Difference between revisions of "Free resolution"
From Encyclopedia of Mathematics
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A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules | A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules | ||
| − | $$F_0\leftarrow F_1\leftarrow\ | + | $$F_0\leftarrow F_1\leftarrow\dotsb\leftarrow F_n\leftarrow\dotsb,$$ |
called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation). | called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation). | ||
Latest revision as of 12:12, 14 February 2020
A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules
$$F_0\leftarrow F_1\leftarrow\dotsb\leftarrow F_n\leftarrow\dotsb,$$
called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation).
Comments
See also Free module.
How to Cite This Entry:
Free resolution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_resolution&oldid=32280
Free resolution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_resolution&oldid=32280
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article