Projective cover
In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.
Definition
[edit]Let be a category and X an object in . A projective cover is a pair (P,p), with P a projective object in and p a superfluous epimorphism in Hom(P, X).
If R is a ring, then in the category of R-modules, a superfluous epimorphism is then an epimorphism such that the kernel of p is a superfluous submodule of P.
Properties
[edit]Projective covers and their superfluous epimorphisms, when they exist, are unique up to isomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledged universal property.
The main effect of p having a superfluous kernel is the following: if N is any proper submodule of P, then .[1] Informally speaking, this shows the superfluous kernel causes P to cover M optimally, that is, no submodule of P would suffice. This does not depend upon the projectivity of P: it is true of all superfluous epimorphisms.
If (P,p) is a projective cover of M, and P' is another projective module with an epimorphism , then there is a split epimorphism α from P' to P such that
Unlike injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have projective covers. A ring R is called left (right) perfect if every left (right) R-module has a projective cover in R-Mod (Mod-R).
A ring is called semiperfect if every finitely generated left (right) R-module has a projective cover in R-Mod (Mod-R). "Semiperfect" is a left-right symmetric property.
A ring is called lift/rad if idempotents lift from R/J to R, where J is the Jacobson radical of R. The property of being lift/rad can be characterized in terms of projective covers: R is lift/rad if and only if direct summands of the R module R/J (as a right or left module) have projective covers.[2]
Examples
[edit]In the category of R modules:
- If M is already a projective module, then the identity map from M to M is a superfluous epimorphism (its kernel being zero). Hence, projective modules always have projective covers.
- If J(R)=0, then a module M has a projective cover if and only if M is already projective.
- In the case that a module M is simple, then it is necessarily the top of its projective cover, if it exists.
- The injective envelope for a module always exists, however over certain rings modules may not have projective covers. For example, the natural map from Z onto Z/2Z is not a projective cover of the Z-module Z/2Z (which in fact has no projective cover). The class of rings which provides all of its right modules with projective covers is the class of right perfect rings.
- Any R-module M has a flat cover, which is equal to the projective cover if M has a projective cover.
See also
[edit]References
[edit]- ^ Proof: Let N be proper in P and suppose p(N)=M. Since ker(p) is superfluous, ker(p)+N≠P. Choose x in P outside of ker(p)+N. By the surjectivity of p, there exists x' in N such that p(x' )=p(x ),, whence x−x' is in ker(p). But then x is in ker(p)+N, a contradiction.
- ^ Anderson & Fuller 1992, p. 302.
- Anderson, Frank Wylie; Fuller, Kent R (1992). Rings and Categories of Modules. Springer. ISBN 0-387-97845-3. Retrieved 2007-03-27.
- Faith, Carl (1976), Algebra. II. Ring theory., Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag
- Lam, T. Y. (2001), A first course in noncommutative rings (2nd ed.), Graduate Texts in Mathematics, 131. Springer-Verlag, ISBN 0-387-95183-0