By Denes Petz

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**Additional resources for An Invitation to the Algebra of Canonical Commutation Relations**

**Sample text**

Indeed, any left CDG-module over B that can be obtained from total CDG-modules of exact triples of CDG-modules over B using the operations of cone and inﬁnite direct sum is a ﬁltered inductive limit of its CDGsubmodules that can be obtained from the total CDG-modules of exact triples of ﬁnitely generated CDG-modules using the operation of cone. 7 that the objects of Dabs (B–modfg ) are compact in Dco (B–mod). The following result (cf. 8) is due to Dmitry Arinkin and is reproduced here with his kind permission.

8 below). Cf. 4. 2. Remark. 5. Assume that all inﬁnite direct sums exist in DG and the class of exact triples in Z 0 (DG♮ ) is closed under inﬁnite direct sums. Whenever there are enough injectives in the exact category Z 0 (DG♮ ) and countable direct sums of injectives have ﬁnite injective dimensions in Z 0 (DG♮ ), the full triangulated subcategory of the homotopy category H 0 (DG) consisting of all the objects whose images are injective in Z 0 (DG♮ ) is equivalent to the coderived category of DG.

By our assumption, there exists a contracting homotopy c : M ′ /M −→ J for f . Denote the related map of graded B # -modules M ′ −→ J of degree −1 also by c. Then h′ = h′′ + c : M ′ −→ J is a contracting homotopy for ι′ extending h. When the graded ring B # is left Noetherian and has a ﬁnite graded left homological dimension, the homotopy category Hot(B–modfgp ) of left CDG-modules over B that are projective and ﬁnitely generated as graded B-modules is equivalent to the absolute derived category of ﬁnitely generated left CDG-modules Dabs (B–modfg ) over B.