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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.

In this paper, we introduce the concepts of product and directsum, star projective and star injective in $H_v$-modules. Weinvestigate generalizations of some notions in homological algebrato prove the five lemma and star projective and star injectivetheorems in $H_v$-modules. We determine equivalent conditions for split sequences in $H_v$-modules and present some related results.

In this paper we explore the snail lemma in a pointed regular category. In particular, we show that under the presence of cokernels of kernels, the validity of the snail lemma is equivalent to subtractivity of the category. As a corollary, this gives that in the more restrictive context of a normal category the validity of the snail lemma is equivalent to the validity of the snake lemma.

Gauss’ lemma is not only critically important in showing that polynomial rings over unique factorization domains retain unique factorization; it unifies valuation theory. It figures centrally in Krull’s classical construction of valued fields with pre-described value groups, and plays a crucial role in our new short proof of the Ohm-Jaffard-Kaplansky theorem on Bezout domains with given lattice-ordered abelian groups. Furthermore, Eisenstein’s criterion on the irreducibility of polynomials as well as Chao’s beautiful extension of Eisenstein’s criterion over arbitrary domains, in particular over Dedekind domains, are also obvious consequences of Gauss’ lemma. We conclude with a new result which provides a Gauss’ lemma for Hermite rings.

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.

Five lemma micromorphological characters including bicellular microhairs, hooks, long cells, prickles, and silica cells of six Sporobolus taxa representing the S. compositus and S. vaginiflorus complexes in Texas were investigated using scanning electron microscopy (SEM). Our study supplements earlier SEM treatments of Sporobolus in Valdes-Reyna & Hatch and Liu et al. and colleagues. The micromorphological differences found by our study support the taxonomic treatment of these two complexes by Peterson et al.

We present a ‘Regular Slice Lemma’ which, given a k-graph G , returns a regular ( k − 1 ) -complex J with respect to which G has useful regularity properties. We believe that many arguments in extremal hypergraph theory are made considerably simpler by using this lemma rather than existing forms of the Strong Hypergraph Regularity Lemma, and advocate its use for this reason.

We obtain a weak version of the Hopf lemma for the invariant Laplacian on the unit ball of the complex n -space. We also show that our result is sharp in some sense. Motivated by this result, we also consider a class of degenerate elliptic operators with the degeneracy depending on the distance to the boundary of the domain. We study the dependence of the validity of Hopf lemma on the degree of degeneracy of the operator. We show that Hopf lemma holds if the degeneracy is small and fails in general if the degeneracy is large. What is more interesting is the critical case for which we show that certain weak version of Hopf lemma holds.

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.

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller’s axioms defining Quillen’s exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen’s “obscure axiom”. We show that some homological results, such as the Short Five Lemma and the 3×3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,..., x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields., 18 pages. A slightly more general version where the quantifiers for the main result are independent of the entries of the input matrix (only depend on its dimensions). Explanations concerning the condition on the k-determinantal in the main result are included

In this paper, we present a generalization of a combinatorial lemma we stated and proved in a recent work. Then we apply the generalized lemma to prove: (1) a theorem on the existence of a zero for an excess demand mapping, (2) the existence of a continuum of zeros for a parameterized excess demand mapping, (3) Sperner’s lemma on labelings of triangulations. Proofs of these results are constructive: they contain algorithms (based on the combinatorial lemma) for the computation of objects of interest or, at least, of their approximations.

Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if (i) $S$ is a Gr\"obner-Shirshov basis in $C(B,N)$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient conformal algebra $C(B,N|S)$, but not conversely. In this paper, by introducing some new definitions of normal $S$-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras which makes the conditions (i) and (ii) equivalent. We show that for each ideal $I$ of $C(B,N)$, $I$ has a unique reduced Gr\"obner-Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg-Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras., Comment: 49 pages

It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the 3 × 3 lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties.

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 consider the multi-point Schwarz-Pick lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick lemma give Schur's continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk., Comment: 14 pages

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.

A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 x 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with "exact forks" and then, the corresponding 3 x 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 x 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors. F.N.R.S. [1.5.016.10F]; South African National Research Foundation; Georgian National Science Foundation [GNSF/ST09_730_3-105]; CMUC/FCT (Portugal)

We formulate and prove a “five lemma”, which unifies two independent generalizations of the classical five lemma in an abelian category: the five lemma in a (modular) semi-exact category in the sense of M. Grandis, and the five lemma in a pointed regular protomodular category in the sense of D. Bourn.

In this paper we investigate some situations in connection with two exact sequences of fuzzy linear maps. Also we obtain a generalization of the work [Theorem 4] of Pan [5], and we study the analogies of The Four Lemma and The Five Lemma of homological algebra. Finally we obtain a special exact sequence.

We prove a Dold–Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold–Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.

We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.

We derive the modified -adic -measures related to -Nasybullin's type lemma.

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.

We introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT, the prime ideal theorem. The relation of various versions of the Lemma to each other and to PIT is also explored.

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.

A Schröder category extends the category of all binary relations among sets, that is, it realises a relatively huge part of predicate logic. On the other hand Urysohn's lemma asserts that every pair of disjoint closed subsets in a $T_4$ topological space can be separated by a continuous function into the reals. Usually the lemma is demonstrated with calculus of elementary set theory. However the structure of this lemma is very interesting from a view point of lattice theory and relational method. This paper gives a relational proof for Urysohn's lemma within Schröder categories.

We give a unified treatment of the Baer sums in the context of efficiently homological categories which, on the one hand, contains any category of groups with multiple operators and more generally any semi-abelian variety and, on the other hand, the category of Hausdorff groups and more generally any category of semi-abelian Hausdorff algebras. This gives rise to a generalized “Euclide's Postulate” and a five terms exact sequence.

Barisan -eksak merupakan perumuman dari barisan eksak yang diperkenalkan oleh Davvaz dan Parnian-Garamaleky. Dalam tulisan ini akan dikaji perumuman dari Lemma Snake dan Lemma Lima yang memanfaatkan sifat-sifat dari barisan -eksak. Kata Kunci : Barisan -eksak, Lemma Snake, Lemma Lima Abstract. -exact sequences was introduced by Davvaz dan Parnian-Garamaleky as a generalization of exact sequences. In this paper, we give some characterizations and properties of -exact sequences. We use these result to find a generalization of Snake Lemma and Five Lemma. Key words Kunci : -exact sequences, Snake Lemma, Five Lemma

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.

In this note a Schwarz lemma for general Euclidean spaces is established. We show that the two-dimensional version of the lemma is equivalent to the Schwarz lemma in the complex plane.

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)

We prove an extension of the regularity lemma with vertex and edge weights which in principle can be applied for arbitrary graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.

The classical 3 × 3 lemma and snake lemma, valid in any abelian category, still hold in any quasi-pointed (the map 0 → 1 is a mono), regular, and protomodular category. Some applications are given, in this abstract context, concerning the denormalization of kernel maps and the normalization of internals groupoids (i.e., associated crossed modules).

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.

The Snake Lemma, fundamental in producing long exact sequences in homology or cohomology theories based on an abelian category, is shown to have a proof formally identical to the familiar one involving abelian groups when the category either has enough projectives or enough injectives.

In this paper, we examine the following question:do two conditions imply the third one in the '2 out of 3' Lemma (see e.g. Lemma 9.5 in [Alp86, p. 68])?We get a positive answer in two implications for p-nilpotent groups (Section 2) and give some counterexamples in case of soluble groups (see Section 3).