Video uploaded again, because of youtube player problems. The first edition was very well accepted and is now one of the leading introductory texts on the advanced undergraduate or beginning graduate level. Serres conjecture, for the most part of the second half of the 20th century, ferred to the famous statement made by j. Projective modules are direct summands of free modules. Serre s problem on projective modules, is very clear and well written and quickly gets the reader properly airborne.
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. Personal reminiscences of the birth of algebraic ktheory. Example of a projective module which is not a direct sum of f. 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. For \if, recall i proved a lemma in the previous solutions which says that hom a m. The serre problem for discrete hodge algebras springerlink. Serres problem, also called serres conjecture, asserts that the implication free module projective module can be reversed for every module over the.
By module we will mean rmodule and by homomorphism respectively isomorphism we will mean homomorphism respectively isomorphism of rmodules, unless explicitly stated otherwise. Torrsn,htam,x in serre subcategories of the category of rmodules. Serres problem on projective modules konrad voelkel 6. His research interests focus on the creation of robust intelligent systems. Let mod be the category of nite dimensional left modules. Aug 17, 2015 this video is a tutorial about the environment modules system used at byu supercomputing. On retracts and retractions of free modules over graded rings. Jun 25, 2011 projective modules are direct summands of free modules this is a bookwork post.
In our approach, projective injective modules, that is modules which are both, projective and injective, play an important role. An algorithm for the quillensuslin theorem for quotients. M a finite rmodule with finite projective dimension, and x an arbitrary rmodule. We can identify an ordinary ntorus rn with its algebra, crn, of continuous complexvalued functions under pointwise multiplication.
The hermite ring conjecture in dimension one sciencedirect. Serres problem on projective modules konrad voelkel. A stronger version of this conjecture specifies the weight and level of the modular form. February 20 themainsourceforthistalkwaslamsbookserre sproblemonprojectivemodules.
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. We would like a means to recognise projective modules p without having to consider all possible surjections and morphisms from p. Projective modules over dedekind domains, february 14, 2010 3 2. The theorem states that every finitely generated projective module over a polynomial.
This ideal is countably generated and projective but not a direct sum of finitely generated submodules. On projective modules over polynomial rings sciencedirect. Vasconcelos1 one of the aims of this paper is to answer the following question. We will write pim for a projective indecomposable amodule. On serres problem on projective semimodules over polynomial semirings. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings.
A finitely generated, locally free module over a domain which is not projective. A finitely generated, locally free module over a domain which. 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. 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. 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. Krull dim r are extended, then finitely generated projective rx 1, x n modules are extended. Projective and injective modules play a crucial role in the study of the cohomology of representations. The third part is an introduction to brauer theory.
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. The following lemma provides this, and shows that the above example is typical. The problem with derived functors is that first of all, categories usually have enough injective objects. M a finite rmodule with finite projective dimension, and x an arbitrary r module. If f is a free rmodule and p f is a submodule then p need not be free even if pis a direct summand of f. Bharath ramsundar is a phd student in computer science at stanford university. Because of their relatively elementary character they are especially suitable for an introduction to commutative algebra. The adjunction formula for the dualizing sheaf, and. When every finitely generated flat module is projective. 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. Iwasawa theory, projective modules, and modular representations.
Find materials for this course in the pages linked along the left. We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps. An module is projective if and only if there exists an module such that is. We give a procedure and describe an algorithm to compute the dimension of a module over laurent polynomial ring.
Introduction to commutative algebra and algebraic geometry 1984th edition. Finite free resolutions and serre s theorem on projective modules 58 7. 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. Get your kindle here, or download a free kindle reading app. Homotopy categories, leavitt path algebras, and gorenstein. 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. Torsion functors of generalized local cohomology modules. 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. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules. The simplest example of a projective module is a free module. Throughout is a ring with 1 and all modules are left modules. Criterion for splitting of a projective module in terms of.
Over additively cancellative division semirings, the categories of semimodules constitute pschreier varieties. The difference between free and projective modules is, in a sense, measured by the algebraic ktheory group k 0 r, see below. Rieffel the noncommutative tori provide probably the most accessible interesting examples of noncommutative differentiable manifolds. Let a be a commutative ring for which projective ideals are finitely generated. For example, in most treatments i know the the definition of projective module is given as either. The coincidence of the class of projective modules and that of free modules has been proved for. Siam journal on control and optimization siam society for. Example of modules that are projective but not free. February 20 themainsourceforthistalkwaslamsbook serre sproblemonprojectivemodules. 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. Fix a ring r and a field k you can take r k c if youd like to.
Introduction to commutative algebra and algebraic geometry. In particular, weibels homological algebra book and hatchers algebraic topology book would contain these proofs perhaps without the messy detail. Serre s conjecture, for the most part of the second half of the 20th century, ferred to the famous statement made by j. I have freely used the language of abelian categories projective modules, grothendieck groups. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory citation. When serre raised this question, this was the only case in which the answer to the above problem was known. We also give an example of a nonfinitely generated projective module over an integral domain which contains no unimodular elements. Reasoning of this sort allows serre to prove geometric results such as those mentioned above as well as homological ones. 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. On projective modules and computation of dimension of a.
Over rings decomposable into a direct sum there always exist projective modules different from free ones. Unimodular elements in projective modules and an analogue of a result of mandal. Given its sophistication, and the fact that working though the book will require serious commitment and proper effort, the payoff is huge. We prove here, among other results, that if r is a commutative noetherian ring and projective rx 1, x n modules of rank. Later similar results were obtained by various researchers for. 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. Raynaudgruson characterized flat and pure morphisms between affine schemes in terms of projective modules.
Local cohomology algebraic introduction geometric applications 2nd. In the geometric setting it is a statement about the triviality of vector bundles on affine space. Projective modules over dedekind domains, february. Lam, 9783540233176, available at book depository with free delivery worldwide. 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. 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. In the rst part of the talk we will consider selforthogonal and exceptional modules. L b is a surjection of bmodules, so since m b is projective. Let a be a commutative ring for which projective ideals are finitely. In this paper, for fixed integers s,t and an arbitrary r module n, we study the membership of. The aim of this paper is to describe these homotopy categories in terms of leavitt path algebras and gorenstein projective modules. We prove the cancellation theorems for projective modules and also prove the qualitative version of laurent polynomial analogue of horrocks theorem. Projective modules over dedekind domains, february 14, 2010.
Citeseerx 1 serres problem and pschreier varieties of. 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. 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. 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. 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.
Serre in 1955, to the effect that one did not know if. In addition, a splitting property for projective modules recently established by gabber, liu and lorenzini is also discussed. Approximation of sheaves on algebraic stacks international. Lamserres problem on projective modules tsit yuen lam department of mat. In this theory, one considers representations of the group algebra a cg of a. Projective modules with finitely many generators are studied in algebraic theory. Various equivalent characterizations of these modules appear below. Vorst, the serre problem for discrete hodge algebras, math. 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, is very clear and well written and quickly gets the reader properly airborne. The notion of a module over a ring r is a generalization of that of a vector space over a. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. 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. Perhaps the most wellknown results concerning pschreier varieties are the following.
Projectiveinjective modules, serre functors and symmetric. In particular one gets very easy but not very satisfying examples by looking at disconnected rings. Introduction to representation theory mit mathematics. Serres problem on projective modules pdf free download. Papers in all of the following papers with more than one author, the authors names. Oct 28, 2011 examples of projective modules posted. How to introduce notions of flat, projective and free modules. Ideals and reality projective modules and number of.