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Hello,

This question might be vague and not thought-through enough.

If we have a real positive number $x$, we can start to write it as a continued fraction: $x = a_0 + \frac{1}{x_1} , \ldots , x_n=a_n + \frac{1}{x_{n+1}}$ where $a_i$ are non-negative integers, $x_i$ non-nogative real numbers less than $1$. So we can write $x=[a_0,a_1,\ldots]$. But we also may write $x=[a_0,\ldots,a_{n-1},x_n]$, i.e. we might decide that our continued fraction is finite, allowing the last term to be non-integer.

If we have a module, we can start taking a projective resolution of it. Again, we can take an infinite projective resolution, or decide to truncate it at some finite level, but then the last term will not be maybe projective.

The last term will be projective if our module was "good", i.e. of small enough (in particular, finite) cohomological dimension. In the continued fraction setting, the last term will be integer if our number was "good", i.e. rational...

Is there any (wild?) relation between rational numbers among real numbers, and projective modules of finite cohomological dimension among modules?

Sasha

# Continued fractions and projective resolutions

Hello,

This question might be vague and not thought-through enough.

If we have a real positive number $x$, we can start to write it as a continued fraction: $x = a_0 + \frac{1}{x_1} , \ldots , x_n=a_n + \frac{1}{x_{n+1}}$ where $a_i$ are non-negative integers, $x_i$ non-nogative real numbers less than $1$. So we can write $x=[a_0,a_1,\ldots]$. But we also may write $x=[a_0,\ldots,a_{n-1},x_n]$, i.e. we might decide that our continued fraction is finite, allowing the last term to be non-integer.

If we have a module, we can start taking a projective resolution of it. Again, we can take an infinite projective resolution, or decide to truncate it at some finite level, but then the last term will not be maybe projective.

The last term will be projective if our module was "good", i.e. of small enough (in particular, finite) cohomological dimension. In the continued fraction setting, the last term will be integer if our number was "good", i.e. rational...

Is there any (wild?) relation between rational numbers among real numbers, and projective modules among modules?

Sasha