BF perfect obstruction theory and intrinsic-normal-cone class

Assume the Axiom of Choice carried by the derived-scheme and cotangent-complex suppliers. Let $X$ be a separated Deligne–Mumford stack locally of finite type over a field $k$, and write $L_X=L_{X\mathrm{/}k}$ for the cotangent complex obtained by étale descent from the scheme complexes on atlases. A perfect obstruction theory is a morphism $\varphi :E^{\bullet }\to L_X$ in the derived category of quasi-coherent étale sheaves such that $E^{\bullet }$ is perfect of amplitude $[-1,0]$, $h^0(\varphi )$ is an isomorphism, and $h^{-1}(\varphi )$ is surjective. Its virtual-rank function is the well-defined locally constant integer-valued function $n:X\to \mathbb{Z}$ given by $n(x)={\mathrm{rk}}_x(E^0)-{\mathrm{rk}}_x(E^{-1})$ in any local two-term vector-bundle presentation. Different open-and-closed components of $X$ may have different virtual ranks. For each integer $r$, the rank locus $X_r=n^{-1}(r)$ is open and closed, and the obstruction theory, its vector-bundle stack and the intrinsic cone restrict to it. This family of loci need not be finite when $X$ is not quasi-compact. No Chow theory of locally finite cycles, or virtual Chow class on a non-finite-type stack, is asserted here; the Chow suppliers below cover finite-type Deligne–Mumford stacks.

For an étale-local closed embedding $U\hookrightarrow M$ into a smooth scheme, its normal cone is $C_{U\mathrm{/}M}={\mathrm{Spec}}_U\left({⨁}_{j\ge 0}{I}^j\mathrm{/}{I}^{j+1}\right)$, where $I$ is the embedding ideal. Quotienting by the action of $T_M{\mathrm{\mid }}_U$ gives local cone-stack charts that glue to the intrinsic normal cone $C_X\hookrightarrow h^1\mathrm{/}{h}^0(L_X^{\vee })$. A perfect obstruction theory induces a closed immersion $C_X\hookrightarrow h^1\mathrm{/}{h}^0((E^{\bullet }{)}^{\vee })$; for a two-term presentation this vector-bundle stack is $[(E^{-1}{)}^{\vee }\mathrm{/}(E^0{)}^{\vee }]$. For the virtual-class construction assume additionally that $X$ is of finite type over $k$; the locally-finite-type hypotheses above suffice for the POT and intrinsic cone, but do not provide a finite fundamental cycle. If the source's stated global two-term vector-bundle resolution $F^{\bullet }=[F^{-1}\to F^0]$ exists, put $F_1=(F^{-1}{)}^{\vee }$ and pull back $C_X$ to obtain a closed cone $C(F^{\bullet })\subset F_1$. Let $X_{a,b}\subset X$ be the open-and-closed locus where $\mathrm{rk}(F^{-1})=a$ and $\mathrm{rk}(F^0)=b$, and let $S$ be the finite set of pairs $(a,b)$ for which this locus is nonempty. Finiteness follows from finite type and local constancy of the two ranks. Write $i_{a,b}:X_{a,b}\hookrightarrow X$, $F_{1,a,b}=F_1{\mathrm{\mid }}_{X_{a,b}}$, and $C_{a,b}=C(F^{\bullet }){\mathrm{\mid }}_{X_{a,b}}$. On this locus the cone cycle has dimension $b$, so its zero-section Gysin image has dimension $b-a$. Define the componentwise Behrend–Fantechi class by $[X,E^{\bullet }{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}:={\sum }_{(a,b)\in S}(i_{a,b}{)}_{\ast }{0}_{F_{1,a,b}}^{!}[C_{a,b}]\in A_{\ast }(X{)}_{\mathbb{Q}}\mathrm{.}$ Here each summand belongs to $A_{b-a}(X{)}_{\mathbb{Q}}$, using the rational Chow and refined-Gysin conventions below. The finite open-and-closed decomposition respects cycles and rational equivalence: each integral cycle and each principal-divisor relation lies in one piece. Equivalently, for the canonical rank loci $X_r=n^{-1}(r)$, the restriction of the virtual class to $X_r$ is homogeneous in $A_r(X_r{)}_{\mathbb{Q}}$. If $n$ is globally constant with value $r$, this recovers $[X,E^{\bullet }{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\in A_r(X{)}_{\mathbb{Q}}$. If $X$ is empty the sum is zero. The definition uses a chosen global resolution; independence on every constant-virtual-rank locus, and hence of the finite sum, is the separate theorem on the A page.

Derived schemes and the cotangent complex of a morphism

The cotangent complex of a morphism of schemes

Classical Deligne–Mumford stack and cotangent complex

Rational Chow groups and zero-section Gysin maps