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This chapter contains some known results and some results deemed to be new, but no proofs. Its goal is to prepare the reader for the more technically involved constructions of the main body of the monograph (where the proofs are given). In particular, we do not have to worry about nonassociativity of the cotensor product and partial definition of the semitensor product here, distinguish between the myriad of notions of absolute/relative coflatness/coprojectivity/injectivity of comodules and analogously for contramodules, etc., because we only consider coalgebras over fields. [ABSTRACT FROM AUTHOR]

Throughout Chapters 1-11, k is a commutative ring. All our rings, bimodules, abelian groups, … will be k-modules; all additive categories will be k-linear. [ABSTRACT FROM AUTHOR]

A complex C· over an exact category [28] A is called exact if it is composed of exact triples Zi → Ci → Zi+1 in A. A complex over A is called acyclic if it is homotopy equivalent to an exact complex (or equivalently, if it is a direct summand of an exact complex). Acyclic complexes form a thick subcategory Acycl(A) of the homotopy category Hot(A) of complexes over A. All acyclic complexes over A are exact if and only if A contains images of idempotent endomorphisms [69]. [ABSTRACT FROM AUTHOR]

Throughout Chapters 3-11, k⌄ is an injective cogenerator of the category of k-modules. One can always take k⌄ = HomZ(k,Q/Z). [ABSTRACT FROM AUTHOR]

By a closed model category we mean a model category in the sense of Hovey [49]. The closed model categories that we will construct are also abelian model categories in the sense of [51], so our results can be viewed as particular cases of the general framework developed in [50]. [ABSTRACT FROM AUTHOR]

Let A and B be associative k-algebras. [ABSTRACT FROM AUTHOR]

Let S be a semialgebra over a coring ϱ. [ABSTRACT FROM AUTHOR]

Let ϱ be a coring over a k-algebra A and D be a coring over a k-algebra B. [ABSTRACT FROM AUTHOR]

Let ϱ → D be a map of corings compatible with a k-algebra map A → B. Let S be a semialgebra over the coring ϱ and T be a semialgebra over the coring D. [ABSTRACT FROM AUTHOR]

Let A be an exact category in which all infinite products exist and the functors of infinite product are exact. A complex C· over A is called Italic if it belongs to the minimal triangulated subcategory Acyclctr(A) of the homotopy category Hot(A) containing all the total complexes of exact triples 'K· → K· → "K· of complexes over A and closed under infinite products. Any contraacyclic complex is acyclic. It follows from the next lemma that any acyclic complex bounded from above is contraacyclic. [ABSTRACT FROM AUTHOR]

Let ϱ be a coring over a k-algebra A. [ABSTRACT FROM AUTHOR]

All the constructions of Chapters 1-10 can be carried out with the category of k-modules replaced by the category of graded k-modules. [ABSTRACT FROM AUTHOR]