Associative Operations - Parallel Programming in Scala
... Associativity is not preserved by mapping In general, if f(x, y) is commutative and h1 (z), h2 (z) are arbitrary functions, then any function defined by g(x, y) = h2 (f(h1 (x), h1 (y))) is equal to h2 (f(h1 (y), h2 (x))) = g(y, x), so it is commutative, but it often loses associativity even if f was ...
... Associativity is not preserved by mapping In general, if f(x, y) is commutative and h1 (z), h2 (z) are arbitrary functions, then any function defined by g(x, y) = h2 (f(h1 (x), h1 (y))) is equal to h2 (f(h1 (y), h2 (x))) = g(y, x), so it is commutative, but it often loses associativity even if f was ...
A somewhat gentle introduction to differential graded commutative
... DG commutative algebra provides a useful framework for proving theorems about rings and modules, the statements of which have no reference to the DG universe. For instance, a standard theorem says the following: Theorem 1.2 ([20, Corollary 1]) Let (R, m) → (S, n) be a flat local ring homomorphism, t ...
... DG commutative algebra provides a useful framework for proving theorems about rings and modules, the statements of which have no reference to the DG universe. For instance, a standard theorem says the following: Theorem 1.2 ([20, Corollary 1]) Let (R, m) → (S, n) be a flat local ring homomorphism, t ...
Lecture Notes
... However, as it is necessary in the subsequent discussion, we can describe the isomorphisms between the two sides in the following way: As a convenience of notation, let Rg be a copy of the ring R, and assume the g-component of GR is (Spec R)g = Spec Rg . For every such component we have a morphism S ...
... However, as it is necessary in the subsequent discussion, we can describe the isomorphisms between the two sides in the following way: As a convenience of notation, let Rg be a copy of the ring R, and assume the g-component of GR is (Spec R)g = Spec Rg . For every such component we have a morphism S ...
Commutative ring objects in pro-categories and generalized Moore
... The starting point for the proof that Moore towers admit E∞ structures is the following algebraic observation. Proposition 1.1. Suppose that R is a commutative ring, S is an R-module, and e ∈ S is an element such that the evaluation map HomR (S, S) → S is an isomorphism. Then S admits a unique binar ...
... The starting point for the proof that Moore towers admit E∞ structures is the following algebraic observation. Proposition 1.1. Suppose that R is a commutative ring, S is an R-module, and e ∈ S is an element such that the evaluation map HomR (S, S) → S is an isomorphism. Then S admits a unique binar ...
Commutative ring objects in pro-categories and generalized Moore spectra June 30, 2013
... The starting point for the proof that Moore towers admit E∞ structures is the following algebraic observation. Proposition 1.1. Suppose that R is a commutative ring, S is an R-module, and e ∈ S is an element such that the evaluation map HomR (S, S) → S is an isomorphism. Then S admits a unique binar ...
... The starting point for the proof that Moore towers admit E∞ structures is the following algebraic observation. Proposition 1.1. Suppose that R is a commutative ring, S is an R-module, and e ∈ S is an element such that the evaluation map HomR (S, S) → S is an isomorphism. Then S admits a unique binar ...
diagram algebras, hecke algebras and decomposition numbers at
... In general, suppose that we know ξi ξj = ξj ξi for all j > i, 1 i < k. Since ξk = σk−1 ξk−1 σk−1 , if j > k, ξj commutes with σk−1 by (2.6.1), and with ξk−1 by induction, and hence with ξk . Hence by induction the ξi all commute. To see that there are no relations among the ξi , one may use the br ...
... In general, suppose that we know ξi ξj = ξj ξi for all j > i, 1 i < k. Since ξk = σk−1 ξk−1 σk−1 , if j > k, ξj commutes with σk−1 by (2.6.1), and with ξk−1 by induction, and hence with ξk . Hence by induction the ξi all commute. To see that there are no relations among the ξi , one may use the br ...
THE ORBIFOLD CHOW RING OF TORIC DELIGNE
... The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the ...
... The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the ...
(pdf)
... consists of the central (A, A)-bimodules X, those for which ax = xa for all a and x. Example 9. Graded version of BR has γ defined with a sign. Leads to differential graded version. Topological version in brave new algebra. BR and other examples arise from anchored bicategories by neglect of structu ...
... consists of the central (A, A)-bimodules X, those for which ax = xa for all a and x. Example 9. Graded version of BR has γ defined with a sign. Leads to differential graded version. Topological version in brave new algebra. BR and other examples arise from anchored bicategories by neglect of structu ...
PLETHYSTIC ALGEBRA Introduction Consider an example from
... Recall that the action of a k-algebra A on a k-module M can be given in three ways: as a map A ⊗k M → M , as a map M → Modk (A, M ), or as a map A → Modk (M, M ). In fact, we have the same choices when defining the multiplication map on A itself. The Witt vector approach to operations on rings follo ...
... Recall that the action of a k-algebra A on a k-module M can be given in three ways: as a map A ⊗k M → M , as a map M → Modk (A, M ), or as a map A → Modk (M, M ). In fact, we have the same choices when defining the multiplication map on A itself. The Witt vector approach to operations on rings follo ...
Homomorphisms - Columbia Math
... Proposition 2.1. Let G1 and G2 be groups and let f : G1 → G2 be a homomorphism. Then (i) If H1 ≤ G1 , the f (H1 ) ≤ G2 . In other words, the image of a subgroup is a subgroup. (ii) If H2 ≤ G2 , the f −1 (H2 ) ≤ G1 . In other words, the preimage of a subgroup is a subgroup. Proof. (i) We need to chec ...
... Proposition 2.1. Let G1 and G2 be groups and let f : G1 → G2 be a homomorphism. Then (i) If H1 ≤ G1 , the f (H1 ) ≤ G2 . In other words, the image of a subgroup is a subgroup. (ii) If H2 ≤ G2 , the f −1 (H2 ) ≤ G1 . In other words, the preimage of a subgroup is a subgroup. Proof. (i) We need to chec ...
structure of abelian quasi-groups
... A definition of an abelian quasi-group which retains the above three properties has previously been given(1). It is a system closed under multiplication, which satisfies the quotient axiom and the generalized associative-commutative law (ab)(cd) = (ac)(bd). It is the purpose of this paper to give a ...
... A definition of an abelian quasi-group which retains the above three properties has previously been given(1). It is a system closed under multiplication, which satisfies the quotient axiom and the generalized associative-commutative law (ab)(cd) = (ac)(bd). It is the purpose of this paper to give a ...
slides
... The existence of such products for each ordered pair of objects (and also what is called a terminal object) gives rise to a monoidal structure on C which is referred to as a cartesian monoidal category. For instance the category of sets with the cartesian product is a cartesian monoidal category, so ...
... The existence of such products for each ordered pair of objects (and also what is called a terminal object) gives rise to a monoidal structure on C which is referred to as a cartesian monoidal category. For instance the category of sets with the cartesian product is a cartesian monoidal category, so ...
Hopf algebras
... 1. The identity functor 1C : C → C, where 1C (C) = C for all objects C ∈ C and 1C (f ) = f for all morphisms f : C → C 0 in C. 2. The constant functor C → D at X, assigns to every object C ∈ C the same fixed object X ∈ D, and assigns to every morphism f in C the identity morphism on X. Remark that d ...
... 1. The identity functor 1C : C → C, where 1C (C) = C for all objects C ∈ C and 1C (f ) = f for all morphisms f : C → C 0 in C. 2. The constant functor C → D at X, assigns to every object C ∈ C the same fixed object X ∈ D, and assigns to every morphism f in C the identity morphism on X. Remark that d ...
Homological algebra
... Proof. We already saw that the hom functor HomZ (−, X) is left exact for any abelian group X. It is also obviously additive which means that (f + g)] = f ] + g ] for all f, g : N → M . I.e., the duality functor induces a homomorphism (of abelian groups): HomR (N, M ) → HomZ (M ∧ , N ∧ ) Duality also ...
... Proof. We already saw that the hom functor HomZ (−, X) is left exact for any abelian group X. It is also obviously additive which means that (f + g)] = f ] + g ] for all f, g : N → M . I.e., the duality functor induces a homomorphism (of abelian groups): HomR (N, M ) → HomZ (M ∧ , N ∧ ) Duality also ...
Algebra I: Section 6. The structure of groups. 6.1 Direct products of
... identity element e and we get xi = yi , contrary to the definition of i. Uniqueness of factorization is proved. We are now ready to state the main result. We frame it in terms of the unique factorization condition (iii), but by 6.1.11 we may replace (iii) with (iii)′ if we wish. 6.1.12 Theorem (In ...
... identity element e and we get xi = yi , contrary to the definition of i. Uniqueness of factorization is proved. We are now ready to state the main result. We frame it in terms of the unique factorization condition (iii), but by 6.1.11 we may replace (iii) with (iii)′ if we wish. 6.1.12 Theorem (In ...
Iterated Bar Complexes of E-infinity Algebras and Homology
... The first purpose of this paper is to give a direct construction of the iterated bar complex, within the framework of [22], but so that we avoid the iteration process of the definition and the use of cofibrant replacements of E∞ -operads. Roughly, we show that the definition of the iterated bar comp ...
... The first purpose of this paper is to give a direct construction of the iterated bar complex, within the framework of [22], but so that we avoid the iteration process of the definition and the use of cofibrant replacements of E∞ -operads. Roughly, we show that the definition of the iterated bar comp ...
WHEN EVERY FINITELY GENERATED FLAT MODULE IS
... associative algebra over a field. Nevertheless there are domains that are neither right nor left S-rings. See Section 3 for all this. From [6] it follows that we may assign to each sequence A1 , A2 , . . . as above a projective right module P such that this sequence converges if and only if P is fini ...
... associative algebra over a field. Nevertheless there are domains that are neither right nor left S-rings. See Section 3 for all this. From [6] it follows that we may assign to each sequence A1 , A2 , . . . as above a projective right module P such that this sequence converges if and only if P is fini ...
EXTERIOR POWERS 1. Introduction Let R be a commutative ring
... commutes, and β is an isomorphism. Proof. This is the usual argument that an object equipped with a map satisfying a universal mapping property is determined up to a unique isomorphism: set two such objects and maps against each other to get maps between the objects in both directions whose composit ...
... commutes, and β is an isomorphism. Proof. This is the usual argument that an object equipped with a map satisfying a universal mapping property is determined up to a unique isomorphism: set two such objects and maps against each other to get maps between the objects in both directions whose composit ...
Page 1 AN INTRODUCTION TO REAL CLIFFORD ALGEBRAS AND
... Real Clifford algebras are associative, unital algebras that arise from a pairing of a finitedimensional real vector space and an associated nondegenerate quadratic form. Herein, all the necessary mathematical background is provided in order to develop some of the theory of real Clifford algebras. T ...
... Real Clifford algebras are associative, unital algebras that arise from a pairing of a finitedimensional real vector space and an associated nondegenerate quadratic form. Herein, all the necessary mathematical background is provided in order to develop some of the theory of real Clifford algebras. T ...
skew-primitive elements of quantum groups and braided lie algebras
... algebraic structures in such a category is a generalization of the study of group graded algebraic structures. We will describe the braid structure in the category of YetterDrinfel'd modules, the concept of a Hopf algebra in this category, and explain the reason why we want to study such braided Hop ...
... algebraic structures in such a category is a generalization of the study of group graded algebraic structures. We will describe the braid structure in the category of YetterDrinfel'd modules, the concept of a Hopf algebra in this category, and explain the reason why we want to study such braided Hop ...
Representation schemes and rigid maximal Cohen
... To prove Karroum’s theorem we may take δr = rα + 1. Indeed, if M is an MCM A-module with w(M ) > r(M )α + 1 then the pigeonhole principle implies that there must exist i0 such that (M/R+ M )i0 +j = 0 for j = 1, . . . , α. Then Lemma 3.2 implies the existence of an MCM submodule of M with MCM quotien ...
... To prove Karroum’s theorem we may take δr = rα + 1. Indeed, if M is an MCM A-module with w(M ) > r(M )α + 1 then the pigeonhole principle implies that there must exist i0 such that (M/R+ M )i0 +j = 0 for j = 1, . . . , α. Then Lemma 3.2 implies the existence of an MCM submodule of M with MCM quotien ...
Morita equivalence for regular algebras
... is compact closed. This gives us a quick construction of a group in which the Brauer group of R embeds. ...
... is compact closed. This gives us a quick construction of a group in which the Brauer group of R embeds. ...
REVIEW OF MONOIDAL CONSTRUCTIONS 1. Strict monoidal
... The map σi corresponds to the injective order preserving map (n−1)+ → n+ that misses the value i + 1. Note that in Γop there is a morphism σij missing in ∆op that simply permutes two elements i and j in a set. In fact, any morphism in Γop is a composition of permuting morphisms and morphisms in the ...
... The map σi corresponds to the injective order preserving map (n−1)+ → n+ that misses the value i + 1. Note that in Γop there is a morphism σij missing in ∆op that simply permutes two elements i and j in a set. In fact, any morphism in Γop is a composition of permuting morphisms and morphisms in the ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.