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Fix indent in AlgebraicCycles.md #4368

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44 changes: 24 additions & 20 deletions docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md
Original file line number Diff line number Diff line change
Expand Up @@ -30,26 +30,30 @@ the variety in question.
For complete and simplicial toric varieties, many things are
known about the Chow ring and algebraic cycles (cf. section 12.5
in [CLS11](@cite):
* By therorem 12.5.3 of [CLS11](@cite), there is an isomorphism
among the Chow ring and the cohomology ring. Note that the
cohomology ring is naturally graded (cf. last paragraph
on page 593 in [CLS11](@cite)). However, the Chow ring
is usually considered as a non-graded ring. To match this general
convention, and in particular the implementation of the Chow ring
for matroids in OSCAR, the toric Chow ring is constructed as a
non-graded ring.
* By therorem 12.5.3 of [CLS11](@cite), the Chow ring is isomorphic
to the quotient of the non-graded Cox ring and a certain ideal.
Specifically, the ideal in question is the sum of the ideal of
linear relations and the Stanley-Reisner ideal.
* It is worth noting that the ideal of linear relations is not
homogeneous with respect to the class group grading of the Cox ring.
In order to construct the cohomology ring, one can introduce a
$\mathbb{Z}$-grading on the Cox ring such that the ideal of linear
relations and the Stanley-Reißner ideal are homogeneous.
* Finally, by lemma 12.5.1 of [CLS11](@cite), generators of the
rational equivalence classes of algebraic cycles are one-to-one to
the cones in the fan of the toric variety.

* By therorem 12.5.3 of [CLS11](@cite), there is an isomorphism
among the Chow ring and the cohomology ring. Note that the
cohomology ring is naturally graded (cf. last paragraph
on page 593 in [CLS11](@cite)). However, the Chow ring
is usually considered as a non-graded ring. To match this general
convention, and in particular the implementation of the Chow ring
for matroids in OSCAR, the toric Chow ring is constructed as a
non-graded ring.

* By therorem 12.5.3 of [CLS11](@cite), the Chow ring is isomorphic
to the quotient of the non-graded Cox ring and a certain ideal.
Specifically, the ideal in question is the sum of the ideal of
linear relations and the Stanley-Reisner ideal.

* It is worth noting that the ideal of linear relations is not
homogeneous with respect to the class group grading of the Cox ring.
In order to construct the cohomology ring, one can introduce a
$\mathbb{Z}$-grading on the Cox ring such that the ideal of linear
relations and the Stanley-Reißner ideal are homogeneous.

* Finally, by lemma 12.5.1 of [CLS11](@cite), generators of the
rational equivalence classes of algebraic cycles are one-to-one to
the cones in the fan of the toric variety.


## Constructors
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