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We study projections in the corona algebra of 
$C\left( X \right)\,\otimes \,K$, where 
$K$ is the 
${{C}^{*}}$-algebra of compact operators on a separable infinite dimensional Hilbert space and 
$X\,=\,[0,\,1],\,[0,\,\infty ),\,(-\infty ,\,\infty ),\,\text{or}\,\text{ }\!\![\!\!\text{ 0,}\,\text{1 }\!\!]\!\!\text{ / }\!\!\{\!\!\text{ 0,}\,\text{1 }\!\!\}\!\!\text{ }$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in 
${{K}_{0}}$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.
