- Schanuel's lemma
In
mathematics , especially in the area of algebra known asmodule theory , Schanuel's lemma 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.tatement
Schanuel's lemma is the following statement:
If 0 → "K" → "P" → "M" → 0 and 0 → "K"' → "P" ' → "M" → 0 are
short exact sequence s of "R"-modules and "P" and "P" ' are projective, then "K" ⊕ "P" ' is isomorphic to "K" ' ⊕ "P."Proof
Define the following
submodule of "P" ⊕ "P" ', where φ : "P" → "M" and φ' : "P" ' → "M"::
The map π : "X" → "P", where π is defined as the projection of the first coordinate of "X" into "P", is surjective. Since φ is surjective, for any "p" "X", one may find a "q" "P" ' such that φ("p") = φ '("q"). This gives ("p","q") "X" with π ("p","q") = "p". Now examine the kernel of the map π :
We may conclude that there is a short exact sequence
:
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
:
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 sequence s. [cite book | author = Lam, T.Y. | title = Lectures on Modules and Rings | publisher = Springer | year = 1999 | id = ISBN 0387984283 pgs. 165-167.]Origins
Stephen Schanuel discovered the argument inIrving Kaplansky 'shomological algebra course at theUniversity 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 Shanuel 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." " [cite book | author = Kaplansky, Irving. | title = Fields and Rings | publisher = University Of Chicago Press | year = 1972 | id = ISBN 0226424510 pgs. 165-168.]Notes
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