Commutative-algebraic techniques, together with methods in operator theory and function theory, are used to study the Dirichlet space over the unit disc. For any invariant subspace M ⊂ script D sign ⊗ ℂN of ℂN-valued Dirichlet space N - dim(script M sign ⊖ zscript M sign) is exactly the shift Samuel multiplicity of script M sign ⊥ = script D sign ⊗ ℂ ⊖ script M sign. The following monotonic property which generalizes a result of Richter is a consequence. If script M sign1 ⊂ script M sign2 ⊂ script D sign ⊗ ℂN are two invariant subspaces, then dim(script M sign1 ⊖zscript M sign1) ≦ dim(script M sign2 ⊖ zscript M sign2). A notion of curvature invariant is introduced for the Dirichlet space, and is shown to be equal to the Fredholm index (up to a sign). A version of the Gauss-Bonnet-Chern formula is also available. A notion of multiplicity is introduced for both invariant and coinvariant subspaces. They are defined by asymptotic formulas and turn out to be integers.