus B)}(\text{add}(A\oplus B))$ with respect to $L$ are equal; now, the elements $[\Omega^{\phi(A\oplus B)}(A)]$ and $[\Omega^{\phi(A\oplus B)}(B)]$ are stably equal since $[A]$ and $[B]$ are, so $[\Omega^{\phi(A\oplus B)}(A)]=[\Omega^{\phi(A\oplus B)}(B)]$, hence $n\leq \phi(A\oplus B)$ by definition of $n$.

Ad 3: That's just the Fitting lemma applied to the homomorphism $X\to X$ and the observation that the automorphism part of it can be cancelled from the exact sequence.