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We construct a right adjoint functor to the Thom functor, i.e., to the functor which assigns the Thom space T ξ T\xi to a vector bundle ξ \xi .

Let A A be a finite-dimensional algebra. The category b m o d A bmod A of finitely generated left A A -modules canonically embeds into the derived category D b ( A ) {D^b}\left ( A \right ) of bounded complexes over b m o d A bmod A and the stable category mod _ Z T ( A ) {\underline {\bmod } ^\mathbb {Z}}T\left ( A \right ) of Z \mathbb {Z} -graded modules over the trivial extension algebra of A A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D b ( A ) {D^b}\left ( A \right ) to mod _ Z T ( A ) \underline {\bmod }^{\mathbb {Z}}T\left ( A \right ) . Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if A A has finite global dimension.

Given a reduced group G, the class of groups A such that A Tor(A, G) is studied. A complete characterization is obtained when G is separable.

Let f : X → Y f:X \to Y be an arbitrary morphism of schemes of finite type over C {\mathbf {C}} , and let f an {f^{{\text {an}}}} be the associated map of analytic spaces. Let S \mathcal {S} be a coherent sheaf on X X . Then ( f ∗ S ) an → f ∗ an ( S an ) {({f_*}\mathcal {S})^{{\text {an}}}} \to f_*^{{\text {an}}}({\mathcal {S}^{{\text {an}}}}) is injective.

Let R denote a strongly prime ring. An explicit construction is given of the radical in R-mod corresponding to the unique maximal proper torsion theory. This radical is characterized in two other ways analogous to known descriptions of the prime radical in rings. If R is a left Ore domain the radical of a module coincides with the torsion submodule. 1. The strongly prime radical. The terminology of radicals in modules is that of Stenstrom [3]. Throughout this paper all rings have a unity and all modules are unital left modules. For a ring R the category of R-modules is denoted by R-mod. A functor a: R-mod -R-mod is called a preradical if a(M) is a submodule of M and a(M)a C a(N) for each morphism M -) N in R-mod. A preradical a is called a radical if a(M/a(M)) = 0 for all M E R-mod. A preradical a is called left exact if a(N) = N n a(M) whenever N C M in R-mod (equivalently, if a is a left exact functor). One method of constructing left exact radicals is given by the following result. PROPOSITION 1. Let OR be any nonempty class of modules closed under isomorphisms. For any module M define a(M) = n{KIK C M, M/K E 9 ). It is assumed that a(M) = M if M/K M 'D for all K C M. Then (1) a[M/a(M)] = Ofor all modules M; (2) if 6O is closed under taking nonzero submodules, a is a radical; (3) if 6O is closed under taking essential extensions, then a(M) n N C a(N) for all submodules N C M. In particular, a is a left exact radical if DX is closed under nonzero submodules and essential extensions. PROOF. The proofs of (1) and (2) are straightforward and so are omitted; the last sentence follows from (2) and (3). To prove (3) let N c M be modules. We must verify that N n a(M) C K whenever N/K E 'DX. By Zorn's lemma, choose W maximal in S = (WIK C W c M, W n N = K). Received by the editors August 4, 1978. AMS (MOS) subject classifications (1970). Primary 16A12; Secondary 16A21.

Let R R and S S be rings with unit. Suppose P P is a free R R -module on β \beta generators, where β \beta is an infinite cardinal number not smaller than the cardinality of R R , and T T is the ring of endomorphisms End ⁡ ( R P ) \operatorname {End}{(_R}P) . Theorem. If R R is left coherent and there exists an exact embedding functor F : R − Mod → S − Mod F:R - \operatorname {Mod} \to S - \operatorname {Mod} , then S F ( R ) R _SF{(R)_R} is a bimodule such that F ( R ) R F{(R)_R} is faithfully flat. Theorem. If F : R − Mod → S − Mod F:R - \operatorname {Mod} \to S - \operatorname {Mod} is an exact embedding functor, then R P T _R{P_T} is a bimodule such that R P _RP is a projective generator (inducing an exact embedding Hom functor from R − Mod R - \operatorname {Mod} into T − Mod T - \operatorname {Mod} ,) and S F ( T ) T _SF{(T)_T} is a bimodule such that F ( T ) T F{(T)_T} is faithfully flat (inducing an exact embedding tensor product functor S F ( T ) ⊗ T _SF(T){ \otimes _T} — from T − Mod T - \operatorname {Mod} into S − Mod S - \operatorname {Mod} .) Theorem. There exists an exact embedding functor R − Mod → S − Mod R - \operatorname {Mod} \to S - \operatorname {Mod} iff there exists an S S -module N N and a unit-preserving ring monomorphism h : End ⁡ ( R P ) → End ⁡ ( S N ) h:\operatorname {End}{(_R}P) \to \operatorname {End}{(_S}N) of their endomorphism rings, such that h h preserves and reflects exact pairs of endomorphisms.

If enough subgroups of a group satisfy the Frattini argument in the group, then normality is a transitive relation within the group. Subgroup functors are used to specify what enough is. 1. Introduction. This article is an outgrowth of the investigation into functors which satisfy the Frattini argument. By a functor, we mean an association to each group G, a collection f{G) of subgroups of G such that if a: G —> G is a monomorphism, then f{Ga) = {Ua\U G f{G)}. Several types of functors have been explicitly formulated in the literature, e.g. Gaschutz functors in (1) and Sylow functions in (10). Of course, the idea has been implicit when associating to a group, its Sylow subgroups, its system normalizers or its /-injectors, where J is a Fitting class. In attempting to better understand the nature of injectors for Fitting classes, the notion of a Fitting functor evolved in (2). A Fitting functor is a functor / which satisfies the additional property: if N < G, then f(N) = {U n N\U G f{G)}. If / is a Fitting functor and G is a group, then / satisfies the Frattini argument in G provided for each U G f(G) and each K < G, G = K ■ NG{U D K). Theorem 7.2 of (2) and Theorem 3.10 of (3) give interesting characterizations of Fitting functors which satisfy the Frattini argument in each finite solvable group. Noting the general nature of the proofs of these results, it seemed of interest to investigate the groups in which a given functor satisfies the Frattini argument. This is the context in which the groups in which normality is a transitive relation appeared. In §2 we forge this connection and in §3 we investigate functors for which our work in §2 is applicable. All groups considered are finite. Any unusual notation will be explained as it is introduced. We use F{G) for the Fitting subgroup of the group G, and Fi{G) is the subgroup

In Chapter III of H. Cartan and S. Eilenberg, Homological algebra, there is the substantial THEOREM 3.1. Let (1) O A' A A" O be an exact sequence. If T is a covariant half exact functor then the sequence (2) * * * Sn-S T(A") SnT(A') SnT(A) Sn T(A") Sn+'T(A') >... is exact. For T contravariant, A' and A" should be interchanged. The proof, which occupies pp. 40-42 of the book, is routine until the critical step of showing that the sequence S1T(A")-*T(A') -->T(A) is exact. The proof of this assertion begins with the last paragraph of p. 41 and goes through most of p. 42. At Cambridge University in 1957 (on a National Science Foundation fellowship) we observed that this part of the proof contains an error. Fortunately it can be made right, as we communicated to Professor Eilenberg at the time. Since we have been asked for this correction several times it seems proper to put it in print. The precise mistake is the sentence on lines 12-15 of p. 42. In the special case being considered, the diagram on p. 41 reduces to 0 0

In this paper we generalize the notion of exact functor to an arbitrary category and show that every small category has a full embedding into a category of all set-valued functors on some small category. The notion of exact is such that this result generalizes the author's exact embedding of regular categories and, indeed, Mitchell's embedding of abelian categories. An example is given of the type of diagram-chasing argument that can be given with this embedding. Introduction. In [Barr] we proved that every small regular category has an exact embedding into a set-valued functor category (C, 5) where C is some small category. In this paper we show that the same result can be proved for any small category when the definition of exact functor is slightly extended. This new definition will agree with the previous one when the category is regular. The embedding which results can be used to chase diagrams in completely arbitrary categories, much as Mitchell's theorem does for abelian categories. As an illustration of this, we derive a theorem of Grothendieck on the descent of pullbacks. 1. Universal regular epimorphisms. In any category a map is called a regular epimorphism if it is the coequalizer of some parallel pair of maps. One usually defines /: X—>Y to be a universal regular epimorphism if for every Y'-+ Y, the fibred product Y' x r X exists and the projection Y' xr X-* Y' is always a regular epimorphism. It is gradually becoming clearer that the universal regular epimorphisms are the "good" epimorphisms. In fact a good case could be made that these should be termed the quotient mappings. In a regular category, every regular epimorphism is universally so. In [Verdier] it is shown how every category X has a full embedding into a topos E where E is the category of sheaves for the so-called canonical topology on X. A brief description of E and the embedding R follows. Received by the editors May 12, 1972. AMS (A/OS) subject classifications (1970). Primary 18A25, 18B15.

This paper provides a new topological proof that a certain sequence involving the inverse limit and first derived functor is short exact.

We obtain a generalization of adjoint functors and triples by axiomatizing the behavior of Albanese varieties, obtain a few basic properties, and show that abelian varieties form a generalized type of triplable category over complete abstract varieties. 1. Basic properties. Let U:63--(t be a functor. We will say that U has a left semiadjoint V if there are an object function V: Obj a, -,Obj 63 and a functor E: 63--Ab, where Ab is the category of abelian groups, such that EB CAut( UB), and the following conditions are satisfied. For every object A of e, there is a fixed map aA:A -*UVA = FA such that f:A -UB implies the existence of unique maps g: VA -B and cEEB such that f =cU(g)aA. Furthermore, if h:B1->B2 and cEEB, then E(h)(c)oU(h) = U(h)oc. We will often write h* or even U(h)* for E(h). EB will be called the group of translations of B. Note first that V becomes a functor in the usual way. Letf:A--B, g:B-*C be maps of G; then V(f) is the unique map in 63 such that aBf=cUV(f)a*A for some ceEVB. We will denote this c by c(f). Clearly V(1A) = IVA. Also acg = c(g) UV(g)axB and we have

In this note, we call a prime idempotent kernel functor a localizing prime if it has the so-called property (T) of Goldman. We generalize a theorem of Heinicke to characterize localizing prime idempotent kernel functors and present an example of a prime idempotent kernel functor on Mod-R, the category of unitary right R-modules, which is not a localizing prime, even though R is a right artinian ring. We shall in most cases follow the terminology in [l1. All rings are assumed to have a unit element and all modules are unitary right modules. Let a be an idempotent kernel functor on Mod-R, i.e., a left exact subfunctor of the identity functor on Mod-R such that a(M/a(M)) = 0 for all M E Mod-R. We denote the module of quotients of M with respect to ax by M., To each R-module S, there is an associated idempotent kernel functor Ts given by the formula rs(M) = Im E M / f(m) 0 O for all f: M E}, where E is the injective hull of S. In case S = RuI for some two-sided ideal I of R, we write , for rs, and M, for the module of quotients of M with respect to PIAn idempotent kernel functor ax on Mod-R is called a prime if ' =rs) where S is a supporting module for a, i.e., S is a-torsion free and S/S' is a-torsion for each nonzero submodule S' of S. A prime idempotent kernel functor a on Mod-R is called a localizing prime if every R.-module is ar-torsion free as R-module. In other words, an idempotent kernel functor a on Mod-R is a localizing prime if and only if a is a prime with Goldman's property (T) (see [l, Theorem 4.3]). In [l1 Goldman has shown that if A is a commutative ring, then an idempotent kernel functor a on Mod-A is a prime if and only if ar = ,up for some Received by the editors June 28, 1973. AMS (MOS) subject classifications (1970). Primary 16A08.

Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-,.HC the homotopy functor to the homotopy category. Let S: C-..SC denote the functor of shape in the sense of Holsztynski for the projection functor H. Every continuous mapping f between spaces gives rise to a shape morphism S(f) in SC, but not every shape morphism is in the image of S. In this paper it is shown that if X is a continuum with x E X and A is a compact connected abelian topological group, then if F is a shape morphism from X to A, then there is a continuous map f:X-'.A such thatf(x)=O and S(f)=F. It is also shown that if f, g: X-A are continuous withf(x)=g(x)=O and S(f)=S(g), then fandg are homotopic. These results are then used to show that there are shape classes of continua containing no locally connected continua and no arcwise connected continua. Some other applications to shape theory are given also. Ihtroduction. Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-iHC the homotopy functor to the homotopy category. Let S: C-).SC denote the functor of shape in the sense of Holsztyn'ski for the projection functor H [5]. Let X and Y be compact Hausdorff spaces. In [6] it is shown that if X and Y are associated with ANR-systems X and Y, respectively, then there is one to one correspondence between Morsc(X, Y) and the homotopy classes of maps of ANRsystems used in the approach of Mardesic and Segal [7]. Thus, our results in this paper will apply to either approach to shape. In the first part of the paper we show that if X is a continuum with x E X and A is a compact connected abelian topological group, then if Fe Morsc(X, A), then there is a continuous f: X-.A with S(f)=F and withf(x)=0. It is also shown that if X and A are as above andf, g:X-+A are continuous with f(x)=g(x)=O and with S(f)=S(g), then f and g are homotopic. These results are clearly related to the results in [6]. Received by the editors November 14, 1972. AMS (MOS) subject classfications (1970). Primary 55D99; Secondary 22B99.