In the branch of abstract mathematics called
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, 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
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
Projective covers and their superfluous epimorphisms, when they exist, are unique up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. The isomorphism need not be unique, however, since the projective property is not a full fledged
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
.
The main effect of ''p'' having a superfluous kernel is the following: if ''N'' is any proper submodule of ''P'', then
.
[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.] 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.
Examples
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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, 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 ''R'' has a projective cover.
See also
*
Projective resolution
References
*
*
*{{citation, last= Lam, first=T. Y., title=A first course in noncommutative rings, edition= 2nd, publisher=Graduate Texts in Mathematics, 131. Springer-Verlag, year=2001, isbn=0-387-95183-0
Category theory
Homological algebra
Module theory