Math 2.0

Algebraic geometry Definition

Algebraic cycle

Builds on Algebraic varietyDimension of an algebraic varietyIrreducible componentFree abelian groupLength of a module

Definition

Let XX be an algebraic variety over a field, or more generally a scheme of finite type over a field. Let r≥0r \ge 0 be an integer. An rr-cycle on XX is a finite formal sum Z=∑ini[Vi]Z = \sum_i n_i [V_i]. Here the nin_i are integers and the ViV_i are irreducible closed subvarieties of XX of dimension rr. The rr-cycles form the free abelian group Zr(X)Z_r(X) on the set of such subvarieties. An algebraic cycle is an element of Z∗(X)=⨁r≥0Zr(X)Z_*(X) = \bigoplus_{r \ge 0} Z_r(X). If XX is equidimensional of dimension nn, one also writes Zj(X)=Zn−j(X)Z^j(X) = Z_{n-j}(X) for the cycles of codimension jj.

Remarks

A cycle is effective if all ni≥0n_i \ge 0. Let W⊂XW \subset X be a closed subscheme whose irreducible components W1,…,WsW_1, \ldots, W_s all have dimension rr. Its fundamental cycle is [W]=∑imi[Wi][W] = \sum_i m_i [W_i], where mim_i is the length of the local ring of WW at the generic point of WiW_i. Example: in X=ℙ2X = \mathbb{P}^2, with LL a line and CC a conic, 2[L]−[C]2[L] - [C] is a 11-cycle. The subscheme defined by x2=0x^2 = 0 is the double line, with fundamental cycle 2[L]2[L] for L={x=0}L = \set{x = 0}. Non-example: the union L1∪L2L_1 \cup L_2 of two distinct lines is not one of the generators, since it is not irreducible; its fundamental cycle is [L1]+[L2][L_1] + [L_2].

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Algebraic variety, concept, no evidence levelAlgebraic varietyconceptBinary operation, definition, no evidence levelBinary operationdefinitionDimension of an algebraic variety, concept, no evidence levelDimension of an algeb…conceptGroup, concept, no evidence levelGroupconceptIrreducible component, concept, no evidence levelIrreducible componentconceptModule over a ring, definition, no evidence levelModule over a ringdefinitionReduced word, definition, no evidence levelReduced worddefinitionCommutativity, definition, no evidence levelCommutativitydefinitionCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionRing, definition, no evidence levelRingdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSubmodule, definition, no evidence levelSubmoduledefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementFree group, definition, no evidence levelFree groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionQuotient group, definition, no evidence levelQuotient groupdefinitionAbelianisation, definition, no evidence levelAbelianisationdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionFree abelian group, definition, no evidence levelFree abelian groupdefinitionComposition series, definition, no evidence levelComposition seriesdefinitionLength of a module, definition, no evidence levelLength of a moduledefinitionAlgebraic cycle, definition, no evidence levelAlgebraic cycledefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic variety concept
  2. Binary operation definition
  3. Dimension of an algebraic variety concept
  4. Group concept
  5. Irreducible component concept
  6. Module over a ring definition
  7. Reduced word definition
  8. Commutativity definition
  9. Commutator of two elements definition
  10. Conjugate elements definition
  11. Group homomorphism definition
  12. Ring definition
  13. Subgroup definition
  14. Submodule definition
  15. Abelian group definition
  16. Group isomorphism definition
  17. Conjugate subgroup definition
  18. Left coset definition
  19. Right coset definition
  20. Subgroup criterion statement
  21. Free group definition
  22. Normal subgroup definition
  23. Intersection of subgroups is a subgroup statement
  24. Simple group definition
  25. Normality via left and right cosets statement
  26. Subgroup generated by a subset definition
  27. Commutator subgroup definition
  28. Quotient group definition
  29. Abelianisation definition
  30. Subnormal series definition
  31. Free abelian group definition
  32. Composition series definition
  33. Length of a module definition
  34. Algebraic cycle definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_cycle
About this page
TagNE0XB0AEDR
URIm2://NE0XB0AEDR
Kinddefinition
WritingW0 stub
MSC14C25
DefinesAlgebraic cycle
UsesAlgebraic variety, Dimension of an algebraic variety, Irreducible component, Free abelian group, Length of a module
Changed2 November 2026

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