Some homological algebra
نویسنده
چکیده
1.1 Adjoint functors Let C and D be categories and let f∗ : C → D and f∗ : D → C be functors. Then f∗ is a right adjoint to f∗ and f∗ is a left adjoint to f∗ if, for each D ∈ D and C ∈ C there is a natural vector space isomorphism HomC(fD,C) Φ −→HomD(D, f∗C). For C ∈ C and D ∈ D define τC = Φ(idf∗C) ∈ HomC(ff∗C,C) and φD = Φ(idf∗D) ∈ HomD(D, f∗fD). In general τC and φD are neither injective nor surjective, see the examples below in ??? and ???.
منابع مشابه
Homological Dimension of Smash Product over Quasitriangular Weak Hopf Algebra
Let (H,R) be a quasitriangular weak Hopf algebra, and A a quantum commutative weak H-module algebra. We establish the relationship of homological dimensions between weak smash product algebra A#H and A under some conditions. As an application, we consider the case of twisted weak Hopf algebra. Mathematics Subject Classification (2010): 16T05
متن کاملHomological Properties of Balanced Cohen-macaulay Algebras
A balanced Cohen-Macaulay algebra is a connected algebra A having a balanced dualizing complex ωA[d] in the sense of Yekutieli (1992) for some integer d and some graded A-A bimodule ωA. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem: Theorem 0.1. Let A be a Noetherian balanced Cohen-Macaulay algebra, and M a nonzero ...
متن کاملSome applications of Gröbner bases in homological algebra
In this paper we make some computations in homological algebra using Gröbner bases for modules over polynomials rings with coefficients in a Noetherian commutative ring. In particular, we show easy procedures for computing the Ext and Tor modules.
متن کاملTowards a higher reasoning level in formalized Homological Algebra
We present a possible solution to some problems to mechanize proofs in Homological Algebra: how to deal with partial functions in a logic of total functions and how to get a level of abstraction that allows the prover to work with morphisms in an equational way.
متن کاملMechanized reasoning in Homological Algebra
We face the problem of obtaining a certified version of a crucial algorithm in the field of Homological Algebra, known as “Perturbation Lemma”. This lemma is intensively used in the software system “Kenzo”, devoted to symbolic computation in Homological Algebra. To this end, we use the proof assistant “Isabelle”. Our main motivations are to increase the knowledge in the algorithmic nature of th...
متن کاملCategories and Homological Algebra
The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...
متن کامل