Return to Question

I am reading the article “The homological theory of maximal Cohen-Macaulay approximations" by Auslander and Buchweitz (pdf on NUMDAM). In this article, the proof Lemma 3.1 constructs a commutative diagram as follows. We work in an Abelian category $\mathbf{C}$ with a full, additively closed, exact subcategory $\mathbf{X}$ (and a few more hypotheses that don’t seem to be relevant). Given exact sequences $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ where the latter is an $\mathbf{X}$-approximation of $A$, the proof says that since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exists an exact commutative diagram like:

My question is: How to calculate Z? Is it a pull back or push out?

(P.S: The above is in the context of an abelian category, in which the existence of projective and/or injective objects is not assumed.)

I am reading the article

• Maurice Auslander, Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay 21-22 mai 1987), Mémoires de la Société Mathématique de France, Série 2, no. 38 (1989), pp. 5-37. doi:https://doi.org/10.24033/msmf.339

In this article, the proof Lemma 3.1 constructs a commutative diagram as follows. We work in an Abelian category $\mathbf{C}$ with a full, additively closed, exact subcategory $\mathbf{X}$ (and a few more hypotheses that don’t seem to be relevant). Given exact sequences $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ where the latter is an $\mathbf{X}$-approximation of $A$, the proof says that since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exists an exact commutative diagram like:

My question is: How to calculate Z? Is it a pull back or push out?

(P.S: The above is in the context of an abelian category, in which the existence of projective and/or injective objects is not assumed.)

I read the article "The Homological Theory of maximal cohen-macaulay approximations" wrote by Auslander and Buchweitz. In this article in Lemma 3.1 in category theory drew commutative diagram by two exact sequence $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ such that above exact sequence is X-approximaion of $A$ and $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ and since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exist an exact commutative diagram like:

My questions are these: How to calculate Z? Is it a pull back or push out? Can anybody help me please? thanks

P.S: All the above were in the context of an abelian category, in which the existence of projective and/or injective objects is not necessary.

I am reading the article “The homological theory of maximal Cohen-Macaulay approximations" by Auslander and Buchweitz (pdf on NUMDAM). In this article, the proof Lemma 3.1 constructs a commutative diagram as follows. We work in an Abelian category $\mathbf{C}$ with a full, additively closed, exact subcategory $\mathbf{X}$ (and a few more hypotheses that don’t seem to be relevant). Given exact sequences $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ where the latter is an $\mathbf{X}$-approximation of $A$, the proof says that since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exists an exact commutative diagram like:

My question is: How to calculate Z? Is it a pull back or push out?

(P.S: The above is in the context of an abelian category, in which the existence of projective and/or injective objects is not assumed.)

I read the article "The Homological Theory of maximal cohen-macaulay approximations" wrote by Auslander and Buchweitz. In this article in Lemma 3.1 in category theory drew commutative diagram by two exact sequence $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ such that above exact sequence is X-approximaion of $A$ and $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ and since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exist an exact commutative diagram like:

My questions are these: How to calculate Z? Is it a pull back or push out? Can anybody help me please? thanks

I read the article "The Homological Theory of maximal cohen-macaulay approximations" wrote by Auslander and Buchweitz. In this article in Lemma 3.1 in category theory drew commutative diagram by two exact sequence $$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$ such that above exact sequence is X-approximaion of $A$ and $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ and since $$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ is an isomorphism, there exist an exact commutative diagram like:

My questions are these: How to calculate Z? Is it a pull back or push out? Can anybody help me please? thanks

P.S: All the above were in the context of an abelian category, in which the existence of projective and/or injective objects is not necessary.