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The main goal of my exposition shall be to

• describe how schemes are a generalization of varieties
• or, more precisely, how varieties are a special case of schemes,
• or, more precisely, how the category of varieties is a subcategory of that of schemes,
• or, to be really precise, how there is a fully faithful functor

from the category of varieties over a field k, into that of schemes over a field k.

In order to be even able to state this result, I will need to say a few words about what varieties and schemes actually are.

Doing so already goes a long way towards proving the above statement, since the definition of a scheme is precisely motivated by the desire to generalize that of a variety.

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Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes . After the proof of the simplicial approximation theorem this approach provided rigour.

The change of name reflected the move to organise topological classes such as cycles modulo boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether,[1] and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf,[2] who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.[3]

A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944.[4]

Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of Euler characteristic theorem and Gauss–Bonnet theorem were obtained by Chern et al. (See digital topology.) In history, a 2D grid cell topology had appeared in Alexandrov-Hopf's book Topologie I (1935).

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In algebraic geometry, a lemniscate is any of several figure-eight or ∞-shaped curves.[1][2] The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning ribbons, [2] or alternatively may refer to the wool from which the ribbons were made.[1]
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