Length functions on modules.

For a unital ring , a length function is a numerical invariant of the left -modules, with non-negative real values and that may attain infinity, with two fundamental properties: additivity on short exact sequences and continuity on injective direct limits.
In a recent paper, I have shown that any length function can be restricted to a so-called Sylvester module rank function, a kind of numerical invariant for the finitely presented left -modules. Furthermore, I have shown that such a rank function is the restriction of a length function if, and only if, it satisfies a specific property called exactness.
In fact, the above connection between length functions and exact Sylvester rank functions can be seen as a particular case of a new theorem of Hanfeng Li. Indeed, Li has shown that any Sylvester rank function can be extended to a “bivariant” length function on left -modules (i.e., a generalization of the concept of a length function, that takes a pair of a module and one of its submodules as an input, and satisfies a “relative additivity” on short exact sequences). Then, a bivariant length function is “absolutely additive”, if and only if it comes from an exact Sylvester rank function or, equivalently, if it “restricts” to a classical length function.

In this talk, after reviewing the above concepts and results, we will show that the bivariant length functions introduced by Li can be seen as classical length functions on the category of additive pre-sheaves of additive functors from finitely presented modules to Abelian groups. In particular, there is a bijection between Sylvester rank functions on and (normalized) length functions on . As a consequence, when suitably extended to locally coherent Grothendieck categories, the theory of Sylvester rank functions is equivalent to that of length functions (in the same way as the theory of injective objects is “equivalent” to that of pure-injectives in this generality).
If time allows, we will give some motivation and applications of the above theory, with connections to infinite group representations and to the classification of suitable ring epimorphisms.

[Li’21] Li, Hanfeng, Bivariant and extended Sylvester rank functions, Journal of the London Mathematical Society 103 1 (2021) 222—249.

[LL’19] Li, Hanfeng and Liang, Bingbing, Sofic mean length, Advances in Mathematics 353 (2019), 802—858.

[Va’68] Vámos, Peter, Length function on modules, University of Sheffield (1968).

[Vi’19] Virili, Simone, On the relation between length functions and exact Sylvester rank functions, Topological Algebra and its Applications 7 1 (2019), 69—74.