P._J._Hilton,_U._Stammbach_auth._A_Course_in_Homological_Algebra.pdf

2015

This article corrects two mistakes in the article \Coarse homol-ogy theories " [5]. AMS Classication 55N35, 55N40; 19K56, 46L85

Mathematics and Statistics, 2021

Polynomial: algebra is essential in commutative algebra since it can serve as a fundamental model for differentiation. For module differentials and Loday's differential commutative graded algebra, simplified homology for polynomial algebra was defined. In this article, the definitions of the simplicial, the cyclic, and the dihedral homology of pure algebra are presented. The definition of the simplicial and the cyclic homology is presented in the Algebra of Polynomials and Laurent's Polynomials. The long exact sequence of both cyclic homology and simplicial homology is presented. The Morita invariance property of cyclic homology was submitted.

2005

Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.

Mathematics and Statistics

Springer eBooks, 1984

Archives of Current Research International

According to Soulie [1], computing the homology of a group is a fundamental question and can be a very difficult task. In his assertion, a complete understanding of all the homology groups of mapping class groups of surfaces and 3-manifolds remains out of reach at present time. It is imperative that we give the universal coefficient theorem the supposed needed attention. In this article, we study some product topologies as well as the kiinneth formula for computing the (co) homology group of product spaces. The paper begins with study on the algebraic background with specific definitions and extends into four theorems considered as the Universal Coefficient Theorem. Though this article does not proof the theorems, yet much is done on some properties of each of these theorems, which is enough for the calculation of (co) homology groups.

homology In this book our attention will be focused on compact, simply connected, dif-ferentiable 4-manifolds. The restriction to the simply connected case certainly rules out many interesting examples :indeed it is well-known that any finitely presented group can occur as the fundamental group of a 4-manifold.Furthermore the techniques we will develop in the body of the book are in reality rather insensitive to the fundamental group and much of our discussion can easily be generalized. The main issues however can be reached more quickly in the simply connected case. We shall see that for many purposes 4-manifolds with trivial fundamental group are of beguiling simplicitly but nevertheless the most basic questions about the differential topology of these manifolds lead us into new uncharted waters where the results described in this book serve at present,as isolated markers. After the fundamental group we have the homology and cohomology groups of a 4-manifold. For a closed oriented 4-manifold Poincare duality gives an isomorphism between homology and cohomology in complementary dimensions i and 4-i. So, when X is simply connected the first and third homology groups vanish and all the homological information is contained in H 2. The universal coefficient theorem for cohomology implies that when H 1 is zero H 2 is a free abelian group.In turn by Poincare duality the homology group is free. There are three concrete ways in which we can realize 2-dimensional ho-mology or cohomology classes on a 4-manifold and it is useful to be able to translate easily between them. The first is complex line bundles complex vector bundles of rank 1. On any space X a line bundle L is determined up to the bundle isomorphism by its Chern class and this sets up a bijection between the isomorphism classes of line bundles. The second realization is by smoothly embedded two-dimensional oriented surfaces in the manifold X. Such a surface carries a fundamental homology class ,given a line bundle L we can choose a general smooth section of the bundle whose zero set is a surface representing the homology class dual to c 1 (L) Third we have the de Rham representation of real cohomology classes by differential forms. Let X be a compact oriented simply connected four-manifold. The Poincare duality isomorphism between homology and cohomology is equivalent to a bi-linear form Q. This is the intersection form of the manifold. It is a unimodular symmetric form.Geometrically two oriented surfaces in X placed in general position will meet in a finite set of points. To each point we associate a sign ±1 according to the matching of the orientations in the isomorphism of the tangent bundles at that point. The intersection number is given by the total number of points counted with the signs. The pairing passes to homology to yield the form Q.Going over to cohomology the form translates into the cup product. Thus the form is an invariant of the oriented homotopy type of X. In terms of de Rham cohomology if w 1 and w 2 are closed 2-forms representing classes dual to the surfaces the intersection number is given by the integral X w 1 ∧ w 2 .

Pacific Journal of Mathematics, 1984

In this paper we introduce the concept of a (d, i)-sequence (d, i E N) in a commutative ring A, noetherian and with identity (cf. Def. 1.1). Let K(z, A) be the Koszul complex on A, with respect to the sequence z = Z],... ,z n : the concept of a (d, z)-sequence is expressed in terms of the structure of HχK(z, A)); in particular, it turns out that z is an (n, z)-sequence iff HχK(z, A)) = 0, and such a condition implies z is a (d, /)-sequence for any d < n. If z u ... ,z Λ is a (d, /)-sequence in h A = A/(z h+] ,... ,z n),d<h < «, then z is seen_to be a (d, /)-sequence in A so, in particular, if HχK(z; d A))-0 in d A 9 then z is a (d, /)-sequence. Moreover, for i = 1, the two conditions are equivalent, so that z is a (d, \)-sequence means precisely that z u ...,z d \s regular in d A. For i> 1, examples show that z is a (d, i)-sequence is a condition strictly weaker than Zj,... ,z Λ is a (d, i)-sequence in h A, and we investigate the relationship between those two properties. In fact, their equivalence allows us to read the depth of a quotient ring A/(z h+ ,,... ,z n) in terms of the Koszul complex K(z; A) and implies, for (d, /)-sequences, properties which are a natural generalization of good properties satisfied by regular sequences, such as the depth-sensitivity of the Koszul complex. A characteristic condition for their equivalence is a kind of weak surjectivity of a natural map acting between syz /+1 (^(z; A)) and syz /+1 (#(z; h A)). From an algebraic form of that weak surjectivity we get some sufficient conditions, in terms of weak regularity of the sequence z h+ι ,...,z n. For instance, if z Λ+1> ... ,z n is a ^/-sequence, or a relative regular sequence, or less, if z h+ ,,... ,z w is a relative regular A-sequence with respect to a convenient set of ideals, then zisa(d, i)-sequence in A implies z u. ..,z h is a(d, i)-sequence in h A. Moreover, if z is a (d, z)-sequence and z d+ι ,...,z n is a regular sequence, then H^Kiz; A)) = 0, while this vanishing implies that it is possible to find x ι9 ...,x n in /=(ZJ,...,Z B) such that Zj,...,^], x l9 ...,x n isa(d, z)-sequence and x d+] >... ,x n is a regular sequence. In the last section we give an interpretation of our results in terms of the behaviour of some systems of linear equations. N. 1. Let A be a noetherian ring (with 1) and z = z,,...,z ll a sequence of elements of A such that (z l9 ...,z n)A ^A. We denote by K(z\ A) the Koszul complex with respect to z, i.e. the differential graded algebra (DGA for short) (cf. [G-L] cap. I for a definition) 0 137 138 CARLA MASSAZA AND ALFIO RAGUSA generated by e o i-1,...,«, with differential d J (e Jχ Λ Ae tj) = ^ 2 (~ l) r+1^ e^ Λ Λ^ Λ Λe z/ Also we write syz ι (K(z\ A)) = ker(d I _ 1) C Λ'^U" for / = 1,... ,n + 1. As in [M-R], for every 1 < / < rf < π, Γ/ π '^ will mean the free ^4-module generated by e Jχ .. Ί-e Λ Λe y , with 1 <j x <-< j\ < n and j) > d, which is a complementary module of Λ'(Ae ι θ ••• θ^e^), briefly ΛU,... έ/ ,in A ι A n 9 so Λ ι A n = ΛUp.^θ Then χ z : Λ ^4"-» Λ ι A λ ... d will be the usual projections, i.e. χ t \ ?, a e, ... I = Λ ^h and, more generally, χf:ΛU^Λ'4.. Λ (A</i) will be like χ z-when we set J = Λ. When z l9 ... ,z n are fixed elements of ^4, we write t A=A/(z t+ι ,...,z n)A for ί = 0,... 9 n (n A = A), and z i E t A means the image of z, by the natural map ρ t \ A-* t A.

Journal of Algebra, 2003

In this paper we apply the Garland-Lepowsky Theorem to compute the homology of the Lie algebra N = sl n (tC [t]). By suitably taking Euler characteristics we derive a theorem of Stembridge which gives a combinatorial decomposition of a virtual character of SL n (C). In his theorem, Stembridge indexes the Schur functions that appear in his decomposition by an integer vector of length n and a permutation in S n . An additional feature of our approach is that we interpret the integer vector and permutation in terms of the affine Weyl group of the Kac-Moody Lie algebra of type A. A second result, given in the Garland-Lepowsky paper (extending an earlier result of Kostant), is an explicit formula for the eigenvalues of the Laplacian in the Koszul complex for computing the homology of N. We give an alternative quite different in nature description of those eigenvalues. The Garland-Lepowsky formula is algebraic, stated in terms of differences of squared norms. Our description is combinatorial, stated in terms of structural features of certain graphs. In order to make this combinatorial interpretation seemingly more natural, we link it directly with the paper of Kostant. We work in this part with the Lie algebra L = T ⊕ N where T is the span of x ⊗ 1 with x taken from the strictly upper triangular matrices in sl n (C).

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022

for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from the category of cochain complexes and chain maps to the category of abelian groups and their homomorphisms). At first glance cohomology appears to be very abstract so it is natural to look for explicit examples. A way to obtain a cochain complex is to apply the operator (functor) Hom Z (-, G) to a chain complex C, where G is any abelian group. Given a fixed abelian group A, for any abelian group B we denote by Hom Z (B, A) the abelian group of all homomorphisms from B to A. Given any two abelian groups B and C, for any homomorphism f : The map Hom Z (f, A) is also denoted by Hom Z (f, id A ) or even Hom Z (f, id). Observe that the effect of Hom Z (f, id) on ϕ is to precompose ϕ with f . If f : B → C and g : C → D are homomorphisms of abelian groups, a simple computation shows that Hom R (g 1.1. EXACT SEQUENCES, CHAIN COMPLEXES, HOMOLOGY, COHOMOLOGY 21 Observe that Hom Z (f, id) and Hom Z (g, id) are composed in the reverse order of the composition of f and g. It is also immediately verified that Hom Z (id A , id) = id Hom Z (A,G) . We say that Hom Z (-, id) is a contravariant functor (from the category of abelian groups and group homomorphisms to itself). Then given a chain complex we can form the cochain complex obtained by applying Hom Z (-, G), and denoted Hom Z (C, G). The coboundary map d p is given by which means that for any f ∈ Hom Z (C p , G), we have Thus, for any (p + 1)-chain c ∈ C p+1 we have We obtain the cohomology groups H p (Hom Z (C, G)) associated with the cochain complex Hom Z (C, G). The cohomology groups H p (Hom Z (C, G)) are also denoted H p (C; G). This process was applied to the simplicial chain complex C * (K) associated with a simplicial complex K by Alexander and Kolmogoroff to obtain the simplicial cochain complex Hom Z (C * (K); G) denoted C * (K; G) and the simplicial cohomology groups H p (K; G) of the simplicial complex K; see Section 5.6. Soon after, this process was applied to the singular chain complex S * (X; Z) of a space X to obtain the singular cochain complex Hom Z (S * (X; Z); G) denoted S * (X; G) and the singular cohomology groups H p (X; G) of the space X; see Section 4.8. Given a continuous map f : X → Y , there is an induced chain map f : S * (Y ; G) → S * (X; G) between the singular cochain complexes S * (Y ; G) and S * (X; G), and thus homomorphisms of cohomology f * : H p (Y ; G) → H p (X; G). Observe the reversal: f is a map from X to Y , but f * maps H p (Y ; G) to H p (X; G). We say that the map X → (H p (X; G)) p≥0 is a contravariant functor from the category of topological spaces and continuous maps to the category of abelian groups and their homomorphisms. So far our homology groups have coefficients in Z, but the process of forming a cochain complex Hom Z (C, G) from a chain complex C allows the use of coefficients in any abelian obtained by applying Hom R (-, G), and denoted Hom R (C, G).