Künneth theorem

The Künneth theorem, named after Hermann Künneth , is a proposition from the mathematical branch of homological algebra . The theorem traces the homology of a tensor product of chain complexes back to the homologies of the chain complexes involved, in a memorable formulation it says that the homology of a tensor product of chain complexes is equal to the tensor product of the homologies except for torsion. Künneth's theorem, which is often simply called the Künneth formula , is a generalization of the universal coefficient theorem .

Tensor products of chain complexes

If and are two chain complexes , then the tensor product is the chain complex with ${\displaystyle (C',d')}$ ${\displaystyle (C'',d'')}$ ${\displaystyle (C',d')\otimes (C'',d'')}$ ${\displaystyle (C,d)}$

${\displaystyle C_{n}=\bigoplus _{i+j=n}C_{i}'\otimes C_{j}''}$

${\displaystyle d_{n}(c_{i}'\otimes c_{j}''):=d_{i}'c_{i}'\otimes c_{j}''+(-1)^{i}c_{i}'\otimes d_{j}''c_{j}''}$, where . ${\displaystyle c_{i}'\in C_{i}',c_{j}''\in C_{j}'',i+j=n}$

If a chain complex is special , which only has a module different from 0 in the 0th position , then the chain complex is ${\displaystyle (C',d')}$ ${\displaystyle A}$ ${\displaystyle (C',d')\otimes (C'',d'')}$

${\displaystyle \ldots \rightarrow A\otimes C_{n}''{\xrightarrow {\mathrm {id} _{A}\otimes d_{n}''}}A\otimes C_{n-1}''\rightarrow \ldots }$.

This chain complex is also written as an abbreviation . ${\displaystyle A\otimes C''}$

The sentence to be introduced here answers the question of how one can calculate the homology of the tensor product from the homology of the chain complexes. In general, the homology of the tensor product is not determined by the homology of and ; further requirements must be placed on the ring and the given chain complexes. The simplest formula for such a dependency would be that the -th homology of the tensor product is isomorphic to the direct sum of the tensor products of the homologies of and . It turns out that this formula has to be extended by the direct sum of the first torsions of the homology groups. ${\displaystyle C\otimes C'}$ ${\displaystyle C}$ ${\displaystyle C'}$ ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle H_{n}(C\otimes C')}$ ${\displaystyle \textstyle \bigoplus _{i+j=n}H_{i}(C')\otimes H_{j}(C'')}$ ${\displaystyle C}$ ${\displaystyle C'}$

Formulation of the sentence

Let and two chain complexes of modules over a principal ideal and one of the chain complexes composed exclusively of flat modules . Then for every integer there is a natural , short exact sequence ${\displaystyle (C',d')}$ ${\displaystyle (C'',d'')}$ ${\displaystyle R}$ ${\displaystyle n}$

${\displaystyle 0\rightarrow \bigoplus _{i+j=n}H_{i}(C')\otimes H_{j}(C'')\rightarrow H_{n}(C'\otimes C'')\rightarrow \bigoplus _{i+j=n-1}\mathrm {Tor} _{1}^{R}(H_{i}(C'),H_{j}(C''))\rightarrow 0}$.

This sequence breaks down, i.e. is isomorphic to a direct sum of the other two components of the sequence, but not naturally. ${\displaystyle H_{n}(C'\otimes C'')}$

meaning

The set of Eilenberg-Zilber performs the calculation of the singular homology of a product of topological spaces back to the tensor of the singular homologies of the spaces involved. Künneth's theorem is the algebraic part still missing in this theorem in order to complete the calculation of the homology of a product space.

The universal coefficient theorem

If the chain complex only has a module different from the zero module at the 0th position , then most of the summands from the Künneth formula above are 0 and the exact sequence is obtained ${\displaystyle C'}$ ${\displaystyle A}$

${\displaystyle 0\rightarrow A\otimes H_{n}(C'')\rightarrow H_{n}(A\otimes C'')\rightarrow \mathrm {Tor} _{1}^{R}(A,H_{n-1}(C''))\rightarrow 0}$,

and that is nothing more than the universal coefficient theorem .

Individual evidence

1. ↑ Peter Hilton and Urs Stammbach: A course in homological algebra. , Springer-Verlag, Graduate Texts in Mathematics, 1970, ISBN 0-387-90032-2 , Chapter V, Theorem 2.1
2. ^ Robert M. Switzer: Algebraic Topology - Homotopy and Homology , Springer Grundlehren der Mathematischen Wissenschaften, Volume 212 (1975), ISBN 3-540-06758-2 , Theorem 13.31