The equivariant Brauer group of a group

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

In this paper, we give a Grobner-Shirshov basis of quantum group of type ℂ3 by using the Ringel-Hall algebra approach. For this, first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type ℂ3 by using an “inductive” method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we compute the “easier” skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first, then “inductive” others from these “easier” ones and this in turn gives Hall polynomials as a byproduct. Then we prove that the set of these relations is closed under composition. So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type ℂ3. Dually, we get a Grobner-Shirshov basis of the negative part of quantum group of type C3. And finally we give a Grobner-Shirshov basis for the whole quantum group of type ℂ3.

Introduction. In the first paper of this series [l], henceforth referred to as I, we define a cohomology theory for a commutative algebra P with coefficients in an P-module M for which H2(R, M) is the group of singular extensions of P by M. In this paper we generalize to a cohomology 22(5, <b, M) where (S, <p) is a singular extension of P and M is an P-module.The second group is again a group of extensions and when (S, <p) = (R, 1B), 22(5, <p, M) -H(R, M) as defined in I.This generalization leads to a surprising result: a connected sequence in the first variable.2. Definitions.Let P be a commutative ring with identity.All rings considered will be unitary P-algebras and all modules will also be unitary P-modules.Let P be an algebra and M an P-module.In I we defined a singular extension 0-M-*S-**R-0 of P by M and morphism of two such extensions.This extension will also be denoted by (5, <f>) or just 5.If f>: (5', cp')->(S, <b) is a morphism we will also use $>: S'-+S to denote the implied algebra morphism.It will be called a surjection or an extension if the implied morphism is a surjection.In any case, ker $, ker <p, and ker </>' are all P-modules.We let CB or just e denote the category of extensions of P and morphisms of extensions.Then an extension of (5, <b) is an exact sequence 0->M ->(5', <p')->*(5, <&)-() where i> is a surjection and M is its kernel.If M and (S-, <p) are held fixed, the notions of equivalence and Baer composition make sense as in I. Then the equivalence classes can be shown to form a group which is denoted by H2(S, <p, M).We denote by PK5, <6, M) the group of derivations of 5 to M (made into an 5-module by <f>).We define an -long exact sequence over 6 to be an exact sequence 0-+M->MB_2-M"_,-*-> Mi -+ (S', <p') -+ (5, <p) - 0 where 0-ker <->(5', <f>')-+(S, <p) -*0 is an extension and 0->M-M"_s - ->Mi->ker $-0 is an ordinary exact sequence of P-modules.An co-long sequence is one like that which has no left end.Again we define equivalence, Baer composition and morphism of long exact

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.

The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their 1960 paper The Brauer Group of a Commutative Ring, building on earlier work of Azumaya. This group coincides with the “classical” Brauer group (cf. Chapter 4) in the case when R is a field. One of the points of extending the theory to rings is that one can relate Brauer groups of fields to Brauer groups of related rings in exact sequences; one then hopes that this will help compute the classical Brauer group. The Brauer group of a commutative ring is also part of a Galois theory of commutative rings. For more on these matters, the reader may consult Galois Theory and Cohomology of Commutative Rings by Chase, Harrison and Rosenberg, The Brauer Group of Commutative Rings by Orzech and Small, Separable Algebras Over Commutative Rings by DeMeyer and Ingraham, or the paper of Auslander and Goldman quoted above.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

The Grothendieck ring of a finite group

Let p be a prime, R a commutative ring. There are two familiar group scheme sequences yielding computations of the flat (or etale) cohomology group Hi(R, Z/pZ): the Kummer exact sequence I~7Z/pZ~G,,~G,,~I, applicable where p is invertible and a primitive p-th root of unity ( is in R, and the ArtinSchreier sequence 1 ~71/pZ~ G , ~ Ga ~ 1 that is valid when p is zero in R (see e.g. [6, pp. 126-127]). In this paper I shall show that these are both special cases of a single sequence of the same kind defined over Z [(]. The two groups involved (besides Z/pZ) have H 1-cohomology computable by familiar K-theory, and hence we have a way of computing H 1 (R, 7Z/pZ) for all Z [(]-algebras R. In particular, the computation works for all rings when p = 2, and we recover the description of etale quadratic algebras derived previously by ad hoc methods (see e.g. [7]). The group schemes occurring in the exact sequence are smoothed versions of congruence subgroups, objects which (I believe) were first described in general in my joint work with Boris Weisfeiler classifying smooth connected group schemes of dimension one [8].

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let A A be a finitely generated algebra over a commutative ring K K with a (possibly infinite) Gröbner base G G on a free algebra F F , that is, A A is the quotient F / I ( G ) F/I(G) with the ideal I ( G ) I(G) of F F generated by G G . Given a Gröbner base H H for an A A -subbimodule L L of the free A A -bimodule A ⋅ X ⋅ A = A K ⊗ K ⋅ X ⊗ K A A \cdot X \cdot A = A_K \otimes K \cdot X \otimes _KA generated by a set X X , we have a morphism ∂ \partial of A A -bimodules from the free A A -bimodule A ⋅ H ⋅ A A \cdot H \cdot A generated by H H to A ⋅ X ⋅ A A \cdot X \cdot A sending the generator [ h ] [h] to the element h ∈ H h \in H . We construct a Gröbner base C C on F ⋅ H ⋅ F F \cdot H \cdot F for the A A -subbimodule Ker( ∂ \partial ) of A ⋅ H ⋅ A A \cdot H \cdot A , and with this C C we have the free A A -bimodule A ⋅ C ⋅ A A \cdot C \cdot A generated by C C and an exact sequence A ⋅ C ⋅ A → A ⋅ H ⋅ A → A ⋅ X ⋅ A A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A . Applying this construction inductively to the A A -bimodule A A itself, we have a free A A -bimodule resolution of A A .

Homology of jet groups

Let A be a unital C*-algebra, n ∈ N ∪ {∞}. It is proved that the isomorphism is isometric for some suitable distances. As an application, the author has the split exact sequence with iA contractive (and isometric if n = ∞) under certain condition of A.

Let R be a commutative Noetherian ring and C a semidualizing module. Based on a kind of Tate \(\mathcal {F}_C\)-resolutions of modules constructed by Hu et al., in this paper, we introduce and investigate Tate homology for R-modules. For modules admitting a Tate \(\mathcal {F}_C\)-resolution, we first connect the Tate homology and corresponding relative homologies via a long exact sequence. Then we show that the vanishing of such Tate homology characterizes the finiteness of \(\mathcal {F}_C\)-projective dimension of modules. Finally, if in addition R is a Cohen–Macaulay ring with a dualizing module, we establish a balance result for such Tate homology.

Do infinite nilpotent groups always have equipotent Abelian subgroups?

We prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group stills valid in the realm of symmetric monoidal categories. To this end, we built some group exact sequences from an adjunction in a bicategory, which are of independent interest. As a particular byproduct of the evolving theory, we prove a version of Hilbert's theorem 90 for cocommutatvie coalgebra coextensions (=surjective homomorphisms).

Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{R} provides its integrals of motion. The construction can be applied to any infinite dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N), the Yangian based on gl(N), we recover by this construction the nonlinear Schrodinger equation and its Y(N) symmetry.