Homological Algebra Modulo a Regular Sequence with Special Attention to Codimension Two

Homological Algebra

Codimension Two Pl Embeddings of Spheres with Nonstandard Regular Neighborhoods

Abstract. For a given polyhedron K ⊂ M the notation RM (K) denotes a regular neighborhood of K in M . We study the following problem: find all pairs (m,k) such that if K is a compact k-polyhedron and M a PL m-manifold, then RM (fK) ∼= RM (gK), for each two homotopic PL embeddings f, g : K → M . We prove that R Sk+2 (S) 6∼= S × D for each k ≥ 2 and some PL sphere S ⊂ S (even for any PL sphere S ...

Categories 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...

Some homological algebra

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...

On a Special Class of Smooth Codimension Two Subvarieties

We work on an algebraically closed field of characteristic zero. By Lefschetz’s theorem, a smooth codimension two subvariety X ⊂ P, n ≥ 4, which is not a complete intersection, lying on a hypersurface Σ, verifies dim(X ∩ Sing(Σ)) ≥ n− 4. In this paper we deal with a situation in which the singular locus of Σ is as large as can be, but, at the same time, the simplest possible: we assume Σ is an ...