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Let $X$ be a scheme, let $\text{Div} (X)$ be its Weil divisors. For two Weil divisors $D,E$ we can easily define their gcd and lcm: $$\gcd(D,E)=\sum\min\{\text{ord}_F D,\text{ord}_F E\}F,\\ \text{lcm}(D,E)=\sum\max\{\text{ord}_F D,\text{ord}_F E\}F,$$ where the sums range over all prime divisors $F$ in $\text{Div}(X)$, and $\text{ord}_F D$ denotes the coefficient of $F$ of $D$.

I wonder if there already are some definitions for gcd and lcm for, say Cartier divisors (this might be easy), subschemes of distinct codimensions, or equivalent, their corresponding ideal sheaves.

A stupid observation is that, if $D,E$ are two small "disjoint" ample divisors, $A$ an ample divisor. If $A-D$ and $A-E$ are ample, then $A-\text{lcm}(D,E)$ remains ample. So I think of the corresponding sheaves $\mathscr{L}(A)\otimes \mathscr{L}(-\text{lcm}(D,E))$ remains ample, then $\mathscr{L}(-\text{lcm}(D,E))$ should be viewed as a sheaf lcm of $\mathscr{L}(-D)$ and $\mathscr{L}(-E)$. I wonder for arbitrary ideal sheaves, something similar is already been defined or what people should expect for such a definition.

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    $\begingroup$ For ideals, natural operations are $I+J$ and $I \cap J$ which can be taken as definition of the gcd and the lcm, respectively. You can certainly do that for arbitrary ideal sheaves but then it depends what type of divisors you want to consider. Effective Cartier divisors are locally given by an equation $f=0$ where $f$ is a non-zero divisor. Taking two such divisors $f=0$ and $g=0$, the problem is that the ideals $(f,g)$ and $(f) \cap (g)$ are not always locally principal e.g. for $f=2$ and $g=x$ in $\mathbf{Z}[x]$, the subscheme corresponding to $(2,x)$ has codimension 2, we may want (...) $\endgroup$
    – François Brunault
    Commented Apr 20, 2023 at 17:10
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    $\begingroup$ (...) to throw it away if we want to stay with Cartier divisors. In general, gcds and lcms already do not always exist in rings. $\endgroup$
    – François Brunault
    Commented Apr 20, 2023 at 17:11
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    $\begingroup$ This actually shows up in a number of papers, under a different name. For Weil divisors on a normal variety, for instance $D \wedge E$ is what you call gcd while $D \vee E$ is what you call lcm. $\endgroup$
    – Karl Schwede
    Commented Apr 22, 2023 at 0:24

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