A new algebraic method for the investigation of vectorbundles on the projective planeClassical terms and summaryIn this work we place a new algebraic method to the investigation of vector bundles, in addition, arbitrary coherent modules on projective spaces ℙ_{n}. To motivate the introduction of the essential term, the Kronecker-module, connected with coherent ℙ_{n}-modules, we want to remind now of the classical results of Serre. Let Grad^{n} be the category of finitely generated modules over the graduated k-algebra of the polynomials, k[x_{0}....x_{n}], whereby k is an algebraically closed field. From a module M in Grad^{n} we can construe a ℙ_{n}-module F in the following way: We define the cuts Γ(ℙ_{nf},F) of F over the open set ℙ_{nf} of ℙ_{n} of the points, where the homogeneous function f∈k[x_{1},...,x _{n}] does not disappear, as the 0-component (M _{f})_{0} of the localization M_{f} of M concerning the set {1, f, f^{2},f^{3}...}. Thereby is determined a coherent ℙ_{n}-module F up to single isomorphism. In particular k[x_{0}...,x_{n}](m), received from k[x_{0}...,x_{n}] by shifting the graduation, is assigned to the locally free module ℙ_{n}-module O(m), a line bundle, with m<0 a power of the Hopf-bundle O(-1). If we name F(m) the tensor-product F⊗O(m) and Γ(F) the global cuts of F, then for a coherent ℙ_{n}-module F the vector space X(F) = ⊕Γ(F(m)) (direct sum) is a finitely generated k[x_{0},...,x_{n}]-module, provided with the natural multiplication, which is defined as follows: We understand x_{i} as cut of O(1) and define x_{i}⋅s for s in Γ(F(m)) as the cut of O(1)⊗F(m) = F(m+1). In this way we receive two functors A: Grad^{n}→Mod and X: Mod→Grad^{n}, whereby Mod designates the category of the coherent ℙ_{n}-module-sheaves. The construction A of ℙ_{n}-modules does not depend however on the homogeneous components on the degrees smaller than some arbitrary constant, and on the other hand there are modules with X(F)_{m} = 0 for all m∈ℤ smaller than some arbitrary constant. Therefore the functors A and X are only then "inverse" if we in Grad^{n} identify objects and morphisms, which agree in all homogeneous components of sufficient high degree. This identification in Grad^{n} is however purely theoretical and practically one works with the objects in Grad^{n}. But then it can occur that e.g. an irreducible graduated module M induces a reducible ℙ_{n}-module A(M). In order to loose-come however from the necessity for the class formation, one must itself in each case limit on ℙ_{n}-modules F, which already are constructable from its cuts Γ(F(m)) for m smaller than a constant. For example the subcategorie Vb_{0}^{n} of Mod_{n} of the locally free globally generated ℙ_{n}-moduls F, where the higher cohomologie of the F(m) for m m≥-n disappears, posseses this characteristic, and we will show that such a module is already determined by two components of X(F), e.g. by Γ(F(-1)) and Γ(F), together with the multiplication of the x_{i}∈k[x_{0},...,x_{n}], which induce n+1 linear mappings x_{i}: Γ(F(-1))→Γ(F). Such an object of two vector spaces V and W with a finite number k of linear mappings V→W is called Kronecker-module with k arrows. We can thus assign to each coherent ℙ_{n}-module F a Kronecker-module κ(F) with n+1 arrows Γ(F(-1))→Γ(F). We develop also a procedure for the global construction of ℙ_{n}-modules M(æ) from Kronecker-modules æ and show that the functor κ on a subcategorie Mod _{0}^{n} of Mod, who contains Vb_{0}^{n}, induce a fully faithful embedding into the category of the Kronecker-modules and is a "cut" for the M. The category Mod _{0}^{n} has the characteristic that there exists for each coherent ℙ_{n}-module F an m∈ℕ, so that F(m) lies in _{}Mod^{}^{}_{0 }^{n}. Therefore one can limit oneself to Mod_{0}^{n}, if one regards ℙ_{n}-modules only up to isomorphism, since F(m) is isomorphic to G (m) exactly if F isomorphic to G. Since we are interested in vector bundles primarily by questions of isomorphism, we examine characteristics of the Kronecker-modules of vector bundles in Mod _{0 }^{n} in this work. First however we give a memory of the "classical" methods for designing a quasiprojektives scheme of isomorphism classes of vector bundles over a non-singular projective curve X. One designs first a family or a scheme of vector bundles with a reductive group operating on it in such a way that its orbits are exactly the isomorphism classes. Then one studies the group operation and marks the open subscheme of the "stable" points, for which a "geometrical" quotient, called moduli-scheme, of the group operation by the moduli-group exists, i.e. essentially: a universal quasiprojektive scheme, whose points parametrize the isomorphism classes. [ D. Mumford, geometric invariantl theory, 0.4 ] Three substantial characteristics of bundles are used on projektive curves X: We are in the situation, to mark the stable ("particularly stable") vector bundles with similar methods on ℙ_{}_{2 }(for arbitrary ranks). Unfortunately a geometrical interpretation of a criterion for stability succeeds only for special (monomial) bundles. The methods used here are surely "developable", but in contrast not for arbitrary bundles over ℙ_{n}, because the three central characteristics used for to the derivation of the condition of stability on ℙ_{n} reliably do not apply for n > 2. The following correspond to the above statements in this work: |