## You are here

Homesyzygy

## Primary tabs

# syzygy

Let $\mathcal{A}$ be an abelian category. Let $M\in\operatorname{Ob}(\mathcal{A})$. If there is an exact sequence

$\xymatrix{0\ar[r]&S\ar[r]&P_{n}\ar[r]&{\cdots}\ar[r]&P_{1}\ar[r]&P_{0}\ar[r]&M% \ar[r]&0},$ |

where each $P_{i}$ is a projective object in $\mathcal{A}$, then we call $S$ an $n$th *syzygy* of $M$.

If $S$ is itself projective, then the projective dimension of $M$, $\operatorname{pd}(M)$, is less than or equal to $n$.

Remark.

1. 2. Any two $n$th syzygies of a given object are projectively equivalent.

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18G10*no label found*16E05

*no label found*13D02

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections