Yes, there is an analogous notion of "Cartier -cycles" and a corresponding sheaf.

Key Idea:

For , Cartier divisors are defined using the quotient sheaf , where is the sheaf of total quotient rings (rational functions) and is the sheaf of invertible regular functions. This sheaf's global sections $\Gamma...

Short answer: not as a single sheaf. The right higher‑codimension replacement of “Cartier divisors” is no longer captured by global sections of one sheaf; instead one needs a (Gersten) complex of sheaves coming from K‑theory whose last term is the sheaf of codimension‑p cycles.

More precisely (assume X is regular, noetherian; e.g. a smooth variety), set Z^p_X := ⊕{x∈X^{(p)}} i{x,*} Z, the sheaf...

Let us recall what makes a Cartier divisor different from a Weil divisor.

• A Weil divisor is just a formal linear combination of codimension-1 subvarieties.
• A Cartier divisor, on the other hand, is a subvariety that can be cut out by ONE equation – i.e. its ideal sheaf is locally generated by a single non–zero‐divisor. “Locally generated by one element” is what is measured by the sheaf
...

Yes, there is a notion of higher codimension Cartier cycles, though the theory becomes more subtle and less universally applicable than for divisors. Let me explain the main approaches.

The Analogue: Higher Chow Groups and K-theory

The most direct analogue of Cartier divisors for higher codimension cycles comes from Milnor K-theory and higher Chow groups.

1. Milnor K-sheaves

For...

This is an excellent and very natural question. The transition from Weil divisors to Cartier divisors is a fundamental step in algebraic geometry, moving from a purely geometric definition (formal sums of subvarieties) to one that is more sheaf-theoretic and tied to local equations. You are asking for the analogous step in higher codimension.

The short answer is: **Yes, such notions exist, but th...

Only in codimension 1 do “Cartier objects’’ behave at all like Weil cycles.
The reason is essentially algebraic: a closed subscheme of codimension 1 can be described (étale-locally, or even Zariski-locally if the scheme is integral and normal) by a single non-zero divisor (i.e. by one equation), and two such local equations differ by a unit. Therefore the transition functions live in 𝒪×, an ab...

Generalizing Cartier Divisors to Higher Codimensions

The question asks about finding a natural generalization of the relationship between Weil divisors and Cartier divisors to higher codimensions.

The Existing Relationship

For codimension 1:

• Weil divisors (Div(X)) are formal sums of codimension 1 subvarieties
• Cartier divisors (Cart(X)) are global sections of $\mathcal{K}_X^{\times}/\ma...

Here is one reasonably concrete way to see that “Cartier‐p–cycles’’ exist exactly by imitating the case p=1, but replacing K₁ by Quillen’s higher K‐theory sheaves. Recall that for p=1 one has on the Zariski site of X the exact sequence of sheaves

0 → 𝒪_X^× → 𝒦_X^× → Cart_X^1 → 0

and by definition Cart^1(X) = Γ(X,Cart_X^1) = coker[Γ(X,𝒪_X^×)→Γ(X,𝒦_X^×)] = principal Weil‐div...