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 whose sections over U are the free abelian group on codimension‑p points of U (so Γ(X, Z^p_X)=Z^p(X)).

There is the Gersten (a.k.a. Bloch–Quillen) complex of Zariski sheaves 0 → K_p → ⊕{x∈X^{(0)}} i{x,} K_p(k(x)) → ⊕{x∈X^{(1)}} i{x,} K_{p-1}(k(x)) → … → → ⊕{x∈X^{(p-1)}} i{x,} K_1(k(x)) → Z^p_X → 0 which is exact for regular X. Here K_i denotes Quillen K‑theory and the differentials are the tame residue maps. For p=1 this collapses to the familiar short exact sequence 0 → O_X^ → K_X^* → Z^1_X → 0, i.e. “Cartier divisors” are K_X^/O_X^.

For p≥2 there is no analogue with only two terms; the last map to Z^p_X is surjective but its kernel is the image of the previous term ⊕ i_{x,*}K_1(k(x)), not the image of a single sheaf map. Taking global sections gives the exact sequence 0 → K_p(X) → K_p(k(X)) → ⊕{x∈X^{(1)}} K{p-1}(k(x)) → … → → ⊕_{x∈X^{(p)}} Z → CH^p(X) → 0, so the Chow group CH^p(X) is the cokernel at the last step, and (Bloch–Quillen) CH^p(X) ≅ H^p(X, K_p) (Zariski).

Thus:

• There is no standard “sheaf of Cartier p‑cycles” whose global sections give a reasonable group Cart^p(X) analogous to Γ(K_X^/O_X^) for p=1.
• The correct replacement is the K‑theoretic Gersten complex, and the higher analogue of “Cartier divisors modulo linear equivalence” is CH^p(X)=H^p(X, K_p), which already involves higher cohomology, not just global sections.

An alternate geometric notion one might consider is “locally complete intersection (lci) p‑cycles,” i.e. cycles represented locally by zero loci of regular sections of rank‑p vector bundles. These do generalize effective Cartier divisors (p=1), but for p≥2 they form a proper subclass of all codimension‑p cycles on a regular scheme, and there is no natural quotient-of-sheaves description for their group either.

References:

• Fulton, Intersection Theory, Appendix B (Gersten complexes and the divisor sequence), §1.6–1.7 for CH^p.
• Weibel, The K‑book, §6.6 (Gersten for regular schemes), §24.1 (Bloch–Quillen isomorphism).