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Although I agree that one can easily decide to not worry about the case of the zero module, but as ashpool points out, it happens that sometimes we end up with the zero module whether we want or not and then each time we need to say (using ashpool's example) if $M/aM\neq 0$, then bluh and if $M/aM=0$ than something else happens.

So, I think there is actually something to be gained from making a definition that makes sense for the zero module (or the zero object in a more general situation). Of course, sometimes the definition that makes one (in)equality work does not work for another. However, one could still say in a paper (less likely in a book I suppose) that we are using the following definition for whatever which is the usual one if the object is not zero and gives this or that when it is zero and makes the following inequality work.

So having philosophized about this let me give a definition of projective dimension that gives $-\infty$ for the zero module.

Definition Let $(R,\mathfrak m,k)$ be a noetherian local ring and $M$ a finite $R$-module. Define the projective dimension of $M$ as $$\mathrm{proj\, dim}_R M:=\sup \left\{ i\in \mathbb{Z}\cup\{\pm\,\infty\} mathbb{Z} \ \vert \ \mathrm{Ext}_R^i (M,k)\neq 0 \right\}. right\},$$ where $\sup$ is taken in $\mathbb{Z}\cup\{\pm\,\infty\}$.

This is actually essentially ashpool's definition (1), except that for $M=0$ it takes the $\sup$ of the empty set. (This may have been what samantha's professor told her). It also makes the change of rings formula to work.

In fact, I would argue that this is the "right" definition anyway, because the point is those Ext groups that are non-zero, not those that are.

Regarding adding the $\{\pm\,\infty\}$ possibilities: We definitely need to allow $+\infty$, so it makes sense to allow $-\infty$ as well, especially that because we need it for $M=0$.

Comment Of course one can start wondering what to do with non-local and/or non-noetherian rings, but I will leave that meditation to the reader.

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Although I agree that one can easily decide to not worry about the case of the zero module, but as ashpool points out, it happens that sometimes we end up with the zero module whether we want or not and then each time we need to say (using ashpool's example) if $M/aM\neq 0$, then bluh and if $M/aM=0$ than something else happens.

So, I think there is actually something to be gained from making a definition that makes sense for the zero module (or the zero object in a more general situation). Of course, sometimes the definition that makes one (in)equality work does not work for another. However, one could still say in a paper (less likely in a book I suppose) that we are using the following definition for whatever which is the usual one if the object is not zero and gives this or that when it is zero and makes the following inequality work.

So having philosophized about this let me give a definition of projective dimension that gives $-\infty$ for the zero module.

Definition Let $(R,\mathfrak m,k)$ be a noetherian local ring and $M$ a finite $R$-module. Define the projective dimension of $M$ as $$\mathrm{proj\, dim}_R M:=\sup \{ i left\{ i\in \mathbb{Z}\cup\{\pm\,\infty\} \ \vert \ \mathrm{Ext}_R^i (M,k)\neq 0 \}. right\}.$$

This is actually essentially ashpool's definition (1), except that for $M=0$ it takes the $\sup$ of the empty set. (This may have been what samantha's professor told her). It also makes the change of rings formula to work.

In fact, I would argue that this is the "right" definition anyway, because the point is those Ext groups that are non-zero, not those that are.

Regarding adding the $\{\pm\,\infty\}$ possibilities: We definitely need to allow $+\infty$, so it makes sense to allow $-\infty$ as well, especially that we need it for $M=0$.

Comment Of course one can start wondering what to do with non-local and/or non-noetherian rings, but I will leave that meditation to the reader.

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Although I agree that one can easily decide to not worry about the case of the zero module, but as ashpool points out, it happens that sometimes we end up with the zero module whether we want or not and then each time we need to say (using ashpool's example) if $M/aM\neq 0$, then bluh and if $M/aM=0$ than something else happens.

So, I think there is actually something to be gained from making a definition that makes sense for the zero module (or the zero object in a more general situation). Of course, sometimes the definition that makes one (in)equality work does not work for another. However, one could still say in a paper (less likely in a book I suppose) that we are using the following definition for whatever which is the usual one if the object is not zero and gives this or that when it is zero and makes the following inequality work.

So having philosophized about this let me give a definition of projective dimension that gives $-\infty$ for the zero module.

Definition Let $(R,\mathfrak m,k)$ be a noetherian local ring and $M$ a finite $R$-module. Define the projective dimension of $M$ as $$\mathrm{proj\, dim}_R M:=\sup \{ i \ \vert \ \mathrm{Ext}_R^i (M,k)\neq 0 \}.$$

This is actually essentially ashpool's definition (1), except that for $M=0$ it takes the $\sup$ of the empty set. (This may have been what samantha's professor told her). It also makes the change of rings formula to work.

In fact, I would argue that this is the "right" definition anyway, because the point is those Ext groups that are non-zero, not those that are.

Comment Of course one can start wondering what to do with non-local and/or non-noetherian rings, but I will leave that meditation to the reader.

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