A direct sum of rmodules l i2i p i is projective i each p i is projective. However, the converse is not true in general, which justi es giving a name to this important class of modules. It is known that every projective finitely generated left module is free or isomorphic to. We have approached the subject simultaneously from two di. If f is a free r module and p f is a submodule then p need not be free even if pis a direct summand of f. If u is a subspace of a vector space v over a division ring, then u is a direct summand of v. Oct 28, 2011 so far we have only given a trivial example of projective modules, i. This property follows directly from the fact that zorns lemma can be used to extend a basis of u to a basis of v. Moreover, if eis a vector bundle over x speca, e 7. S m by letting is be the function which takes value 1. Vasconcelos1 one of the aims of this paper is to answer the following question.
Projective modules over commutative rings have nice properties. Chasles et m obius study the most general grenoble universities 3. Over rings decomposable into a direct sum there always exist projective modules different from free ones. These are the modules that behave very much like vector spaces. We characterize rings over which every projective module is a direct sum of. Projective modules with finitely many generators are studied in algebraic theory. Vector bundles on projective space university of michigan. A left r module is isomorphic to a direct summand of a free left r module if and only if it is projective.
Since every module is a factor of a free module, we have trivially the following important 3. Coreflexive modules and semidualizing modules with finite projective dimension tang, xi and huang, zhaoyong, taiwanese journal of mathematics, 2017. Free modules, projective, and injective modules springerlink. How to introduce notions of flat, projective and free modules. Let m be a projective module of constant finite rank i.
Bass cancellation theorem for stably free modules 1. Another large class of projective modules were shown to. However, formatting rules can vary widely between applications and fields of interest or study. A nice reference for this material is the book by silverman and tate 6. Example of modules that are projective but not free. Projective geometry and homological algebra david eisenbud. Then using the splitting lemma on the short exact sequence. The following lemma provides this, and shows that the above example is typical. Projective modules over dedekind domains, february 14, 2010 3 2. Projective modules over dedekind domains, february. What is the insight of quillens proof that all projective.
Projective modules over polyhedral semirings sciencedirect. Projective and injective modules play a crucial role in the study of the cohomology of representations. But over a pid, any submodule of a free module is free. Frobenius categories, gorenstein algebras and rational surface singularities osamu iyama, martin kalck, michael wemyss, and dong yang dedicated to ragnarolaf buchweitz on the occasion of his 60th birthday. Free modules, gln and stably free modules if r is a.
A direct sum of r modules l i2i p i is projective i each p i is projective. Projective modules are direct summands of free modules. Pdf torsion free and projective modules hyman bass. One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. Jun 25, 2011 projective modules are direct summands of free modules this is a bookwork post.
Commutative algebratorsionfree, flat, projective and free modules. Clearly t is an additive functor from the category of jcvector bundles over x to the category of cxmodules. In order to prevent bots from posting comments, we would like you to prove that you are human. The coproduct of any family of projective left r modules is projective. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings. The annihilator of in is defined by we say that is faithful if.
In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. Various equivalent characterizations of these modules appear below. Mcgovern, gena puninski, and philipp rothmaler abstract. Pdf the question is addressed of when all pureprojective modules are direct sums. The fact that every r module is a quotient of a projective r module is usually stated as the category of left r modules has enough projectives. Lady april 17, 1998 in the earlier chapter your author has tried to promote the point of view that the category of nite rank torsion free modules under quasihomomorphisms is the appropriate environment for studying torsion free modules. Then it is the direct summand of a free module, say m m0 f. Kaplansky proved in 3, theorem 2 that over a local 1 ring any projective module is free. Then mis projective if and only if m p is a free a p module for each prime p a.
Then finitely generated, projective, right or left a modules are free. The localization of a projective module is a projective module over the localized ring. Suppose first that is projective and let by corollary 1, there exists an module and a free module such that but then and thus is projective by corollary 1 again. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules.
Projective modules and vector bundles the basic objects studied in algebraic ktheory are projective modules over a ring, and vector bundles over schemes. If r is the integer above, then krm is a projective module of rank 1 and by 1. Now if is projective denote the free module over by. This part ii is devoted to a detailed study of all the 3dimensional local. Thus a projective module is locally free in the sense that its localization at every prime ideal is free over the corresponding localization of. Charles weibel, an introduction to homological algebra, section 2. Some of the methods are modeltheoretic, and the techniques developed using these may be of interest in their own right. In this thesis, we study the theory of projective and injective modules. In this case the concept of the rank of a module can be extended to projective modules as follows.
If p is a nonzero element of a projective left r module p then there. These notes arose from a onesemester course in the foundations of projective geometry, given at harvard in the fall term of 19661967. Then, i would show that projective modules are just direct summands of free modules. 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. In particular one gets very easy but not very satisfying examples by looking at disconnected rings. Jsag 5 20, 26 32 the journal of software for algebra and geometry computing free bases for projective modules brett barwick and branden stone abstract. 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. Finally, we sk, for wht r re 11 nonfinitely generated proiective modules uniformly big. So z 2 and z 3 are projective but not free z 6 modules. Even weaker generalisations are flat modules, which still have the property that tensoring with them preserves exact sequences, and torsionfree modules. An amodule f is free if and only if it has a ksubspace s such that any linear map from s to an amodule m can be extended to an amodule morphism from f to m. Under the interpretation of modules as generalized vector bundles a module being locally free corresponds to the corresponding bundle being locally trivial bundle, hence a fiber bundle. It is proved that this is the case over hereditary noetherian rings.
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. An a module f is free if and only if it has a ksubspace s such that any linear map from s to an a module m can be extended to an a module morphism from f to m. Projective modules are direct summands of free modules, so one can choose an injection into a free module and use the basis of this one to prove something for the projective module. The above results for projective modules all have duals for injective modules. Classical results in linear algebra state that every vector space has a basis, and that the rank or dimension of a vector space is independent of the choice of basis. We would like a means to recognise projective modules p without having to consider all possible surjections and morphisms from p. Let x be a free group or monoid, let r be a principal ideal domain, and let a rn, the group ying. Gorensteinprojective and semigorensteinprojective modules. Serre in 1955, to the effect that one did not know if.
For every left rmodule rm there is a projective module rp and an repimorphism pm0. Then is projective if and only if is projective for all. Our rst observation is simply that if r is a graded ring, then r is a graded module over itself. The quillensuslin package for macaulay2 provides the ability to compute a free basis for. The simplest example of a projective module is a free module. Thus z2z and z3z are nonfree modules isomorphic to direct summands of the free module z6z. It involves some of the most basic notions in commutative algebra, and really sounds as though it should be easy the graded case, for example, is easy, but its not. The coincidence of the class of projective modules and that of free modules has been proved for. S m by letting is be the function which takes value 1 at s. Its also nite, so by the previous problem its free, as. Let ank be the weyl algebra, with k a field of characteristic zero. Because m is a projective module it is a direct summand of a free r module.
This follows since a finitely generated flat module is projective if and only if. You can do this by filling in the name of the current tag in the following input field. I theorem i is related to a question raised by eilenberg and ganea 4. Projective modules and vector bundles 3 when r is commutative, every unimodular row of length 2 may be completed. The analysis extends to the case of the semiring of convex, piecewiseaffine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group. As with arbitrary modules, most graded modules are constructed by considering submodules, direct sums, quotients and localizations of other graded modules. Let r be a commutative noetherian ring of krull dimension d. Projective projective modules are direct summands of free modules. We provide an introduction to many of the homological commands in macaulay 2 modules, free resolutions, ext and tor. So far we have only given a trivial example of projective modules, i. Then we define projective modules as those which make this functor exact.
Projective and injective modules arise quite abundantly in nature. A sufficient condition for the rank of a free module over a ring to be uniquely defined is the existence of a homomorphism into a skewfield. For example, all free modules that we know of, are projective modules. Projective modules and prime submodules article pdf available in czechoslovak mathematical journal 562. Free a free r module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring r.
In particular, weibels homological algebra book and hatchers algebraic topology book would contain these proofs perhaps without the messy detail. So z 2 and z 3 are projective but not free z 6modules. I was worrying too much, because of the following theorem bourbaki, commutative algebra, section ii. I would also discuss free resolutions since this is a very elegant machinery. Serres conjecture, for the most part of the second half of the 20th century, ferred to the famous statement made by j. A finitely generated, locally free module over a domain which is not projective.
Pureprojective modules journal of the london mathematical. Since a trivial bundle corresponds to a free module, a locally free module is such that its localization to any maximal ideal is a free module. 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. A over z of an r module m and an abelian group a is. The r module ris projective, because one element namely, 1 can be lifted. Projective and injective model structure on ch r shlomi agmon. I classify projective modules over idempotent semirings that are free on a monoid.