Projective module
In
Every free module is a projective module, but the
Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
Definitions
Lifting property
The usual
The advantage of this definition of "projective" is that it can be carried out in
Split-exact sequences
A module P is projective if and only if every
is a
on the summand h(P). Equivalently,Direct summands of free modules
A module P is projective if and only if there is another module Q such that the direct sum of P and Q is a free module.
Exactness
An R-module P is projective if and only if the covariant
Dual basis
A module P is projective if and only if there exists a set and a set such that for every x in P, fi (x) is only nonzero for finitely many i, and .
Elementary examples and properties
The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:
- Direct sums and direct summands of projective modules are projective.
- If e = e2 is an idempotent in the ring R, then Re is a projective left module over R.
Let be the direct product of two rings and which is a ring for the componentwise operations. Let and Then and are idempotents, and belong to the
Relation to other module-theoretic properties
The relation of projective modules to free and flat modules is subsumed in the following diagram of module properties:
The left-to-right implications are true over any ring, although some authors define torsion-free modules only over a domain. The right-to-left implications are true over the rings labeling them. There may be other rings over which they are true. For example, the implication labeled "local ring or PID" is also true for (multivariate) polynomial rings over a field: this is the Quillen–Suslin theorem.
Projective vs. free modules
Any free module is projective. The converse is true in the following cases:
- if R is a field or skew field: any module is free in this case.
- if the ring R is a submoduleof a free module over a principal ideal domain is free.
- if the ring R is a local ring. This fact is the basis of the intuition of "locally free = projective". This fact is easy to prove for finitely generated projective modules. In general, it is due to Kaplansky (1958); see Kaplansky's theorem on projective modules.
In general though, projective modules need not be free:
- Over a direct product of rings R × S where R and S are nonzerorings, both R × 0 and 0 × S are non-free projective modules.
- Over a Dedekind domain a non-principal ideal is always a projective module that is not a free module.
- Over a matrix ring Mn(R), the natural module R n is projective but is not free when n > 1.
- Over a semisimple ring, every module is projective, but a nonzero proper left (or right) ideal is not a free module. Thus the only semisimple rings for which all projectives are free are division rings.
The difference between free and projective modules is, in a sense, measured by the algebraic K-theory group K0(R); see below.
Projective vs. flat modules
Every projective module is flat.[1] The converse is in general not true: the abelian group Q is a Z-module that is flat, but not projective.[2]
Conversely, a
Govorov (1965) and Lazard (1969) proved that a module M is flat if and only if it is a direct limit of finitely-generated free modules.
In general, the precise relation between flatness and projectivity was established by Raynaud & Gruson (1971) (see also Drinfeld (2006) and Braunling, Groechenig & Wolfson (2016)) who showed that a module M is projective if and only if it satisfies the following conditions:
- M is flat,
- M is a direct sum of countably generated modules,
- M satisfies a certain Mittag-Leffler-type condition.
This characterization can be used to show that if is a
The category of projective modules
Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called left hereditary.
Quotients of projective modules also need not be projective, for example Z/n is a quotient of Z, but not torsion-free, hence not flat, and therefore not projective.
The category of finitely generated projective modules over a ring is an exact category. (See also algebraic K-theory).
Projective resolutions
Given a module, M, a projective resolution of M is an infinite exact sequence of modules
- ··· → Pn → ··· → P2 → P1 → P0 → M → 0,
with all the Pi s projective. Every module possesses a projective resolution. In fact a free resolution (resolution by free modules) exists. The exact sequence of projective modules may sometimes be abbreviated to P(M) → M → 0 or P• → M → 0. A classic example of a projective resolution is given by the Koszul complex of a regular sequence, which is a free resolution of the ideal generated by the sequence.
The length of a finite resolution is the index n such that Pn is
Projective modules over commutative rings
Projective modules over commutative rings have nice properties.
The localization of a projective module is a projective module over the localized ring. A projective module over a local ring is free. Thus a projective module is locally free (in the sense that its localization at every prime ideal is free over the corresponding localization of the ring).
The converse is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative Noetherian ring is locally free if and only if it is projective.
However, there are examples of finitely generated modules over a non-Noetherian ring that are locally free and not projective. For instance, a Boolean ring has all of its localizations isomorphic to F2, the field of two elements, so any module over a Boolean ring is locally free, but there are some non-projective modules over Boolean rings. One example is R/I where R is a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module R/I is locally free since R is Boolean (and it is finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because I is not a principal ideal. (If a quotient module R/I, for any commutative ring R and ideal I, is a projective R-module then I is principal.)
However, it is true that for
- is flat.
- is projective.
- is free as -module for every maximal ideal of R.
- is free as -module for every prime ideal of R.
- There exist generating the unit idealsuch that is free as -module for each i.
- is a locally free sheaf on (where is the sheaf associated toM.)
Moreover, if R is a Noetherian integral domain, then, by Nakayama's lemma, these conditions are equivalent to
- The dimension of the -vector space is the same for all prime ideals of R, where is the residue field at .[6] That is to say, M has constant rank (as defined below).
Let A be a commutative ring. If B is a (possibly non-commutative) A-
Rank
Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free -module . It is a locally constant function on X. In particular, if X is connected (that is if R has no other idempotents than 0 and 1), then P has constant rank.
Vector bundles and locally free modules
This section needs additional citations for verification. (July 2008) |
A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of
Vector bundles are locally free. If there is some notion of "localization" that can be carried over to modules, such as the usual
Projective modules over a polynomial ring
The
Since every projective module over a principal ideal domain is free, one might ask this question: if R is a commutative ring such that every (finitely generated) projective R-module is free, then is every (finitely generated) projective R[X]-module free? The answer is no. A counterexample occurs with R equal to the local ring of the curve y2 = x3 at the origin. Thus the Quillen–Suslin theorem could never be proved by a simple induction on the number of variables.
See also
- Projective cover
- Schanuel's lemma
- Bass cancellation theorem
- Modular representation theory
Notes
- ^ Hazewinkel; et al. (2004). "Corollary 5.4.5". Algebras, Rings and Modules, Part 1. p. 131.
- ^ Hazewinkel; et al. (2004). "Remark after Corollary 5.4.5". Algebras, Rings and Modules, Part 1. pp. 131–132.
- ^ Cohn 2003, Corollary 4.6.4
- ^ "Section 10.95 (05A4): Descending properties of modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-11-03.
- ^ Exercises 4.11 and 4.12 and Corollary 6.6 of David Eisenbud, Commutative Algebra with a view towards Algebraic Geometry, GTM 150, Springer-Verlag, 1995. Also, Milne 1980
- ^ That is, is the residue field of the local ring .
- ^ Bourbaki, Algèbre commutative 1989, Ch II, §5, Exercise 4
- .
References
- William A. Adkins; Steven H. Weintraub (1992). Algebra: An Approach via Module Theory. Springer. Sec 3.5.
- Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3.
- Nicolas Bourbaki, Commutative algebra, Ch. II, §5
- Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse (2016), "Tate objects in exact categories", S2CID 118374422
- ISBN 1-85233-667-6.
- Drinfeld, Vladimir (2006), "Infinite-dimensional vector bundles in algebraic geometry: an introduction", in Pavel Etingof; Vladimir Retakh; I. M. Singer (eds.), The Unity of Mathematics, Birkhäuser Boston, pp. 263–304, MR 2181808
- Govorov, V. E. (1965), "On flat modules (Russian)", Siberian Math. J., 6: 300–304
- ISBN 978-1-4020-2690-4.
- Kaplansky, Irving (1958), "Projective modules", MR 0100017
- ISBN 0-201-55540-9.
- Lazard, D. (1969), "Autour de la platitude",
- Milne, James (1980). Étale cohomology. Princeton Univ. Press. ISBN 0-691-08238-3.
- Donald S. Passman (2004) A Course in Ring Theory, especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, ISBN 0-8218-3680-3.
- Raynaud, Michel; Gruson, Laurent (1971), "Critères de platitude et de projectivité. Techniques de "platification" d'un module", S2CID 117528099
- Interscience Publishers.
- Charles Weibel, The K-book: An introduction to algebraic K-theory