Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with effective motives. My questions are:

  1. Does there exist any 'philosophical' concept of effectivity related with these examples?

  2. Do there exist any other parallel notions of effectivity (for Galois representations? for Hodge modules? for mixed motivic sheaves?)?


To Q1.

Given a effective Hodge structure $V = \oplus_{p+q=k} V^{p,q} $ of weight $k$ one can define the $m$-th Tate-Twist $V(m)^{p,q} = V^{p+m,q+m}$. This a Hodge structure of weight $k-2m$. Note that in general $m$ can be any integer. But restricting to natural numbers, there will be a $m$ which produces a twist that makes $V(m)$ vanish , since effective means that for $p,q$ $< 0$ every $V^{p,q}$ vanishes and we assumed $V$ to be effective.

That means that Tate twisting by natural numbers induces some kind of torsion the "on" Hodge structures. At least this is how i understand it.

In the category of effective Chow-Motive there is also Tate-Twisting by the Tate-Motive. Some authors have the Lefschetz-Motive as one contained in the decomposition of $\mathbb{P}^1$ and define the Tate-Motive as its formal inverse to generate the (general) Chow-Motives,while others do it vice versa like Manin in his 1968 paper "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS".

In the Chow-Motives,which are triples $(X,\rho,n)$,while $\rho$ is a projector on $X$ and $n$ its degree, there is the following rule:

$(X,\rho,n) \otimes (y,\pi,m) = (X\bigsqcup Y,\rho \times \pi,n+m)$.

The Chow-Motives contain the effective Chow-Motives as subcategory. Lets agree that the Tate-Motive is the one having negative degree. Then Twisting by the Tate-Motive is lowering the degree of every Motive in the Chow-Motives. A motive that is embedded into this category by being also a effective Motive, would in this case become none effective.

In both cases the twisting somehow extends the set of objects that are looked on to a less special case,that might enable new results.


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