  The representation theory of the symmetric group has been intensively studied
for over 100 years and is one of the gems of modern mathematics. The full
transformation monoid $\mathfrak T_n$ (the monoid of all self-maps of an
$n$-element set) is the monoid analogue of the symmetric group. The
investigation of its representation theory was begun by Hewitt and Zuckerman in
1957. Its character table was computed by Putcha in 1996 and its representation
type was determined in a series of papers by Ponizovski{\u\i}, Putcha and
Ringel between 1987 and 2000. From their work, one can deduce that the global
dimension of $\mathbb C\mathfrak T_n$ is $n-1$ for $n=1,2,3,4$. We prove in
this paper that the global dimension is $n-1$ for all $n\geq 1$ and, moreover,
we provide an explicit minimal projective resolution of the trivial module of
length $n-1$.
  In an appendix with V.~Mazorchuk we compute the indecomposable tilting
modules of $\mathbb C\mathfrak T_n$ with respect to Putcha's quasi-hereditary
structure and the Ringel dual (up to Morita equivalence).
