From e92f4574a8edaf1c12eb1bff6f8e671b6d8ad4ae Mon Sep 17 00:00:00 2001 From: Erik Paemurru <143521159+paemurru@users.noreply.github.com> Date: Fri, 29 Nov 2024 14:57:30 +0100 Subject: [PATCH 1/2] Fix indent in AlgebraicCycles.md It looks correct on GitHub but is incorrect in https://docs.oscar-system.org/dev/AlgebraicGeometry/ToricVarieties/AlgebraicCycles/ --- .../ToricVarieties/AlgebraicCycles.md | 32 +++++++++---------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md b/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md index f8816d400e42..4b70ebf4ea62 100644 --- a/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md +++ b/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md @@ -31,25 +31,25 @@ 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. + 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. + 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. + 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. + rational equivalence classes of algebraic cycles are one-to-one to + the cones in the fan of the toric variety. ## Constructors From 0e7be24b500b28af9b8aef337c222a559d2d9a2b Mon Sep 17 00:00:00 2001 From: Erik Paemurru <143521159+paemurru@users.noreply.github.com> Date: Fri, 29 Nov 2024 15:37:38 +0100 Subject: [PATCH 2/2] Update AlgebraicCycles.md --- .../ToricVarieties/AlgebraicCycles.md | 44 ++++++++++--------- 1 file changed, 24 insertions(+), 20 deletions(-) diff --git a/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md b/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md index 4b70ebf4ea62..0b77aa96d75b 100644 --- a/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md +++ b/docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md @@ -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