Tag Archives: projective modules

Commutative Algebra 35

Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 34

Nakayama’s Lemma The following is a short statement which has far-reaching applications. Since its main applications are for local rings, we will state the result in this context. Throughout this section, is a fixed local ring. Theorem (Nakayama’s Lemma). Let … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 33

Snake Lemma Let us introduce a useful tool for computing kernels and cokernels in a complicated diagram of modules. Although it is only marginally useful for now, it will become a major tool in homological algebra. Snake Lemma. Suppose we … Continue reading

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 31

Flat Modules Recall from proposition 3 here: for an A-module M, is a right-exact functor. Definition. We say M is flat over A (or A-flat) if is an exact functor, equivalently, if A flat A-algebra is an A-algebra which is flat as … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 27

Free Modules All modules are over a fixed ring A. We already mentioned finite free modules earlier. Here we will consider general free modules. Definition. Let be any set. The free A-module on I is a direct sum of copies of … Continue reading

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

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

Posted in Notes | Tagged , , , , , , | Leave a comment

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

Posted in Notes | Tagged , , , , | Leave a comment

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

Posted in Notes | Tagged , , , , , | Leave a comment

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

Posted in Notes | Tagged , , , | Leave a comment

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

Posted in Notes | Tagged , , , , , | 2 Comments