...of homological algebra [[#References|[2]]]. In the 1960s, interest in non-Abelian categories grew, as a result of problems in logic, general algebra, topolog
...31–294</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper & Row (
43 KB (6,447 words) - 09:17, 26 March 2023
...ense) by K. Gödel (see [[Gödel incompleteness theorem|Gödel incompleteness theorem]]). Positive results (using techniques that Hilbert would not have allowed)
...[#References|[a15]]], [[#References|[a29]]]), in the form of the following theorem: Every locally Euclidean topological group is a Lie group and even a real-a
29 KB (4,109 words) - 19:54, 18 March 2018
expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions $ < 8 $
expresses the theorem on the existence of a Lipschitz structure on an arbitrary $ \mathop{\rm
15 KB (2,155 words) - 14:56, 7 June 2020
The basic structure theorem [[#References|[a1]]] is that of Kac–Moody algebras. Let $\mathfrak { g }
...ng $D_i$ is a derivation, and adjoining these to $\mathfrak h $ defines an Abelian algebra $\mathfrak{h} ^ {e }$. The simple root $\alpha_i$ can be interprete
14 KB (2,218 words) - 10:27, 11 November 2023
has a normal Abelian subgroup $ \Gamma ^ {*} $
...group is a torsion-free crystallographic subgroup. An Auslander–Kuranishi theorem says that each crystallographic group arises as a crystallographic subgroup
18 KB (2,658 words) - 19:36, 18 January 2024
...A$, and $A \times \{ 0 \}$ is isomorphic to the vector bundle $A$ with the Abelian Lie algebroid structure (zero bracket and zero anchor); the prolongation Li
There is no analogue to Lie's third theorem (cf. also [[Lie theorem|Lie theorem]]) in the case of Lie algebroids, since not every Lie algebroid can be inte
12 KB (1,916 words) - 16:55, 1 July 2020
...stem in the algebra of differential polynomials. The Ritt–Raudenbush basis theorem states that all perfect differential ideals are obtained in this way (a dif
...theorem in a polynomial ring, the essential feature of the Ritt–Raudenbush theorem is that the ideals are perfect, i.e. differential ideals (even perfect diff
30 KB (4,468 words) - 18:44, 17 December 2019
be the group algebra of a discrete Abelian group $ G $,
this last example leads to a proof of the well-known Wiener theorem: If the function $ \widehat{f} ( t) $
18 KB (2,806 words) - 03:47, 25 February 2022
...re CW-complexes (Whitehead's theorem, cf. [[CW-complex|CW-complex]]). This theorem is based on the facts that: 1) a mapping $ f: \ X \rightarrow Y $
dimensional cochain group over an Abelian group $ \pi $ (
31 KB (4,636 words) - 12:07, 15 December 2019
is an Abelian group. The homotopy equivalence $ \omega _ {n} : K( \pi , n) \rightarrow
...\mathbf Z ) \simeq \mathop{\rm BU} \times \mathbf Z $ (Bott's periodicity theorem), and one obtains the spectrum of spaces $ \{ \dots, U, \mathop{\rm BU}
9 KB (1,281 words) - 14:40, 21 March 2022
...e 18th century (cf. [[Algebra, fundamental theorem of|Algebra, fundamental theorem of]]). Finally, it was established by N.H. Abel in 1824 that equations of d
...y called addition and multiplication. A ring is defined by [[Abelian group|Abelian group]] axioms for the addition, and by distributive laws for multiplicatio
17 KB (2,478 words) - 16:09, 1 April 2020
induces an isomorphism of homology with any local coefficient system of Abelian groups;
...p"> D. McDuff, G.B. Segal, "Homotopy fibrations and the "group completion" theorem" ''Invent. Math.'' , '''31''' (1976) pp. 279–284</TD></TR>
7 KB (1,010 words) - 16:47, 17 March 2023
...presentation of a given group is finitely generated (the first fundamental theorem of the theory of invariants) and to determine a system of basic invariants.
...prove the existence of a finite basis of syzygies (the second fundamental theorem of the theory of invariants) and to find it.
22 KB (3,406 words) - 07:08, 6 May 2022
...$k$ and let $\mathcal{L}$ be a locally free sheaf on $X$. Serre's duality theorem states that the finite-dimensional cohomology (vector) spaces $H^i(X,\mathc
[[divisor]] $D$ on $X$, this theorem establishes the equality
64 KB (9,418 words) - 12:44, 8 February 2020
...|[a3]]], [[#References|[a19]]], [[#References|[a20]]]. Darboux's classical theorem [[#References|[a1]]], [[#References|[a2]]] was formulated for the equation
...atveev [[#References|[a4]]] for associative rings. The formulation of this theorem contains the natural generalization of the Darboux transformation in the sp
14 KB (2,023 words) - 17:02, 1 July 2020
...ular parametrization allows the construction of points on $E$ defined over Abelian extensions of certain imaginary quadratic fields. This fact was exploited b
...i}}||valign="top"| A. Wiles, "Modular elliptic curves and Fermat's last theorem" ''Ann. of Math.'', '''141''' : 2–3 (1995) pp. 443–551 {{MR|133303
7 KB (1,047 words) - 01:52, 18 July 2022
...hereditary categories $\mathcal{H}$, that is, $\mathcal{H}$ is a connected Abelian $k$-category with vanishing Yoneda functor $\operatorname{Ext} ^ { 2 } ( .,
...[a13]</td> <td valign="top"> D. Happel, R. Reiten, S.O. Smalø, "Tilting in abelian categories and quasitilted algebras" ''Memoirs Amer. Math. Soc.'' , '''575'
16 KB (2,221 words) - 09:47, 11 November 2023
of all Abelian groups; the Burnside variety $ \mathfrak B _ {n} $
The Oates–Powell theorem says that the variety generated by the finite groups is Cross. As a corolla
9 KB (1,421 words) - 08:28, 6 June 2020
...mined by the Gauss–Bonnet formula (cf. [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]]). Among the closed surfaces, only the sphere $ S ^ {2} $
the Hadamard–Cartan theorem); moreover, for any point $ x \in M ^ {n} $
25 KB (3,732 words) - 22:15, 7 June 2020
...in it starting from the fourth are groups and starting from the seventh, [[Abelian group]]s. See <ref name ="Dold" /> <ref name ="Spanier" />.
...one is closed and reproducing, then it is unflattened (the Krein–Shmul'yan theorem).
13 KB (2,188 words) - 13:05, 10 June 2016