Support ideals, mixed products and pullback

For $E\in {\mathrm{Perf}}_T(X)$ and $F\in {\mathrm{Perf}}_S(X)$, at $x\mathrm{\notin }T\cap S$ at least one factor has acyclic stalk. A derived tensor with a zero object is zero: compute with K-flat stalk representatives and use preservation of quasi-isomorphisms. Hence $(E{\otimes }^{L}F{)}_x$ is acyclic there. This proves the mixed pairing, including the special case of a $\mathrm{Perf}(X)$ factor by taking its support bound to be $X$. It is exact in each variable by [F1], since the support subcategories inherit finite stable operations.

The module associativity and the unit ${\mathcal{O}}_X$ action are restrictions of the ambient coherent tensor. These operations land in the supported full subcategories by step 1.1, so their higher coherences also restrict. The unit for the action lies in $\mathrm{Perf}(X)$; it is not asserted to lie in ${\mathrm{Perf}}_T(X)$. This proves the precise tensor-ideal and module statements.

For $T\subseteq Y$ and $E\in {\mathrm{Perf}}_T(Y)$, if $x\mathrm{\notin }{f}^{-1}T$ and $y=f(x)$, then $E_y\simeq 0$. The stalk formula $(L{f}^{\ast }E{)}_x\simeq {\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}^L{E}_y$, obtained from the K-flat pullback construction, makes $(L{f}^{\ast }E{)}_x$ zero. Thus pullback has the stated support. Its symmetric monoidal comparison from [F1] commutes with the ambient actions and mixed products; fullness retains these coherences on supported objects. This proves the Statement for arbitrary subsets, using open restriction only when the complement is open.