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Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_X\right)}_{X\in }\in {\mathbb{N}}^{}$ such that $M\cong {⨁}_{X\in }{X}^{m_X}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ${̃}_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_x\right)}_{x\in X}\in {\mathbb{N}}^X$ such that $M\cong {⨁}_{x\in X}{X}_x^{m_x}$ is studied. A precise formula for $di{m}_{k}Ho{m}_{\Lambda }(M,X)$, for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....