The first chapter of this [paper](http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf) (Gersten’s conjecture and the homology of schemes) defines a "Poincaré duality theory with supports" is a *twisted cohomology* theory satisfying certain properties on page 184.


The second chapter of this paper mentioned some examples. The first one is the étale cohomology (page 185) which is well-known for its extra structure of Galois representation by Tate twist. But the second one is the algebraic de Rham cohomology (page 187). What is the twist structure they mean on it? They referred to Hartshorne's paper, but I have not seen any twist construction in it.



Twist cohomology theory of algebraic de Rham cohomology
Nothing in the axioms says that the Tate twist should be nontrivial: it could be that $H^k(X,n)=H^k (X,m) $ for all $n,m $. This will be the case in both algebraic de Rham and Betti cohomology.


If you want the Tate twist in the de Rham theory to be nontrivial, you must keep track of extra structure, for example the comparison isomorphism with Betti cohomology, which changes under Tate twist.


