Schanuel's lemma
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the area of algebra known as module theory, Schanuel's lemma, named after
Stephen Schanuel Stephen H. Schanuel (14 July 1933 – 21 July 2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory. Life While he was a graduate student at University of Chicago, ...
, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting.


Statement

Schanuel's lemma is the following statement: If 0 → ''K'' → ''P'' → ''M'' → 0 and 0 → ''K′'' → ''P′'' → ''M'' → 0 are short exact sequences of ''R''-modules and ''P'' and ''P′'' are projective, then ''K'' ⊕ ''P′'' is
isomorphic 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 is ...
to ''K′'' ⊕ ''P''.


Proof

Define the following submodule of ''P'' ⊕ ''P′'', where φ : ''P'' → ''M'' and φ′ : ''P′'' → ''M'': :X = \. The map π : ''X'' → ''P'', where π is defined as the projection of the first coordinate of ''X'' into ''P'', is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. Since φ′ is surjective, for any ''p'' in ''P'', one may find a ''q'' in ''P′'' such that φ(''p'') = φ′(''q''). This gives (''p'',''q'') \in ''X'' with π(''p'',''q'') = ''p''. Now examine the kernel of the map π: : \begin \ker \pi &= \ \\ & = \ \\ & \cong \ker \phi' \cong K'. \end We may conclude that there is a short exact sequence :0 \rightarrow K' \rightarrow X \rightarrow P \rightarrow 0. Since ''P'' is projective this sequence splits, so ''X'' ≅ ''K′'' ⊕ ''P''. Similarly, we can write another map π : ''X'' → ''P′'', and the same argument as above shows that there is another short exact sequence :0 \rightarrow K \rightarrow X \rightarrow P' \rightarrow 0, and so ''X'' ≅ ''P′'' ⊕ ''K''. Combining the two equivalences for ''X'' gives the desired result.


Long exact sequences

The above argument may also be generalized to long exact sequences.


Origins

Stephen Schanuel Stephen H. Schanuel (14 July 1933 – 21 July 2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory. Life While he was a graduate student at University of Chicago, ...
discovered the argument in Irving Kaplansky's
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
course at the University of Chicago in Autumn of 1958. Kaplansky writes: :''Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added that, although the statement was so simple and straightforward, it would be a while before we proved it. Steve Schanuel spoke up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma."''


Notes

{{reflist Homological algebra Module theory