@@ -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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+
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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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+
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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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+
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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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+
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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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