The Derived Module Category
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Linear algebra over a field is rendered tractable by the fact that every module over a field is free; that is, has a spanning set of linearly independent vectors. Over more general rings, when a module M is not free we make a first approximation to its being free by taking a surjective homomorphism φ:F→M where F is free. The kernel Ker(φ) may then be regarded as the ‘first derivative’ module of M. These considerations may be made precise by working in the ‘derived module category’, the quotient of the category of Λ-modules obtained by quotienting out by morphisms which factorize through a projective. The invariants of this first derivative lead to a nonstandard definition of module cohomology.
KeywordsExact Sequence Isomorphism Class Short Exact Sequence Natural Transformation Projective Module
- 44.Humphreys, J.J.A.M.: Algebraic properties of semi-simple lattices and related groups. Ph.D. Thesis, University College London (2006) Google Scholar