= Basic hereditary algebra path-algebra theorem
{title2=$A\cong kQ$}
Let $A$ be a finite-dimensional <basic algebra> over an <algebraically closed field> and a <hereditary ring>. The radicals of its indecomposable projectives are projective and satisfy $JP_i\cong\bigoplus_jP_j^{a_{ij}}$, where $a_{ij}$ are the <Ext quiver> arrow multiplicities. Hence $\dim P_i=1+\sum_ja_{ij}\dim P_j$, forcing strict dimension decrease along arrows and therefore acyclicity. Lifting an arrow basis in $J/J^2$ gives a surjective <path algebra> homomorphism $kQ\to A$. The number of paths starting at $i$ satisfies the same dimension recursion. Equality of total dimensions proves the homomorphism is an isomorphism. Sums of vertex <idempotents> over connected components are precisely the primitive <central idempotents>, so components give the blocks.
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