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Category Archives: Notes
Modular Representation Theory (IV)
Continuing our discussion of modular representation theory, we will now discuss block theory. Previously, we saw that in any ring R, there is at most one way to write where is a set of orthogonal and centrally primitive idempotents. If such an … Continue reading
Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an R-module M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
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Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
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Modular Representation Theory (III)
Let’s work out some explicit examples of modular characters. First, we have a summary of the main results. Let be the modular characters of the simple k[G]-modules; they form a basis of Let be those of the projective indecomposable k[G]-modules; they form a basis … Continue reading
Modular Representation Theory (II)
We continue our discussion of modular representations; recall that all modules are finitely-generated even if we do not explicitly say so. First, we introduce a new notation: for each projective finitely-generated k[G]-module P, we have a unique projective finitely-generated R[G]-module denoted for which … Continue reading
Modular Representation Theory (I)
Let K be a field and G a finite group. We know that when char(K) does not divide |G|, the group algebra K[G] is semisimple. Conversely we have: Proposition. If char(K) divides |G|, then K[G] is not semisimple. Proof Let , a two-sided … Continue reading
Projective Modules and the Grothendieck Group
This is a continuation of the previous article. Throughout this article, R is an artinian ring (and hence noetherian) and all modules are finitely-generated. Let K(R) be the Grothendieck group of all finitely-generated R-modules; K(R) is the free abelian group generated by [M] for simple … Continue reading
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Tagged artinian, composition series, grothendieck group, projective modules
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Projective Modules and Artinian Rings
Projective Modules Recall that Hom(M, -) is left-exact: for any module M and exact , we get an exact sequence Definition. A module M is projective if Hom(M, -) is exact, i.e. if for any surjective N→N”, the resulting HomR(M, N) → HomR(M, N”) is … Continue reading
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Tagged artinian, free modules, left-exact, projective modules, semisimple rings, splitting lemma
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Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
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Tagged bimodules, hom functor, left-exact, modules, right-exact, tensor products
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Tensor Product and Linear Algebra
Tensor products can be rather intimidating for first-timers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finite-dimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading
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Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
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Hom Functor
Fret not if you’re unfamiliar with the term functor; it’s a concept in category theory we will use implicitly without delving into the specific definition. This topic is, unfortunately, a little on the dry side but it’s a necessary evil to get … Continue reading
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Tagged bimodules, hom functor, left modules, left-exact, modules, right modules
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Mathematics and Such 