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Fix indent in AlgebraicCycles.md (#4368)
* Fix indent in AlgebraicCycles.md It looks correct on GitHub but is incorrect in https://docs.oscar-system.org/dev/AlgebraicGeometry/ToricVarieties/AlgebraicCycles/ * Update AlgebraicCycles.md
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docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md

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@@ -30,26 +30,30 @@ the variety in question.
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For complete and simplicial toric varieties, many things are
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known about the Chow ring and algebraic cycles (cf. section 12.5
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in [CLS11](@cite):
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* By therorem 12.5.3 of [CLS11](@cite), there is an isomorphism
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among the Chow ring and the cohomology ring. Note that the
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cohomology ring is naturally graded (cf. last paragraph
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on page 593 in [CLS11](@cite)). However, the Chow ring
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is usually considered as a non-graded ring. To match this general
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convention, and in particular the implementation of the Chow ring
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for matroids in OSCAR, the toric Chow ring is constructed as a
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non-graded ring.
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* By therorem 12.5.3 of [CLS11](@cite), the Chow ring is isomorphic
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to the quotient of the non-graded Cox ring and a certain ideal.
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Specifically, the ideal in question is the sum of the ideal of
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linear relations and the Stanley-Reisner ideal.
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* It is worth noting that the ideal of linear relations is not
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homogeneous with respect to the class group grading of the Cox ring.
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In order to construct the cohomology ring, one can introduce a
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$\mathbb{Z}$-grading on the Cox ring such that the ideal of linear
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relations and the Stanley-Reißner ideal are homogeneous.
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* Finally, by lemma 12.5.1 of [CLS11](@cite), generators of the
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rational equivalence classes of algebraic cycles are one-to-one to
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the cones in the fan of the toric variety.
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* By therorem 12.5.3 of [CLS11](@cite), there is an isomorphism
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among the Chow ring and the cohomology ring. Note that the
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cohomology ring is naturally graded (cf. last paragraph
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on page 593 in [CLS11](@cite)). However, the Chow ring
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is usually considered as a non-graded ring. To match this general
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convention, and in particular the implementation of the Chow ring
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for matroids in OSCAR, the toric Chow ring is constructed as a
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non-graded ring.
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* By therorem 12.5.3 of [CLS11](@cite), the Chow ring is isomorphic
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to the quotient of the non-graded Cox ring and a certain ideal.
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Specifically, the ideal in question is the sum of the ideal of
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linear relations and the Stanley-Reisner ideal.
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* It is worth noting that the ideal of linear relations is not
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homogeneous with respect to the class group grading of the Cox ring.
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In order to construct the cohomology ring, one can introduce a
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$\mathbb{Z}$-grading on the Cox ring such that the ideal of linear
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relations and the Stanley-Reißner ideal are homogeneous.
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* Finally, by lemma 12.5.1 of [CLS11](@cite), generators of the
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rational equivalence classes of algebraic cycles are one-to-one to
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the cones in the fan of the toric variety.
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## Constructors

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