Perhaps the most wellknown results concerning pschreier varieties are the following. Some open problems for modules of projective dimension two and algebras of global dimension two dieter happel let be a nite dimensional algebra over some algebraically closed eld k. Projectiveinjective modules, serre functors and symmetric. The difference between free and projective modules is, in a sense, measured by the algebraic ktheory group k 0 r, see below.
Lam, 9783540233176, available at book depository with free delivery worldwide. In mathematics, serre s modularity conjecture, introduced by jeanpierre serre 1975, 1987, states that an odd, irreducible, twodimensional galois representation over a finite field arises from a modular form. Just as in the above problem, if we show that whenever m i2im i, then mis at if and only if each m i is, then we will see that atness of projective modules is equivalent to atness of a, which is, dare i say, obvious. These informal reminiscences, presented at the ictp 2002 conference on algebraic ktheory, recount the trajectory in the authors early research, from work on the serre conjecture on projective modules over polynomial algebras, via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic ktheory. Video uploaded again, because of youtube player problems. Fix a ring r and a field k you can take r k c if youd like to. In our approach, projective injective modules, that is modules which are both, projective and injective, play an important role. As is the case with all of lams books, serre s problem on projective modules, is very clear and wellwritten, as already pointed out, and quickly gets the reader properly airborne. Assuming the axiom of choice, then by the basis theorem every module over a field is a free module and hence in particular every module over a field is a projective module by prop. A finitely generated, locally free module over a domain which is not projective. The reason this seems simple is that there are many equivalent definitions of projective module, and what you give as the definition is usually a property that is shown to be equivalent.
We also give an example of a nonfinitely generated projective module over an integral domain which contains no unimodular elements. The following lemma provides this, and shows that the above example is typical. M a finite rmodule with finite projective dimension, and x an arbitrary r module. Over rings decomposable into a direct sum there always exist projective modules different from free ones. They control the serre functor in the case of a quasihereditary algebra having a double centraliser with respect to a projective injective module whose endomorphism ring.
In this paper, for fixed integers s,t and an arbitrary r module n, we study the membership of. Introduction to commutative algebra and algebraic geometry. Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field k of positive characteristic p, necessarily a prime number. Finite free resolutions and serre s theorem on projective modules 58 7. On serres problem on projective semimodules over polynomial semirings. The aim of this paper is to describe these homotopy categories in terms of leavitt path algebras and gorenstein projective modules. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings. In particular, weibels homological algebra book and hatchers algebraic topology book would contain these proofs perhaps without the messy detail. Lamserres problem on projective modules tsit yuen lam department of mat. Krull dim r are extended, then finitely generated projective rx 1, x n modules are extended. The third part is an introduction to brauer theory. Projective modules over dedekind domains, february 14, 2010.
Siam journal on control and optimization siam society for. February 20 themainsourceforthistalkwaslamsbook serre sproblemonprojectivemodules. On retracts and retractions of free modules over graded rings. We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps. Serre s a course in arithmetic is a concentrated, modern introduction to basically three areas of number theory, quadratic forms, dirichlets density theorem, and modular forms. The notion of a module over a ring r is a generalization of that of a vector space over a. By module we will mean rmodule and by homomorphism respectively isomorphism we will mean homomorphism respectively isomorphism of rmodules, unless explicitly stated otherwise.
We prove here, among other results, that if r is a commutative noetherian ring and projective rx 1, x n modules of rank. The coincidence of the class of projective modules and that of free modules has been proved for. In the geometric setting it is a statement about the triviality of vector bundles on affine space. In a similar fashion, the homotopy category k ac aproj of acyclic complexes of projective a modules is related to the singularity category of the opposite algebra a op of a. Example of modules that are projective but not free. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n.
Papers in all of the following papers with more than one author, the authors names. Over additively cancellative division semirings, the categories of semimodules constitute pschreier varieties. We will write pim for a projective indecomposable amodule. When serre raised this question, this was the only case in which the answer to the above problem was known.
Torsion functors of generalized local cohomology modules. Serres problem on projective modules konrad voelkel. Serres problem, also called serres conjecture, asserts that the implication free module projective module can be reversed for every module over the. On projective modules over polynomial rings sciencedirect.
Raynaudgruson characterized flat and pure morphisms between affine schemes in terms of projective modules. An algorithm for the quillensuslin theorem for quotients. Serre in 1955, to the effect that one did not know if. We would like a means to recognise projective modules p without having to consider all possible surjections and morphisms from p. Serre s problem on projective modules, is very clear and well written and quickly gets the reader properly airborne.
In particular one gets very easy but not very satisfying examples by looking at disconnected rings. Introduction to commutative algebra and algebraic geometry 1984th edition. Projective modules with finitely many generators are studied in algebraic theory. A finitely generated, locally free module over a domain which.
I have freely used the language of abelian categories projective modules, grothendieck groups. You might also want to take a look at the article when every projective module is a direct sum of finitely generated modules by w. The theorem states that every finitely generated projective module over a polynomial. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules. The adjunction formula for the dualizing sheaf, and. Throughout is a ring with 1 and all modules are left modules. The serre problem for discrete hodge algebras springerlink. Projective modules are direct summands of free modules. Serres problem on projective modules, is very clear and well written and quickly gets the reader properly airborne. Aug 17, 2015 this video is a tutorial about the environment modules system used at byu supercomputing. Vasconcelos1 one of the aims of this paper is to answer the following question. In addition, a splitting property for projective modules recently established by gabber, liu and lorenzini is also discussed.
On projective modules and computation of dimension of a. Personal reminiscences of the birth of algebraic ktheory. Unimodular elements in projective modules and an analogue of a result of mandal. Torrsn,htam,x in serre subcategories of the category of rmodules. In this paper we compute a free basis of a finitely generated projective module over rx1,xn, r a principal ideal domain with additional properties, test the freeness for projective modules. We prove the cancellation theorems for projective modules and also prove the qualitative version of laurent polynomial analogue of horrocks theorem.
Let a be a commutative ring for which projective ideals are finitely generated. The first edition was very well accepted and is now one of the leading introductory texts on the advanced undergraduate or beginning graduate level. February 20 themainsourceforthistalkwaslamsbookserre sproblemonprojectivemodules. Find materials for this course in the pages linked along the left. The problem with derived functors is that first of all, categories usually have enough injective objects. Criterion for splitting of a projective module in terms of. Let mod be the category of nite dimensional left modules. Local cohomology algebraic introduction geometric applications 2nd. Oct 28, 2011 examples of projective modules posted. Very often the same definition of serre subcategory is used in an arbitrary abelian category a a we will say in that case weakly serre subcategory. Serres problem on projective modules konrad voelkel 6. Because of their relatively elementary character they are especially suitable for an introduction to commutative algebra. Example of a projective module which is not a direct sum of f. Finally, we conclude the talk by discussing some consequences of these results for the serre s problem in the context of polynomial semirings and presenting some open problems and directions for further investigations.
Projective modules over local rings before we can consider projective modules over dedekind domains, we will consider the case of projective modules over noetherian local rings. They control the serre functor in the case of a quasihereditary algebra having a double centraliser with respect to a projective injective module whose endomorphism ring is a symmetric algebra. We can identify an ordinary ntorus rn with its algebra, crn, of continuous complexvalued functions under pointwise multiplication. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory citation. Jun 25, 2011 projective modules are direct summands of free modules this is a bookwork post. Iwasawa theory, projective modules, and modular representations. For \if, recall i proved a lemma in the previous solutions which says that hom a m. Later similar results were obtained by various researchers for. Various equivalent characterizations of these modules appear below. L b is a surjection of bmodules, so since m b is projective. Serres conjecture, for the most part of the second half of the 20th century, ferred to the famous statement made by j. Projective and injective modules play a crucial role in the study of the cohomology of representations. An module is projective if and only if there exists an module such that is.
The quillensuslin theorem, also known as serres problem or serres conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial. Given its sophistication, and the fact that working though the book will require serious commitment and proper effort, the payoff is huge. In the rst part of the talk we will consider selforthogonal and exceptional modules. How to introduce notions of flat, projective and free modules. This ideal is countably generated and projective but not a direct sum of finitely generated submodules.
A stronger version of this conjecture specifies the weight and level of the modular form. Reasoning of this sort allows serre to prove geometric results such as those mentioned above as well as homological ones. An important related fact worth mentioning is that it has been known well before the settlement of serre s conjecture since 1958 that finitely generated projective modules over a are stably free, i. Serres problem on projective modules pdf free download. For example, in most treatments i know the the definition of projective module is given as either. The quillensuslin theorem, also known as serre s problem or serre s conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. Besides giving an introduction to commutative algebra the theory of c mutative rings this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. As is the case with all of lams books, serres problem on projective modules, is very clear and wellwritten, as already pointed out, and quickly gets the reader properly airborne. We give a procedure and describe an algorithm to compute the dimension of a module over laurent polynomial ring.
M a finite rmodule with finite projective dimension, and x an arbitrary rmodule. Serre s conjecture, for the most part of the second half of the 20th century, ferred to the famous statement made by j. In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules that is, modules with basis vectors over a ring, by keeping some of the main properties of free modules. Ideals and reality projective modules and number of. Introduction to representation theory mit mathematics. Get your kindle here, or download a free kindle reading app. The hermite ring conjecture in dimension one sciencedirect. Vorst, the serre problem for discrete hodge algebras, math. In this theory, one considers representations of the group algebra a cg of a. Rieffel the noncommutative tori provide probably the most accessible interesting examples of noncommutative differentiable manifolds. The simplest example of a projective module is a free module.
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