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In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one. As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.