Some exact sequences associated with adjunctions in bicategories. Applications

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

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.

We consider the Brauer group BM'(k, G) of a group G (finite or infinite) over a commutative ring k with identity. A split exact sequence 1 → Br'(k) → BM'(k, G) → Gal(k, G) → 1 is obtained. This generalizes the Frohlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of G to the infinite case of G. Here Br'(k) is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.

The purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

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].

Classical algebraic K-theory: The functors K0, K1, K2

A fibration, in the Zariski topology, of algebraic varieties leads to an exact sequence of cohomology of the relative units functor. Under suitable hypotheses, the exact sequence can be interpreted as a sequence including the Picard groups of the base variety, the total space, and the fibre. Let X Y be a fibration with fibre F of algebraic varieties (over an algebraically closed field) which is locally trivial in the Zariski topology on Y. If Y and F are smooth and F is rational, Fossum and Iverson [2, Proposition 5, p. 81 constructed an exact sequence relating Pic(Y), Pic(X), and Pic(F). The purpose of the present note is to interpret that sequence in terms of the cohomology, in the Zariski topology, of the relative units sheaf. Besides explaining the Fossum-Iverson sequence, this interpretation helps isolate the hypotheses for the existence of the sequence, which leads to some mild generalizations of the original result. We fix throughout the following conventions: k denotes an algebraically closed field and all varieties considered are over k and irreducible. Unless otherwise specified, all sheaves are for the Zariski topology, and all sheaf cohomology is in that topology. If V is a variety, U an open subset of V and Y a sheaf on V, then (ff)IU denotes Yf restricted to U. If A is an abelian group then (A)v denotes the constant sheaf on V associated to A. We recall that the higher cohomology groups of constant sheaves on V are trivial, and, hence by descent, that all locally constant sheaves on V are constant. Gm denotes the units sheaf on V. Definition 1. Let V be a variety. Uk V denotes the presheaf cokernel of ((k*)v G v) We refer to U k, V as the sheaf of relative units on V. Lemma 2. Let V be a variety. (i) Uk V is a sheaf on V. (ii) H(V, Uk V) = H(V, Gm ) for all i > 1. (iii) Pic(V) = H1(V, UkV)Received by the editors October 25, 1974. AMS (MOS) subject classifications (1970). Primary 14D99, 14F10, 14F20.

Descent theory and Amitsur cohomology of triples

Let R be a commutative ring. We show that a finitely generated R-module M is reflexive (equivalently, linearly compact) if and only if R/annR(M) is reflexive (equivalently, linearly compact). If R is a quasi-noetherian ring, we prove that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. These properly generalize results of Belshoff, Enochs and Garcia Rozas. A ring extension S of R is considered and we show that under some minor condition an S-module is reflexive if and only if it is reflexive as an R-module. Throughout this paper, rings are commutative with identity and modules are unitary. The terminology and notations not defined here can be found in Anderson and Fuller [1]. Let R be a ring and be the minimal injective cogenerator of the category of R-modules, where is an irredundant set of representatives of the simple R-modules. If M is an R-module we let . Following Belshoff, Enochs and Garcia Rozas [2], we call M a (Matlis) reflexive R-module if the evaluation map is an isomorphism. Note that eM is always injective, since U is a cogenerator of R-modules. We recall that an R-module M is linearly compact if every finitely solvable family (where each ) is solvable. For the convenience of the reader we record some basic properties of linearly compact modules and reflexive modules as follows (see [[6], Sec. 3] and [2]): (1) Every artinian module is linearly compact; (2) an infinite direct sum of non-zero modules is not linearly compact (reflexive); (3) If N is a submodule of a module M, then M is linearly compact (reflexive) if and only if both N and M/N are linearly compact (reflexive); consequently, a finite direct sum of modules is linearly compact (reflexive) if and only if each summand is linearly compact (reflexive); and (4) If M is an R-module and IM=0 for an ideal I of R, then M is linearly compact (reflexive) as an R-module if and only if it is linearly compact (reflexive) as an (R/I)-module. This paper consists of two sections. In Sec. 1, we show that the following are equivalent for a finitely generated R-module M: (1) M is reflexive; (2) M is linearly compact; (3) is linearly compact; (4) is reflexive. If R is a quasi-noetherian ring, i.e., the ideal annR(M) is finitely generated for each finitely cogenerated reflexive R-module M, we show that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. In Sec. 2, we consider a ring extension S of R. We prove that is the minimal injective cogenerator of S-modules and an S-module M is reflexive as an S-module if and only if M is reflexive as an R-module, provided that one of the following three conditions is satisfied: (1) S is finitely generated as an R-module, (2) S is linearly compact as an R-module, or (3) S=RQ is a split extension with Q nilpotent.

The property of a generating set of a polynomial ideal or of an ideal of a free associative algebra over a commutative ring to be its Gröbner basis is kept by a flat (but not any) extension of the ground ring. The converse is proved in the broader context of standard bases for filtered modules.

For an extension A ⊆ B of commutative rings, we present a sufficient condition for the ring [[A S, ≤ ]] of generalized power series to be weakly normal (resp., strongly t-closed) in [[B S, ≤ ]], where (S, ≤) be a torsion-free cancellative strictly ordered monoid.As a corollary, it can be applied to the ring of power series in infinitely many indeterminates as well as in finite indeterminates.

In [1] the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence.Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three meaningful relative groups, and we analyze the corresponding definition of relative Cheeger-Simons characters in each case.We then extract another definition, corresponding to the one fitting into a long exact sequence in [1].Finally, we show how the explicit formulas for the holonomy, the transgression maps and the integration can be extended to the relative case.

I ° Introduction This talk is to present a few results on cohomology of groups arising from an investigation into the cohomological behaviour of inverse semigroups ([6], [7], [8]). In this context, relative cohomology in the sense of Auslander ([i]) is just a special case of a more general situation, and various long exact sequences, some of which are well-known ([9], [ii]) and have been established by various authors individually, can be obtained by one and the same method. The starting point for our consideration is the concept of a semilattice of groups, that is a functor S from a semilattice E regarded as a category to the category Gv of groups. Thus for e, f ( E , e ~_ f , there is exactly one group homomorphism ~e,f : S(e) ~ S(f) such that ~e,e is the identity on S(e) ,

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the Jones polynomial of knots.