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Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in $\mathbb{R}$ with values in the category $Sets$ of sets be constant. In a first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In a second part, we extend Kashiwara's result on constant functors by replacing the category $Sets$ with the $\infty$-category of spaces and apply it to generalize the extension lemma to $\infty$-sheaves, the $\infty$-categorical version of sheaves. Finally, we define the micro-support of sheaves with values in a stable $(\infty,1)$-category.

Tucker and Ky Fan’s lemma are combinatorial analogs of the Borsuk–Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan’s lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT–manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT–manifolds.

Structural results on the Direct Factor Problem, the Classification Problem and the Isomorphism Problem are proved for modular group algebras of certain sorts of abelian groups. Mathematics Subject Classification 2010: 20C07, 16S34, 16U60, 20K10, 20K21.

We prove the classical Phillips Lemma in the setting of measures defined on effect algebras of sets. This leads to several Vitali-Hahn-Saks-type results for these measures.

We study the question of the validity in a quasi-Abelian category of some diagram lemma proved by Kuz’minov and Shvedov for Abelian groups and used by them as a tool for calculating the reduced Lp-cohomology of Riemannian manifolds. Bibliography: 14 titles.

We study the global dimension of the category of objects of an abelian category carrying an action of a free partially commutative monoid. We calculate this dimension in the case that the abelian category has infinite coproducts and enough projectives. Previously the author solved the same problem for abelian categories with exact coproducts.

We study the problem of the validity of the classical five- and nine-lemmas in a P-semiabelian category.

We study the validity of the Snake Lemma (the existence and exactness of the Ker-Cokersequence) in a P-semi-abelian category. We also obtain a generalization of the Snake Lemma in a quasiabelian category.

In order to use the trace formula of Arthur–Selberg in the twisted case, we need to prove the “twisted weighted fundamental lemma”, that is a sophisticated version of the fundamental lemma. Here, we prove that this twisted weighted fundamental lemma follows from two others lemmas, where the torsion has disappeared: the weighted fundamental lemma for Lie algebras and a “non-standard weighted fundamental lemma”, concerning Lie algebras too.

Extending the Szemeredi Regularity Lemma for graphs, P. Frank and Rodl [2002] stablished a 3-graph Regularity Lemma guaranteeing that all large triple systems admit partitions of their edge sets into constantly many classes where most classes consist of regularly distributed edges. Many applications of this lemma require a companion Counting Lemma [Nagle and Rodl, 2003] allowing one to estimate the number of copies of K/sub k//sup 3/ in a "dense and regular" environment created by the 3-graph Regularity Lemma. Combined applications of these lemmas are known as the 3-graph Regularity Method. In this paper, we provide an algorithmic version of the 3-graph Regularity Lemma which, as we show, is compatible with a Counting Lemma. We also discuss some applications. For general k-uniform hypergraphs, Regularity and Counting Lemmas were recently established by Gowers [2005] and by Nagle et al., [2005]. We believe the arguments here provide a basis toward a general algorithmic hypergraph regularity method.

The historical development of Hensel's lemma is briefly discussed ( Section 1 ). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given ( Section 3 ). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots ( Section 4 ) and continuity of factors ( Section 6 ) are demonstrated. Using this and a general Krasner's lemma ( Section 7 ), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible ( Section 8 ). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.

The categories dual to the category of Abelian groups (including the category of compact Abelian groups) are considered. In these categories, structure theorems on injective and projective objects are proved, and some projective coverings are calculated. In the category of compact Abelian groups, the notion of connected hull is introduced; some results on connected hulls are obtained and examples are given. Bibliography: 7 titles.

Szemeredi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rodl proved an analogue of Szemeredi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemeredi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rodl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erdos–Stone–Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work., Simons Foundation (Fellowship), David & Lucile Packard Foundation (Fellowship), Alfred P. Sloan Foundation (Fellowship), NEC Corporation (Fellowship), National Science Foundation (U.S.) (Grant DMS-1069197), Akamai Foundation (Presidential Fellowship), Microsoft Research (PhD Fellowship)

In this paper, we study the homological algebra of the category T c of locally convex topological vector spaces from the point of view of derived categories. We start by showing that T c is a quasi-abelian category in which products and direct sums are exact. This allows us to derive projective and inductive limit functors and to clarify their homological properties. In particular, we obtain strictness and acyclicity criteria. Next, we establish that the category formed by the separated objects of T c is quasi-abelian and has the same derived category as T c. Since complete objects of T c do not form a quasi-abelian category, we are lead to introduce the notion of cohomological completeness and to study the derived completion functor. Our main result in this context is an equivalence between the subcategory of D(T c) formed by cohomologically complete complexes and the derived category of the category of pro-Banach spaces. We show also that, under suitable assumptions, we can reduce the computation of Ext’s in T c to their computation in Ban by means of derived projective limits. We conclude the paper by studying derived duality functors.

We prove a sharpening of the Dedekind-Mertens Lemma relating the contents of two polynomials to the content of their product. We show that for a polynomial g the integer 1 +Fdeg(g) in the Dedekind-Mertens Lemma may be replaced by the number of local generators of the content of g. We also raise a question concerning the converse.

The interpretation of the extension groups as Yoneda classes of exact sequences turns out necessary only for having the possibility of using the existence [2] of a complete embedding, preserving the values of Extq, of an arbitrary abelian category into an abelian category with enough injectives. Since there is an embedding taking the relative extension groups into those absolute, Theorem 4.2 easily translates to the relative case with the help of [7, Lemma 2.4]. The author cherishes the hope that the spectral sequence applied to the functorsA[a] andB[b] of Remark 2.2 will be of service in computing the global dimension of the category of functors and solving other relevant problems.

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives., to appear in Indiana Univ. Math. J., 21 pages

A bijective proof is given of a recurrence for the function counting the number of binary operations which endow a finite set with the structure of an abelian group. The proof depends on a lemma in “labelled homological algebra” and provides a simple route to a “curious result” of Philip Hall.

We prove a ``general shrinking lemma'' that resembles the Schwarz--Pick--Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that {\em all} distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz--Pick--Ahlfors-type lemmas to the comparison theorems of Riemannian geometry.

In Chermak and Delgado's paper "A measuring argument for finite groups", a certain "measuring lemma" was shown to hold. This lemma has been successfully applied in many recent papers. We generalize this lemma by expanding the discussion from groups acting on groups to groups acting on sets. As applications, we obtain the main results of several earlier papers.

A set-theoretic lemma is introduced and various applications are given, including: (1) a result of Erdos and Hajnal on the coloring number of a graph; (2) game characterizations of the coloring number of a graph; (3) K. Alster's result that a point-countable collection of open, compact scattered spaces has a point-finite clopen refinement; (4) normal, locally compact, metacompact spaces which are scattered of finite height are paracompact. 1. Introduction. In this paper, a purely set-theoretic lemma is presented which appears to be the "common part" of K. Alster's result (A) that a point-countable collection of compact scattered spaces has a point-finite clopen (open and closed) refinement, and a result of Erdos and Hajnal (EH) on the coloring number of a graph. The author has found this lemma to be quite useful (see (G2j, for example). In this paper, we will show how the lemma can be used to prove the above results, and we will also give some other applications. One such application is a "game characterization" of the coloring number of a graph, from which the Erdos-Hajnal result easily follows. Other applications are partial answers to the question of whether metalindelkf spaces are preserved by closed maps, and Tall's question of whether normal, locally compact, metacompact spaces are paracompact. 2. The lemma. In this section, we state and prove the main lemma. In the lemma, 2M is the set of all subsets of M, and Aa / A means that, for some ordinal K

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

We prove a version of the five lemma which is useful for the study of boundary value problems for partial differential equations. The results are given in the category B \mathcal {B} of Banach spaces and bounded linear operators, and all conditions are stated modulo an arbitrary ideal of B \mathcal {B} . We also show that the results are valid in more general categories.

ABHYANKAR's Lemma provides a simple method of producing unramified extensions of number fields. By application of the lemma we simultaneously simplify and strengthen some recent results of ISHIDA concerning unramified abelian extensions and class numbers.

Christine W. Ayoub has defined diagrams for abelian groups and has shown that the structure of an abelian group A is determined by certain subgroups Ti of A which arise naturally from a given diagram for A. In this paper it is shown that a given diagram for A is formed from certain elementary diagrams for the Ti.

The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined. In particular, a group of projective dimension 2 2 is exhibited.

In this note the following generalization of Fatou’s lemma is proved: Lemma. Let ( f n ) n − 1 ∞ ({f_n})_{n - 1}^\infty be a sequence of integrable functions on a measure space S S with values in R + d R_ + ^d , the nonnegative orthant of a d d -dimensional Euclidean space, for which ∫ f n → a ∈ R + d \int {{f_n} \to a \in R_ + ^d} . Then there exists an integrable function f f , from S S to R + d R_ + ^d , such that a.e. f ( s ) f(s) is a limit point of ( f n ( s ) ) n − 1 ∞ ({f_n}(s))_{n - 1}^\infty and ∫ f ≦ a \int {f \leqq a} .

In [1] the category of relations in an abelian category was studied. A relation from A to B appeared as an equivalence class of pairs of -morphisms (ϕ, ψ),Moreover, this equivalence class may be written as the composite , where is canonically embedded in and is the image of Γ in the canonical involution on . Every equivalence class has an essentially unique minimal representative (ϕ0,ψ0), characterized by the property thatwhereand if and only if the squareis exact.

This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.

In w 1 we consider the situation: L/K is a finite separable field extension, A is an abelian variety over L, and A, is the abelian variety over K obtained from A by restriction of scalars. We study the arithmetic properties of A, relative to those of A, and in particular show that the conjectures of Birch and Swinnerton-Dyer hold for A if and only if they hold for A, . In w 2 we study certain twisted products of abelian varieties and use our results to show that the conjectures of Birch and Swinnerton-Dyer are true for a large class of twisted constant elliptic curves over function fields. In w we develop a method of handling abelian varieties over a number field K which are of CM-type but which do not have all their complex multiplications defined over K. In particular we compute under quite general conditions the conductors and zeta functions of such abelian varieties and so verify Serre's conjecture [12] on the form of the functional equation. Similar, but less complete, results have been obtained by Deuring [1] for elliptic curves and Shimura [15] for abelian varieties.

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, thenis an isomorphism.

tain a. Theorems 1-3 describe the structure of K/M in detail, and Theorems 4-6, in view of this structure, give existence proofs more precise than that trivially given by Zorn's Lemma. The methods throughout are those of classical Galois theory; Lemma 2 (and possibly Lemma 4) are known, but, lacking suitable references, we give proofs of both. All fields to be considered are subfields of K, and the characteristic of K is q>O. LEMMA 1. If AM is maximal without a, then K is an algebraic extension of M.

In this article we study locally compact abelian (LCA) groups from the viewpoint of derived categories, using that their category is quasi-abelian in the sense of J.-P. Schneiders. We define a well-behaved derived Hom-complex with values in the derived category of Hausdorff topological abelian groups. Furthermore we introduce a smallness condition for LCA groups and show that such groups have a natural tensor product and internal Hom which both admit derived versions., 18 pages, AMSLaTeX

We prove: If H is a Fuchs 5 group, then for all groups G containing H it follows that H = H 1 ⊕ H 2 , G = H 1 ⊕ G 2 H = {H_1} \oplus {H_2},G = {H_1} \oplus {G_2} such that H 2 ⊂ G 2 {H_2} \subset {G_2} and | G 2 | ⩽ | G / H | ⋅ ℵ 0 |{G_2}| \leqslant |G/H| \cdot {\aleph _0} . There are a variety of applications.

M. N. Dyer has pointed out that the proof of the key Lemma on "hopfian" rings in w of [1] is incorrect. As I have been unable to find a correct argument, the results on pages 465-469 are moot. (Corollaries 2 and 3 on page 470 are true as it is easy to see that the lemma holds for any commutative ring, while the results in w use only Kaplansky's original theorem, and not the lemma.) I hope that some ring-theorist may be able to prove the lemma. In the first line of page 469, the map from /-/~(C*) to Homr (H2(C,I, F) given by the universal coefficient spectral sequence is, a priori, only a monomorphism. However the theorem is still true (without any essential change in the argument), modulo the lemma. A. Suciu has pointed out that the map �9 in line 1 of page 471 should be replaced by its square 4 2, to ensure that the mapping torus M be orientable. I am grateful to Dyer and Suciu for their observations.