We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices, and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined. An application to the correspondence between complexes of representations of an infinite-dimensional Lie algebra on the complementary central charge levels ($c$ and $26-c$ for the Virasoro) is worked out.
Deep Dive into Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures.
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theo
This monograph grew out of the author's attempts to understand the definitions of semi-infinite (co)homology of associative algebras that had been proposed in the literature and particularly in the works of S. Arkhipov [1,2] (see also [13,43]). Roughly speaking, the semi-infinite cohomology is defined for a Lie or associative algebra-like object which is split in two halves; the semi-infinite cohomology has the features of a homology theory (left derived functor) along one half of the variables and a cohomology theory (right derived functor) along the other half.
In the Lie algebra case, the splitting in two halves only has to be chosen up to a finite-dimensional space; in particular, the homology of a finite-dimensional Lie algebra only differs from its cohomology by a shift of the homological degree and a twist of the module of coefficients. So one can define the semi-infinite homology of a Tate (locally linearly compact) Lie algebra [7] (see also [3]); it depends, to be precise, on the choice of a compact open vector subspace in the Lie algebra, but when the subspace changes it undergoes only a dimensional shift and a determinantal twist. Let us emphasize that what is often called the “semi-infinite cohomology” of Lie algebras should be thought of as their semi-infinite homology, from our point of view. What we call the semi-infinite cohomology of Tate Lie algebras is a different and dual functor, defined in this book (see Appendix D).
In the associative case, people usually considered an algebra A with two subalgebras N and B such that N โ B โ A and there is a grading on A for which N is positively graded and locally finite-dimensional, while B is nonpositively graded. We show that both the grading and the second subalgebra B are redundant; all one needs is an associative algebra R, a subalgebra K in R, and a coalgebra C dual to K. Certain flatness/projectivity and “integrability” conditions have to be imposed on this data. If they are satisfied, the tensor product S = C โ K R has a semialgebra structure and all the machinery described below can be applied.
Furthermore, we propose the following general setting for semi-infinite (co)homology of associative algebraic structures. Let C be a coalgebra over a field k. Then C-C-bicomodules form a tensor category with respect to the operation of cotensor product over C; the categories of left and right C-comodules are module categories over this tensor category. Let S be a ring object in this tensor category; we call such an object a semialgebra over C (due to it being “an algebra in half of the variables and a coalgebra in the other half”). One can consider module objects over S in the module categories of left and right C-comodules; these are called left and right S-semimodules. The categories of left and right semimodules are only abelian if S is an injective right and left C-comodule, respectively; let us suppose that it is. There is a natural operation of semitensor product of a right semimodule and a left semimodule over S; it can be thought of as a mixture of the cotensor product in the direction of C and the tensor product in the direction of S relative to C. This functor is neither left, nor right exact. Its double-sided derived functor SemiTor is suggested as the associative version of semi-infinite homology theory.
Before describing the functor SemiHom (whose derived functor SemiExt provides the associative version of semi-infinite cohomology), let us discuss a little bit of abstract nonsense. Let E be an (associative, but noncommutative) tensor category, M be a left module category over it, N be a right module category, and K be a category such that there is a pairing between the module categories M and N over E taking values in K. This means that there are multiplication functors E ร E โ E, E ร M โ M, N ร E โ N, and N ร M โ K and associativity constraints for ternary multiplications
and N ร E ร M โ K satisfying the appropriate pentagonal diagram equations. Let A be a ring object in E. Then one can consider the category A E A of A-A-bimodules in E, the category A M of left A-modules in M, and the category N A of right A-modules in N. If the categories E, M, N, and K are abelian, there are functors of tensor product over A, making A E A into a tensor category, A M and N A into left and right module categories over A E A , and providing a pairing N A ร A M -โ K. These new tensor structures are associative whenever the original multiplication functors were right exact.
Suppose that we want to iterate this construction, considering a coring object C in A E A , the categories of C-C-bicomodules in A E A and C-comodules in A M and N A , etc. Since the functors of tensor product over A are not left exact in general, the cotensor products over C will be only associative under certain (co)flatness conditions. If one makes the next step and considers a ring object S in the category of C-C-bicomodules in A E A , one discovers that the functors of tensor
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