Explore: Cycles And Subschemes

to the set of closed subschemes of P n × S {\displaystyle \mathbb {P} ^{n}\times S} that are flat over S. The closed subschemes of P n × S {\displaystyle

{Z} \cdot [V],} where the sum is over closed integral subschemes V of X. The groups of cycles for varying r together form a group Z ∗ X = ⨁ r Z r X

open subschemes of Y are mapped to open subschemes of X (and similarly for closed). For example, finitely presented flat morphisms are open and proper

equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined

scheme of closed subschemes of dimension k-1 and degree d inside P n − 1 {\displaystyle \mathbb {P} ^{n-1}} . Each closed subscheme determines an effective

between the set of closed subschemes of Y ×k D, flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NX Y). If i : X ↪ Y

Arithmetic and Geometry of Algebraic Cycles. Section 5. doi:10.1007/978-94-011-4098-0_5. ISBN 978-0-7923-6194-7. Bloch, Spencer (1986), "Algebraic cycles and higher

{\displaystyle Z_{i}(X)} of i {\displaystyle i} -dimensional cycles (or i {\displaystyle i} -cycles, for short) on X {\displaystyle X} is the free abelian group

are Cartier divisors. Topologically, Weil divisors correspond to homology cycles, while Cartier divisors correspond to cohomology classes defined by line

In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber product