$\mathbb{P}_{+}\times_{X}\mathbb{P}(\mathscr{W}^{\vee})$, $0\leq i\leq n-1$:

$\overline{\mathscr{G}}_{i-1}(\mathscr{E})\to\overline{\mathscr{G}}_{i}(\mathscr{E})\to\Sigma^{i}\,\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee})\boxtimes_{X}(\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}(-i)\otimes_{\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}}\mathscr{E})$

for which $\overline{\mathscr{G}}_{-1}(\mathscr{E})\simeq 0$ and
$\overline{\mathscr{G}}_{n-1}(\mathscr{E})\simeq
j_{+\,*}\,j_{+}^{*}(\mathscr{O}_{\mathbb{P}_{+}}\boxtimes_{X}\mathscr{E})$.
From the pullback square

${\mathbb{P}_{+}\times_{X}\mathbb{P}(\mathscr{W}^{\vee})}$${\mathbb{P}(\mathscr{W}^{\vee})}$${\mathbb{P}_{+}}$${X,}$$\scriptstyle{q_{+}}$$\scriptstyle{p_{+}}$$\scriptstyle{\mathrm{pr}_{\mathbb{P}(\mathscr{W}^{\vee})}}$$\scriptstyle{\pi_{+}}$

we obtain that there are canonical equivalences

$\displaystyle
q_{+*}\big{(}\Sigma^{i}\,\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee})\boxtimes_{X}(\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}(-i)\otimes_{\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}}\mathscr{E})\big{)}$
$\displaystyle\simeq\Sigma^{i}\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee})\otimes_{\mathscr{O}_{\mathbb{P}_{+}}}\pi_{+}^{*}\big{(}\mathrm{pr}_{\mathbb{P}(\mathscr{W}^{\vee})\,*}(\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}(-i)\otimes_{\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}}\mathscr{E})\big{)}$
$\displaystyle\simeq\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee})\otimes_{\mathscr{O}_{\mathbb{P}_{+}}}\pi_{+}^{*}\big{(}\Sigma^{i}\,\underline{\mathrm{Map}}_{X}(\mathscr{O}_{\mathbb{P}(\mathscr{W}^{\vee})}(i),\mathscr{E})\big{)}.$

Therefore, by pushing-forward the above fiber sequence along $q_{+}$ and
setting
$\mathscr{G}_{i}(\mathscr{E})=q_{+\,*}\overline{\mathscr{G}}_{i}(\mathscr{E})\in\mathrm{Perf}(\mathbb{P}_{+})$,
we obtain the desired fiber sequences of assertion (1). ∎

###### Proof of Theorem 7.5 in the local situation.

We first show that the functor $\Phi$ is fully faithful. By virtue of Lemma
7.11, it suffices to show that the natural transformation
$\Phi^{L}\circ\Phi(\mathscr{E}_{i})\to\mathscr{E}_{i}$ is an equivalence for
all $\mathscr{E}_{i}=\iota_{-}^{*}(\bigwedge^{i}\mathscr{R}_{-})$, where
$0\leq i\leq m-1$. By virtue of Lemma 7.12 (1) (applied to
$\mathscr{E}=\bigwedge^{i}\mathscr{R}_{-}$) we obtain canonical equivalences
$\mathscr{G}_{n-1}(\bigwedge^{i}\mathscr{R}_{-})\simeq\Phi(\mathscr{E}_{i})$
and

$\displaystyle\mathscr{G}_{0}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\simeq\ldots\simeq\mathscr{G}_{i-1}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\simeq
0;$
$\displaystyle\mathscr{G}_{i}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\simeq\ldots\simeq\mathscr{G}_{m-1}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\simeq\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee}).$

Furthermore, for $0\leq j\leq n-m-1$, there are fiber sequences in
$\operatorname{QCoh}(\mathbb{P}_{+})$:

$\mathscr{G}_{m-1+j}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\to\mathscr{G}_{m+j}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\to\iota_{+}^{*}(\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee})\otimes_{\mathscr{O}_{\mathbb{P}_{+}}}\pi_{+}^{*}(\mathscr{M}_{m+j}(\bigwedge\nolimits^{i}\mathscr{R}_{-})).$

By virtue of Lemma 7.12 (2i) (applied to the case where
$\mathscr{E}=\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee}$), we obtain

$\Phi^{L}\big{(}\iota_{+}^{*}(\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee})\otimes_{\mathscr{O}_{\mathbb{P}_{+}}}\pi_{+}^{*}(\mathscr{M}_{m+j}(\bigwedge\nolimits^{i}\mathscr{R}_{-}))\big{)}\simeq
0\quad\text{for}\quad 0\leq j\leq n-m-1.$

Therefore, we obtain canonical equilvalences

$\Phi^{L}\circ\Phi(\mathscr{E}_{i})=\Phi^{L}(\mathscr{G}_{n-1}(\bigwedge^{i}\mathscr{R}_{-}))\simeq\ldots\simeq\Phi^{L}(\mathscr{G}_{m-1}(\bigwedge^{i}\mathscr{R}_{-}))\simeq\Phi^{L}(\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee}))\simeq\mathscr{E}_{i},$

where the last equivalence follows from Lemma 7.12 (2i). This proves that
$\Phi$ is fully faithful.

Next, we prove that the essential images of $\Phi$ and $\Psi_{j}$ are
semiorthogonal, where $1\leq j\leq\delta$. For all
$\mathscr{A}\in\operatorname{QCoh}(X)$ and
$\mathscr{B}\in\operatorname{QCoh}(\mathbb{P}_{-})$, we wish to show that the
mapping space

$\mathrm{Map}_{\operatorname{QCoh}(\mathbb{P}_{+})}(\Psi_{j}(\mathscr{A}),\Phi(\mathscr{B}))\simeq\mathrm{Map}_{\operatorname{QCoh}(\mathbb{P}_{-})}(\Phi^{L}(\pi_{+}^{*}(\mathscr{A})\otimes\mathscr{O}_{+}(j)),\mathscr{B})$

is contractible. Since
$\Phi^{L}(\pi_{+}^{*}(\mathscr{A})\otimes\mathscr{O}_{+}(j)\simeq\pi_{+}^{*}(\mathscr{A})\otimes\Phi^{L}(\iota_{+}^{*}(\mathscr{O}_{\mathbb{P}(\mathscr{V})}(j)))$,
it suffices to show

$\Phi^{L}(\iota_{+}^{*}(\mathscr{O}_{\mathbb{P}(\mathscr{V})}(j)))\simeq
0\qquad\text{for all}\qquad 1\leq j\leq n-m.$

This follows directly from Lemma 7.12 (2ii).

Finally, let $\mathcal{C}$ denote the smallest stable subcategory of
$\operatorname{QCoh}(\mathbb{P}_{+})$ that contains the essential images
$\operatorname{Im}\Phi$ and $\\{\operatorname{Im}\Psi_{j}\\}_{1\leq
j\leq\delta}$. We wish to show that
$\mathcal{C}=\operatorname{QCoh}(\mathbb{P}_{+})$. By virtue of Lemma 7.11, it
suffices to show that for each $0\leq i\leq n-1$,
$\iota_{+}^{*}(\bigwedge^{i}\mathscr{R}_{+}^{\vee})\in\mathcal{C}$. Since
$\mathscr{R}_{+}$ is a vector bundle on $\mathbb{P}(\mathscr{V})$ of rank
$(n-1)$, we obtain that for all $0\leq j\leq\delta-1$,

$\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee}\simeq(\bigwedge\nolimits^{\delta-1-j}\mathscr{R}_{+})\otimes\mathscr{O}_{\mathbb{P}(\mathscr{V})}(1)\otimes(\mathrm{pr}_{\mathbb{P}(\mathscr{V})}^{*}\det\mathscr{V}).$

By virtue of Proposition 6.19, the sequence of vector bundles

$\bigwedge\nolimits^{\delta-1}\mathscr{R}_{+},\ldots,\mathscr{R}_{+},\mathscr{O}_{\mathbb{P}(\mathscr{V})}$

is left dual to the relative exceptional sequence
$\mathscr{O}_{\mathbb{P}(\mathscr{V})},\mathscr{O}_{\mathbb{P}(\mathscr{V})}(1),\ldots,\mathscr{O}_{\mathbb{P}(\mathscr{V})}(\delta-1)$.
Therefore, by virtue of Remark 6.18, each
$\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee}$ is an iterated extensions of
objects of the form
$\mathscr{O}_{\mathbb{P}(\mathscr{V})}(s)\otimes(\mathrm{pr}_{\mathbb{P}(\mathscr{V})}^{*}\mathscr{F}_{s})$,
where $1\leq s\leq\delta$ and $\mathscr{F}_{s}\in\operatorname{QCoh}(X)$. In
particular, we obtain that for all $\mathscr{M}\in\operatorname{QCoh}(X)$,

$\iota_{+}^{*}(\bigwedge\nolimits^{m+j}\mathscr{R}_{+}^{\vee})\otimes_{\mathscr{O}_{\mathbb{P}_{+}}}(\pi_{+}^{*}\mathscr{M})\in\mathcal{C}\quad\text{for}\quad
0\leq j\leq\delta-1.$

On the other hand, if we consider $\Phi(\mathscr{E}_{i})$, where
$\mathscr{E}_{i}=\iota_{-}^{*}(\bigwedge^{i}\mathscr{R}_{-})$, then
$\mathscr{G}_{n-1}(\bigwedge^{i}\mathscr{R}_{-})\simeq\Phi(\mathscr{E}_{i})\in\mathcal{C}$
by construction. By inverse induction and the fact that $\mathcal{C}$ is
stable under the formation of fibers, we obtain that
$\mathscr{G}_{m+k-1}(\bigwedge^{i}\mathscr{R}_{-})\in\mathcal{C}$ for all
$0\leq k\leq\delta$. In particular, we obtain

$\iota_{+}^{*}(\bigwedge\nolimits^{i}\mathscr{R}_{+}^{\vee})\simeq\mathscr{G}_{m-1}(\bigwedge\nolimits^{i}\mathscr{R}_{-})\in\mathcal{C}$

for all $0\leq i\leq m-1$. This proves
$\mathcal{C}=\operatorname{QCoh}(\mathbb{P}_{+})$. Combined with the proof of
the preceding subsection, this completes the proof of Theorem 7.5. ∎

###  7.3. Applications to classical examples

In this subsection, we apply Theorem 7.5 to classical examples. We only list
examples in lower dimensions in each situation; all these results have direct
analogues in higher dimensions.

####  7.3.1. Criteria for derived projectivizations and symmetric powers being
classical

We first give criteria for the derived projectivizations and symmetric powers
being classical. For simplicity of exposition, we make the following
assumption throughout this subsection:

  1. $(\star)$

We let $X$ be an integral Cohen–Macaulay scheme, let $\mathscr{W}$ and
$\mathscr{V}$ be vector bundles on $X$ of rank $m$ and $n$ where $1\leq m\leq
n$, and let $\sigma\colon\mathscr{W}\to\mathscr{V}$ be a map of
$\mathscr{O}_{X}$-modules which is injective at the generic point of $X$. Let
$\mathscr{F}=\operatorname{cofib}(\mathscr{W}\xrightarrow{\sigma}\mathscr{V})\in\mathrm{Perf}(X)^{\mathrm{cn}}$.

Under assumption ($\star$ ‣ 7.3.1), $\Sigma\mathscr{F}^{\vee}$ is represented
by the shifted dual complex

$\Sigma\mathscr{F}^{\vee}\simeq\operatorname{cofib}(\mathscr{V}^{\vee}\xrightarrow{\sigma^{\vee}}\mathscr{W}^{\vee})\in\mathrm{Perf}(X)^{\mathrm{cn}}.$

(Thus by possibly replacing $\mathscr{F}$ with $\Sigma\mathscr{F}^{\vee}$,
there is no loss of generality by assuming $m\leq n$.)

######  Remark 7.13.

The assumption ($\star$ ‣ 7.3.1) is not necessary: all the results of this
section remain true for a general scheme $X$ and a discrete finite type quasi-
coherent sheaf $\mathscr{F}$ on $X$ of homological dimension $\leq 1$, as long
as one replaces the conditions on codimensions of degeneracy loci with
conditions on polynomial depths of the corresponding Fitting ideals; see [J21,
§2].

######  Notation 7.14.

Under assumption ($\star$ ‣ 7.3.1), for each integer $i\geq 0$, we let
$X_{i}\subseteq X$ denote the closed subscheme of $X$ defined by all
$(m+1-i)\times(m+1-i)$-minors of the map $\sigma$; Equivalently, $X_{i}$ is
$i$-th degeneracy locus of the discrete sheaf $\operatorname{Coker}(\sigma)$,
characterized by the following property: for each point $x\in X_{i}$ (closed
or not), $\operatorname{rank}(\operatorname{Coker}(\sigma)|_{x})\geq n-m+i$.
Therefore we have a sequence of closed subschemes of $X$:

$\emptyset=X_{m+1}\subseteq X_{m}\subseteq\cdots\subseteq
X_{i}\subseteq\cdots\subseteq X_{1}\subseteq X.$

For a nonempty closed subscheme $Z\subseteq X$, we let
$\operatorname{codim}_{Z}(X)$ denote the minimal of the codimensions of
irreducible components of $Z$ inside $X$ (since $X$ is Noetherian and
equidimensional, $\operatorname{codim}_{Z}(X)$ is well-defined). In
particular, for an integer $d$, $\operatorname{codim}_{X}(Z)\geq d$ holds if
and only if $\operatorname{codim}_{X}(Z_{j})\geq d$ holds for each irreducible
component $Z_{j}$ of $Z$.

######  Remark 7.15 (Expected codimensions).

In the situation of Notation 7.14, a theorem of Macaulay–Eagon–Northcott
implies that for any irreducible component $Z$ of $X_{i}$,

$\operatorname{codim}_{X}(Z)\leq(n-m+i)i\qquad i=1,2,\ldots,m.$

Moreover, a theorem of Hochster–Eagon implies that, if
$\operatorname{codim}_{X}(X_{i})$ achieves the maximal possible value
$(n-m+i)i$ (that is, $X_{i}$ is equidimensional and each irreducible component
of $X_{i}$ has codimension $(n-m+i)i$ in $X$), then $X_{i}$ is a Cohen-
Macaulay subscheme (see, for example, [Ei, Exercise 10.9, §18.5, & Theorem
18.18] for these statements for commutative rings, and [J21, §2.2] for a
summary in the geometric situation). The maximal possible value $(n-m+i)i$ for
$\operatorname{codim}_{X}(X_{i})$ is often referred to as the expected
codimension of $i$-th degneracy locus $X_{i}$ in $X$; it is achieved in the
situation when $\sigma$ is a “generic” map.

First, we give criteria for the derived projectivizations to be classical.

######  Lemma 7.16.

Under assumption ($\star$ ‣ 7.3.1) and let $X_{i}$ be defined as in Notation
7.14. Then:

  1. (1)

$\mathbb{P}(\mathscr{F})$ is classical if and only if the following condition
is satisfied:

($a$) $\operatorname{codim}_{X}(X_{i})\geq i\quad\text{for}\quad
i=1,2,\ldots,m.$

If $\operatorname{rk}\mathscr{F}=n-m\geq 1$, then $\mathbb{P}(\mathscr{F})$ is
classical and irreducible if and only if the following condition is satisfied:

($a^{\prime}$) $\operatorname{codim}_{X}(X_{i})\geq i+1\quad\text{for}\quad
i=1,2,\ldots,m.$

  2. (2)

$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is classical if and only if the
following condition is satisfied:

($b$) $\operatorname{codim}_{X}(X_{i})\geq n-m+i\quad\text{for}\quad
i=1,2,\ldots,m.$

$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is classical and irreducible if and
only if the following condition is satisfied:

($b^{\prime}$)
$\operatorname{codim}_{X}(X_{1})=n-m+1\quad\text{and}\quad\operatorname{codim}_{X}(X_{i})\geq
n-m+i+1\quad\text{for}\quad i=2,\ldots,m.$

  3. (3)

Let $\widehat{Z}$ be the universal incidence locus of
$\mathbb{P}(\mathscr{F})$ and $\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ over $X$
of Notation 7.4. Then $\widehat{Z}$ is classical if and only if the following
condition is satisfied:

($c$) $\operatorname{codim}_{X}(X_{i})\geq n-m+2i-1\quad\text{for}\quad
i=1,2,\ldots,m.$

$\widehat{Z}$ is classical and irreducible if and only if the following
condition is satisfied:

($c^{\prime}$)
$\operatorname{codim}_{X}(X_{1})=n-m+1\quad\text{and}\quad\operatorname{codim}_{X}(X_{i})\geq
n-m+2i\quad\text{for}\quad i=2,\ldots,m.$

(Notice that if $\widehat{Z}$ is classical, then $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are classical, and $\widehat{Z}$ is
canonically equivalent to the classical fiber product
$(\mathbb{P}(\mathscr{F})\times_{X}\mathbb{P}(\Sigma\mathscr{F}^{\vee}))_{\mathrm{cl}}$.)

###### Proof.

We first prove the assertions about $\mathbb{P}(\mathscr{F})$. By virtue of
Proposition 4.33, the closed immersion
$\mathbb{P}(\mathscr{F})\hookrightarrow\mathbb{P}(\mathscr{V})$ is the derived
zero locus of a section $s$ of the vector bundle
$\mathscr{W}^{\vee}\otimes\mathscr{O}_{\mathbb{P}(\mathscr{V})}(1)$. Since
$\mathbb{P}(\mathscr{V})$ is an integral Cohen–Macaulay scheme, therefore for
each irreducible component $Z$ of $\mathbb{P}_{\mathrm{cl}}(\mathscr{F})$,
$\operatorname{codim}_{\mathbb{P}(\mathscr{V})}(Z)\leq m$. Moreover,
$\mathbb{P}(\mathscr{F})$ is classical if and only if $s$ is a regular
section, if and only if
$\operatorname{codim}_{\mathbb{P}(\mathscr{V})}(\mathbb{P}_{\mathrm{cl}}(\mathscr{F}))\geq
m$. Since the restriction of the projection from the classical
projectivization $\mathbb{P}_{\mathrm{cl}}(\mathscr{F})\to X$ is smooth with
fiber $\mathbb{P}^{n-m+i-1}$ over the subscheme $X_{i}\backslash X_{i+1}$
(Remark 4.22), it follows that
$\operatorname{codim}_{\mathbb{P}(\mathscr{V})}(\mathbb{P}_{\mathrm{cl}}(\mathscr{F}))\geq
m$ if and only if condition ($a$) is satisfied. If we assume $n-m\geq 1$ and
$\mathbb{P}(\mathscr{F})$ is classical, then $\mathrm{pr}^{-1}(X\backslash
X_{1})$ is a $\mathbb{P}^{n-m-1}$-bundle over $X\backslash X_{1}$. Hence
$\mathrm{pr}^{-1}(X\backslash X_{1})$ has codimension $m$ inside
$\mathbb{P}(\mathscr{V})$. Therefore, in this case, $\mathbb{P}(\mathscr{F})$
is irreducible if and only if ($a^{\prime}$) is satisfied. The same argument
proves the assertions about $\mathbb{P}(\Sigma\mathscr{F}^{\vee})$. By virtue
of Lemma 7.3, the assertions about $\widehat{Z}$ can be proved in a similar
manner. ∎

######  Remark 7.17.

In the situation of Lemma 7.16:

  * •

If $\operatorname{rk}\mathscr{F}=n-m\geq 1$, then we have the following
implications:

$\eqref{eqn:condition:PF}\impliedby\eqref{eqn:condition:irr:PF}\impliedby\eqref{eqn:condition:PK}\impliedby\eqref{eqn:condition:irr:PK}\impliedby\eqref{eqn:condition:hatZ}\impliedby\eqref{eqn:condition:irr:hatZ}.$

In general, each of the above implications “$\impliedby$” is strict (for
example, if $\operatorname{rk}\mathscr{F}$ and $m$ are large enough, say, if
$\operatorname{rk}\mathscr{F}\geq 2$ and $m\geq 3$).

  * •

If $\operatorname{rk}\mathscr{F}=n-m=0$, then we have the following
implications:

$\eqref{eqn:condition:PF}\iff\eqref{eqn:condition:PK}\impliedby\eqref{eqn:condition:irr:PK}\impliedby\eqref{eqn:condition:hatZ}\impliedby\eqref{eqn:condition:irr:hatZ}.$

In general, each of the above implications “$\impliedby$” is strict (for
example, if $m\geq 3$).

It is easy to construct numerous counter-examples for the inverse of eahc of
the above implications “$\impliedby$”, and we will consider some of such
examples in the next subsection.

Next, we give criteria for the derived symmetric powers to be classical.

######  Lemma 7.18.

Under assumption ($\star$ ‣ 7.3.1) and let $X_{i}$ be defined as in Notation
7.14. Then:

  1. (1)

Let $d\geq 0$ be an integer. Then the canonical map

$\operatorname{Sym}^{d}_{X}(\mathscr{F})\to\pi_{0}(\operatorname{Sym}^{d}_{X}(\mathscr{F}))\simeq
S^{d}_{X}(\operatorname{Coker}(\sigma))$

is an equivalence if and only if the following condition is satisfied:

$\operatorname{codim}_{X}(X_{i})\geq i\quad\text{for}\quad
i=1,2,\ldots,\min\\{m,d\\}.$

  2. (2)

Let $d\geq 0$ be an integer.

    1. (i)

If $d\geq n-m+1$. Then the canonical map

$\operatorname{Sym}^{d}_{X}(\Sigma\mathscr{F}^{\vee})\to\pi_{0}(\operatorname{Sym}^{d}_{X}(\Sigma\mathscr{F}^{\vee}))\simeq
S^{d}_{X}(\operatorname{Coker}(\sigma^{\vee}))$

is an equivalence if and only if the following condition is satisfied:

$\operatorname{codim}_{X}(X_{i})\geq n-m+i\quad\text{for}\quad
i=1,2,\ldots,\min\\{m,d\\}.$

    2. (ii)

If $0\leq d\leq n-m$. Assume that the following condition is satisfied:

$\operatorname{codim}_{X}(X_{1})=n-m+1.$

Then there are canonical equivalences

$\pi_{i}(\operatorname{Sym}^{d}_{X}(\Sigma\mathscr{F}^{\vee}))\simeq\begin{cases}\operatorname{Sym}^{d}_{\mathrm{cl}}(\operatorname{Coker}(\sigma^{\vee}))&i=0.\\\
\bigwedge\nolimits^{n-m-d}_{\mathrm{cl}}(\operatorname{Coker}(\sigma))\otimes(\det\mathscr{F})^{\vee}&i=d.\\\
0&\text{otherwise}.\end{cases}$

###### Proof.

We first prove assertion (1). We will assume $n>m$, since the case $n=m$ can
be reduced to assertion (2). By virtue of Corollary 2.30,
$\operatorname{Sym}_{X}^{d}(\mathscr{F})$ is canonically represented by the
complex of vector bundles
$\mathbf{S}^{d}(X,\mathscr{W}\xrightarrow{\sigma}\mathscr{V})$ (defined by
formula (2.2)). For each $0\leq i\leq\min\\{m,d\\}$, we let $F_{i}$ denote the
$i$th term of the complex $\mathbf{S}^{d}(X,\sigma)$, let $d_{i}\colon
F_{i}\to F_{i-1}$ denote the differential of $\mathbf{S}^{d}(X,\sigma)$, and
let $r_{i}=\sum_{j=i}^{d}(-1)^{j-i}\operatorname{rank}F_{j}$ be the “expected
rank” of $d_{i}$. Let $\mathscr{I}_{X_{i}}$ denote the ideal of
$X_{i}\subseteq X$, and let $I_{r_{i}}(d_{i})$ denote the ideal generated by
$(r_{i}\times r_{i})$-minors of the differential $d_{i}$ (see [Ei, §20.2]).
From the same argument as the proof of [Ei, Theorem A2.1], we know that

$\sqrt{I_{r_{i}}(d_{i})}=\sqrt{\mathscr{I}_{X_{i}}}\quad\text{for}\quad
i=1,2,\ldots,\min\\{m,d\\}.$

(This can also be deduced from the general result [AT19, Theorem 6.1$(b)$]).
In particular,

$0pt(I_{r_{i}}(d_{i}))=0pt(\mathscr{I}_{X_{i}})=\operatorname{codim}_{X}(X_{i})$

since $X$ is Cohen–Macaulay. On the other hand, from Buchsbaum–Eisenbud’s
acyclicity criterion ([Ei, Theorem 20.9]), $\mathbf{S}^{d}(X,\sigma)$ is a
resolution of $S^{d}_{X}(\operatorname{Coker}(\sigma))$ if and only if

$0pt(I_{r_{i}}(d_{i}))\geq i\quad\text{for}\quad i=1,2,\ldots,\min\\{m,d\\}.$

This proves assertion (1) in the case where $n>m$.

We next prove assertion (2i). By virtue of Corollary 2.30,
$\operatorname{Sym}_{X}^{d}(\Sigma\mathscr{F}^{\vee})$ is canonically
represented by
$\mathbf{S}^{d}(X,\mathscr{V}^{\vee}\xrightarrow{\sigma^{\vee}}\mathscr{V}^{\vee})$.
For each $0\leq i\leq\min\\{m,d\\}$, we let $F_{i}^{\prime}$ denote the $i$th
term of the complex $\mathbf{S}^{d}(X,\sigma^{\vee})$, let
$d_{i}^{\prime}\colon F_{i}^{\prime}\to F_{i-1}^{\prime}$ denote the
differential, and let
$r_{i}^{\prime}=\sum_{j=i}^{d}(-1)^{j-i}\operatorname{rank}F_{j}^{\prime}$.
From the proof of [Ei, Theorem A2.1], we know that

$\sqrt{I_{r_{i}^{\prime}}(d_{i}^{\prime})}=\begin{cases}\sqrt{\mathscr{I}_{X_{1}}}&\text{if}\quad
1\leq i\leq n-m+1\\\ \sqrt{\mathscr{I}_{X_{i-n+m}}}&\text{if}\quad n-m+2\leq
i\leq\min\\{m,d\\}.\end{cases}$

(This can also be deduced from [AT19, Theorem 6.1 $(a)$ & $(b)$].) Therefore,

$0pt(I_{r_{i}^{\prime}}(d_{i}^{\prime}))=\begin{cases}\operatorname{codim}_{X}(X_{1})&\text{if}\quad
1\leq i\leq n-m+1\\\ \operatorname{codim}_{X}(X_{i-n+m})&\text{if}\quad
n-m+2\leq i\leq\min\\{m,d\\},\end{cases}$

since $X$ is Cohen–Macaulay. By Buchsbaum–Eisenbud’s acyclicity criterion
again, we know that $\mathbf{S}^{d}(X,\sigma^{\vee})$ is a resolution of
$S_{X}^{d}(\operatorname{Coker}(\sigma^{\vee}))$ if and only if

$0pt(I_{r_{i}^{\prime}}(d_{i}^{\prime}))\geq i\quad\text{for}\quad
i=1,2,\ldots,\min\\{m,d\\}.$

This proves assertion (2i) (and assertion (1) in the case where $n=m$).

Finally, to prove assertion (2ii), we consider the Eagon–Northcott complex
$\mathbf{EN}_{d}(X,\sigma)$ of Remark 5.9. Similarly as above, the desired
assertions follow from [Ei, Theorem A2.1] (where the complex $\mathcal{C}^{i}$
there is our $\mathbf{EN}_{i}(X,\sigma)$) and Buchsbaum–Eisenbud’s acyclicity
criterion. ∎

####  7.3.2. Examples violating ($a$)

By virtue of Lemma 7.16, as long as the pair $(X,\mathscr{F})$ in the
situation of ($\star$ ‣ 7.3.1) violates the condition ($a$),
$\mathbb{P}(\mathscr{F})$ is not classical. For example:

######  Example 7.19.

Let $X=\mathbb{A}^{1}=\operatorname{Spec}\mathbb{Z}[x]$, and
$\mathscr{F}=\mathscr{O}_{\mathbf{0}}^{\oplus k}$, where
$\mathbf{0}\in\mathbb{A}^{1}$ is the closed point define by the equation
$x=0$, $k\geq 1$ is an integer, and $\mathscr{O}_{\mathbf{0}}$ is the
structure sheaf of $\mathbf{0}$ regarded as a discrete
$\mathscr{O}_{X}$-module. In this case,
$(X_{1})_{{\operatorname{red}}}=\cdots=(X_{k-1})_{{\operatorname{red}}}=X_{k}=\\{\mathbf{0}\\}.$
Therefore, by virtue of Lemma 7.16 and Lemma 7.18, we obtain:

  * •

If $k=1$, then
$\mathbb{P}(\mathscr{O}_{\mathbf{0}})\simeq\mathbb{P}_{\mathrm{cl}}(\mathscr{O}_{\mathbf{0}})\simeq\\{\mathbf{0}\\}\subseteq\mathbb{A}^{1}$
is classical, and there is a canonical isomorphism
$\operatorname{Sym}_{\mathbb{A}^{1}}^{d}(\mathscr{O}_{\mathbf{0}})\simeq\operatorname{Sym}_{\mathrm{cl},\mathbb{A}^{1}}^{d}(\mathscr{O}_{\mathbf{0}})$
for all integers $d\geq 0$;

  * •

If $k\geq 2$, then $\mathbb{P}(\mathscr{O}_{\mathbf{0}})$ is not classical,
and the canonical map
$\operatorname{Sym}_{\mathbb{A}^{1}}^{d}(\mathscr{O}_{\mathbf{0}}^{\oplus
k})\to\operatorname{Sym}_{\mathrm{cl},\mathbb{A}^{1}}^{d}(\mathscr{O}_{\mathbf{0}}^{\oplus
k})$ is not an equivalence as long as $d\geq 2$.

More concretely, we can identify the classical projectivization with a
projective space

$\mathbb{P}_{\mathrm{cl}}(\mathscr{O}_{\mathbf{0}}^{\oplus
k})\simeq\mathbb{P}_{\mathbb{Z}}^{k-1}\to\\{\mathbf{0}\\}\subseteq\mathbb{A}^{1}.$

On the other hand, by virtue of Proposition 4.33, we can identify the derived
projectivization $\mathbb{P}(\mathscr{O}_{\mathbf{0}}^{\oplus k})$ as the
derived zero locus of
$\mathbb{A}^{1}\times\mathbb{P}^{k-1}=\operatorname{Spec}\mathbb{Z}[x]\times\operatorname{Proj}\mathbb{Z}[X_{1},\ldots,X_{k}]$
defined by the $k$ equations $xX_{1}=xX_{2}=\cdots=xX_{k}=0$. This derived
zero locus has underlying classical scheme
$\\{\mathbf{0}\\}\times\mathbb{P}^{k-1}$ defined by the equation $x=0$; but as
long as $k\geq 2$, it is equipped with a non-trivial derived structure.

####  7.3.3. Examples satisfying ($a$) but violating ($a^{\prime}$) and ($b$)

By virtue of Lemma 7.16, as long as the pair $(X,\mathscr{F})$ in the
situation of ($\star$ ‣ 7.3.1) satisfies condition ($a$) but violate condition
($a^{\prime}$) and ($b$), $\mathbb{P}(\mathscr{F})$ is classical but
reducible, and $\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is not classical. There
are numerous examples.

######  Example 7.20.

Let $X=\mathbb{A}^{1}=\operatorname{Spec}\mathbb{Z}[x]$, and let $\mathscr{F}$
denote the cokernel of the map

$\mathscr{O}_{\mathbb{A}^{1}}\xrightarrow{(x,x)^{T}}\mathscr{O}_{\mathbb{A}^{1}}^{\oplus
2}.$

  * •

By virtue of Lemma 7.16, $\mathbb{P}(\mathscr{F})$ is classical but
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is not.

  * •

By virtue of Lemma 7.18, the canonical map
$\operatorname{Sym}_{\mathbb{A}^{1}}^{d}(\mathscr{F})\to\operatorname{Sym}_{\mathrm{cl},\mathbb{A}^{1}}^{d}(\mathscr{F})$
is an equivalence for every $d\geq 0$, but the canonical map
$\operatorname{Sym}^{d}(\Sigma\mathscr{F}^{\vee})\to\operatorname{Sym}_{\mathrm{cl}}^{d}(\mathscr{O}_{\mathbf{0}})$
is not an equivalence as long as $d\geq 1$ (the case $d\geq 2$ is Lemma 7.18
(2i) and the case $d=1$ is obvious).

More concretely, $\mathbb{P}(\mathscr{F})\simeq\mathbb{A}^{1}\coprod_{\rm
pt}\mathbb{P}^{1}$ is a nodal curve which is the union of $\mathbb{A}^{1}$ and
$\mathbb{P}^{1}$ along a point, and $\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is
isomorphic the derived zero locus in $\mathbb{A}^{1}$ which is “cut out by the
equation $(x=0)$ twice” (Proposition 4.33). By virtue of Example 4.35,
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})\simeq\operatorname{Spec}\mathbb{Z}[\varepsilon]$,
where
$\mathbb{Z}[\varepsilon]=\operatorname{Sym}_{\mathbb{Z}}^{*}(\Sigma\,\mathbb{Z})\simeq\mathbb{Z}\oplus\mathbb{Z}\varepsilon$
is the ring of derived dual numbers (where we informally think of
$\varepsilon$ as the generator in degree $1$; beware that
$\mathbb{Z}[\varepsilon]$ is not isomorphic to the classical dual number
$\mathbb{Z}[x]/(x^{2})$). In particular,
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ has underlying classical scheme
$\\{\mathbf{0}\\}=\operatorname{Spec}\mathbb{Z}\simeq\operatorname{Spec}\mathbb{Z}[x]/(x)$,
but with a non-trivial derived structure given by the extra equation $(x=0)$.

Moreover, by virtue of Theorem 7.5, we obtain semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}\big{(}\mathbb{A}^{1}\coprod\nolimits_{\rm
pt}\mathbb{P}^{1}\big{)}$
$\displaystyle=\big{\langle}\operatorname{QCoh}(\mathbb{A}^{1}),~{}\operatorname{QCoh}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}\mathbb{A}^{1}\coprod\nolimits_{\rm
pt}\mathbb{P}^{1}\big{)}$
$\displaystyle=\big{\langle}\mathrm{Perf}(\mathbb{A}^{1}),~{}\mathrm{Perf}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle}.$

We could consider the following global version of Example 7.20:

######  Example 7.21 (Reducible schemes).

Let $X$ be a scheme, let $D\subseteq X$ be an effective Cartier divisor, and
let $f\in\operatorname{H}^{0}(X,\mathscr{O}(D))$ be the section determined by
$D$. Let $\mathscr{F}=\mathscr{O}_{X}\oplus\mathscr{O}_{D}$. Then:

  * •

By virtue of Proposition 4.33, $\mathbb{P}(\mathscr{F})\subseteq
X\times\mathbb{P}^{1}=X\times\operatorname{Proj}\mathbb{Z}[X_{0},X_{1}]$ is
the divisor defined by the equation $(X_{1}f=0)$. Spelling this out, we see
that $\mathbb{P}(\mathscr{F})$ is classical and is union of $X$ and
$\mathbb{P}_{D}^{1}=D\times\mathbb{P}^{1}$ along the divisor $D$.

  * •

By virtue of Example 4.35, we obtain
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})\simeq\mathbb{V}_{D}(\Sigma\,\mathscr{O}_{D}(-D))$
is not classical (and is equipped with a nontrivial derived structure
everywhere). More concretely, in any open subset $U\subseteq X$ for which
$\mathscr{O}_{X}(-D)|_{U}$ is a trivial line bundle, there is an equivalence
$\mathbb{P}_{U}(\Sigma\mathscr{F}^{\vee})\simeq(D\cap
U)\times\operatorname{Spec}\mathbb{Z}[\varepsilon]$, where
$\mathbb{Z}[\varepsilon]=\operatorname{Sym}_{\mathbb{Z}}^{*}(\Sigma\mathbb{Z})$
is the ring of derived dual numbers. More informally, we could think of
$\mathbb{V}_{D}(\Sigma\,\mathscr{O}_{D}(-D))$ as the total space ${\rm
Tot}(\mathcal{N}_{D/X}[-1])$ of $(-1)$-shifted normal bundle of $D$ in $X$, in
the sense [ACH, PTVV].

By virtue of Theorem 7.5, we obtain semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}\big{)}$
$\displaystyle=\big{\langle}\operatorname{QCoh}(X),~{}\operatorname{QCoh}(\mathbb{V}_{D}(\Sigma\,\mathscr{O}_{D}(-D)))\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}\big{)}$
$\displaystyle=\big{\langle}\mathrm{Perf}(X),~{}\mathrm{Perf}(\mathbb{V}_{D}(\Sigma\,\mathscr{O}_{D}(-D)))\big{\rangle}.$

######  Example 7.22 (Attaching $\mathbb{P}^{1}$’s to smooth points of
curves).

In the above Example 7.21, if we let $X=C$ be a reduced curve over a field
$\kappa$ (that is, a reduced separated finite type scheme over $\kappa$ whose
irreducible components all have dimension $1$), and let $p\in C(\kappa)$ be a
non-singular rational point, and let
$\mathscr{F}=\mathscr{O}_{C}\oplus\mathscr{O}_{p}$, then we obtain
semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}\big{(}C\coprod\nolimits_{p}\mathbb{P}^{1}\big{)}$
$\displaystyle=\big{\langle}\operatorname{QCoh}(C),~{}\operatorname{QCoh}(\operatorname{Spec}\kappa[\varepsilon])\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}C\coprod\nolimits_{p}\mathbb{P}^{1}\big{)}$
$\displaystyle=\big{\langle}\mathrm{Perf}(C),~{}\mathrm{Perf}(\operatorname{Spec}\kappa[\varepsilon])\big{\rangle},$

where
$\kappa[\varepsilon]=\operatorname{Sym}_{\kappa}(\Sigma\,\kappa)=\mathbb{Z}[\varepsilon]\otimes_{\mathbb{Z}}\kappa$
is the ring of derived dual numbers over $\kappa$; These semiorthogonal
decompositions agree with the results in [Ku21, Example 5.2, Theorem 5.12].

We could also repeat the above process: set $C_{0}=C$, $p_{0}=p$, and let
$C_{i}$ be a curve obtained by attaching a $\mathbb{P}^{1}$ to a nonsingular
$\kappa$-point $p_{i-1}\in C_{i-1}(\kappa)$ of $C_{i-1}$. Then for an integer
$n\geq 1$, $C_{n}$ is the union of $C$ with finite number of trees of
$\mathbb{P}^{1}$’s for which the total number of $\mathbb{P}^{1}$ is $n$. We
therefore obtain semiorthogonal decompositions:

$\displaystyle\operatorname{QCoh}(C_{n})$
$\displaystyle=\big{\langle}\operatorname{QCoh}(C),~{}\text{$n$-copies of
$\operatorname{QCoh}(\operatorname{Spec}\kappa[\varepsilon])$}\big{\rangle},$
$\displaystyle\mathrm{Perf}(C_{n})$
$\displaystyle=\big{\langle}\mathrm{Perf}(C),~{}\text{$n$-copies of
$\mathrm{Perf}(\operatorname{Spec}\kappa[\varepsilon])$}\big{\rangle}.$

However, the morphism spaces of the latter components depend on the
configurations of the trees of $\mathbb{P}^{1}$’s. For example, in the case
where $C_{n}$ is the union of $C$ and a chain $\Gamma_{n}$ of
$\mathbb{P}^{1}$’s, the endomorphism algebra of the latter component (of a
choice of tilting bundle on $\Gamma_{n}$) is described explicitly in [Bur,
Theorem 2.1]. In general, the endomorphism algebras of “the contributions” of
the new attached trees of $\mathbb{P}^{1}$’s can be described similarly; see
[KPS21, Theorem 4.9].

We could also consider the following variant555I thank Michel van Garrel for
asking about the relationship between the above example and the schemes in the
construction of deformation to normal cones. of Example 7.21 :

######  Variant 7.23 (Deformation to normal cones).

Similar to Example 7.21, we let $X$ be a scheme and let $D\subseteq X$ be an
effective Cartier divisor. Now we consider
$\mathscr{F}^{\prime}=\mathscr{O}_{X}\oplus\mathscr{O}_{D}(D)$ instead of
$\mathscr{F}=\mathscr{O}_{X}\oplus\mathscr{O}_{D}$. Then
$\mathbb{P}(\mathscr{F}^{\prime})=X\coprod_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})$,
where $\mathcal{N}=\mathcal{N}_{D/X}$ is the normal bundle, and
$\mathbb{P}(\Sigma\,\mathscr{F}^{\prime\vee})=D\times\operatorname{Spec}\mathbb{Z}[\varepsilon]$.
Therefore, Theorem 7.5 implies there are semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})\big{)}$
$\displaystyle=\big{\langle}\operatorname{QCoh}(X),~{}\operatorname{QCoh}(D)\otimes\operatorname{QCoh}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})\big{)}$
$\displaystyle=\big{\langle}\mathrm{Perf}(X),~{}\mathrm{Perf}(D)\otimes\mathrm{Perf}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle}.$

Notice that the reducible scheme
$X\coprod_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})$ in the above
formulae is precisely the special fiber appearing in the construction of
deformation to normal cones applied to $D\subseteq X$. More concretely, let
$M=\operatorname{Bl}_{D\times\\{\infty\\}}(X\times\mathbb{P}^{1})$ be the
total space in the construction of deformation to normal cones of [Ful, §5.1],
and let $\rho\colon M\to\mathbb{P}^{1}$ be the natural projection. Then for
$t\neq\infty\in\mathbb{P}^{1}$, $\rho^{-1}(t)\simeq X$, and
$\rho^{-1}(\infty)\simeq
X\coprod_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})\simeq\mathbb{P}_{X}(\mathscr{F}^{\prime})$.

######  Remark 7.24 (Derived blowups).

In view of Remark 7.8, we could view the semiorthogonal decompositions in
Examples 7.21 and 7.23 as blowup formulae, where
$\mathbb{P}(\mathscr{F})=X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}$ and
$\mathbb{P}(\mathscr{F}^{\prime})=X\coprod\nolimits_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})$
are the derived blowups (in the sense of [KR18, He21]) of $X$ along the
virtual codimension two closed derived subschemes
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})={\rm
Tot}(\mathcal{N}_{D/X}[-1])\subseteq X$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\prime\vee})=D\times\operatorname{Spec}\mathbb{Z}[\varepsilon]\subseteq
X$, respectively.

If a variety $X$ over a field $\kappa$ has locally complete intersection
singularities, then the cotangent complex $L_{X/\kappa}$ has Tor-amplitude
$\leq 1$. In many situations, the pair $(X,\mathscr{E}=L_{X/\kappa})$
satisfies ($a$). For example, we could consider the cases of nodal curves:

######  Example 7.25 (Contracting rational bridges).

Let $\kappa$ be a field, and let
$C\subseteq\mathbb{A}_{\kappa}^{2}=\operatorname{Spec}\kappa[x,y]$ be the
nodal curve defined by the equation $xy=0$. Then the cotangent complex
$\mathscr{F}=L_{C/\kappa}\simeq\Omega_{C/\kappa}^{1}$ is represented by the
complex of locally free sheaves:

$\left[\mathscr{O}_{C}\xrightarrow{(y,x)^{T}}\mathscr{O}_{C}{\rm
d}x\oplus\mathscr{O}_{C}{\rm d}y\right].$

In this case, by virtue of Proposition 4.33, $\mathbb{P}(L_{C/\kappa})$ is the
classical reducible scheme defined the equation $yX_{0}+xX_{1}=0$ in
$C\times\operatorname{Proj}\kappa[X_{0},X_{1}]$. In particular,
$\widetilde{C}=\mathbb{P}(\Omega_{C/\kappa}^{1})$ is the total transform of
$C\subseteq\mathbb{A}^{2}$ along the blowup
$\operatorname{Bl}_{0}\mathbb{A}^{2}\to\mathbb{A}^{2}$, where $o=(0,0)$ is the
origin. Then projection morphism $\pi\colon\widetilde{C}\to C$ is an
isomorphism over $C\backslash\\{o\\}$ and the exceptional locus
$E=\pi^{-1}(\o)$ is a projective line $\mathbb{P}^{1}_{\kappa}$ (called the
rational bridge) with multiplicity two666I thank Martin Kalck for bringing my
attention to the multiplicity issue.. The underlying morphism of reduced
schemes $\widetilde{C}^{\rm red}=\mathbb{P}(\Omega_{C/\kappa}^{1})^{\rm
red}\to C$ is the contraction of the rational bridge
$\mathbb{P}_{\kappa}^{1}$. On the other hand, $\mathbb{P}(\Sigma
L_{C/\kappa}^{\vee})$ is the derived zero locus in $\mathbb{A}^{2}$ cut out by
the three equations $x=0,y=0$ and $xy=0$, hence we obtain $\mathbb{P}(\Sigma
L_{C/\kappa}^{\vee})\simeq\operatorname{Spec}(\kappa[\varepsilon]\otimes_{\kappa}\kappa[\delta])$,
where
$\kappa[\varepsilon]=\operatorname{Sym}_{\kappa}(\Sigma\,\kappa)=\mathbb{Z}[\varepsilon]\otimes_{\mathbb{Z}}\kappa$
is the ring of derived dual numbers over $\kappa$ and
$\kappa[\delta]=\kappa[\delta]/(\delta^{2})$ denote the ring of usual dual
numbers. Theorem 7.5 implies semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}(\widetilde{C})$
$\displaystyle=\big{\langle}\pi^{*}\operatorname{QCoh}(C),~{}\operatorname{QCoh}\big{(}\operatorname{Spec}(\kappa[\varepsilon]\otimes_{\kappa}\kappa[\delta])\big{)}\big{\rangle},$
$\displaystyle\mathrm{Perf}(\widetilde{C})$
$\displaystyle=\big{\langle}\pi^{*}\mathrm{Perf}(C),~{}\mathrm{Perf}\big{(}\operatorname{Spec}(\kappa[\varepsilon]\otimes_{\kappa}\kappa[\delta])\big{)}\big{\rangle}.$

One can generalize this example by considering any nodal curve $C/\kappa$ (or
a family of nodal curves $\mathscr{C}/S$) and the projectivization
$\widetilde{C}:=\mathbb{P}(\Omega_{C/\kappa}^{1})\to C$ (or
$\widetilde{\mathscr{C}}:=\mathbb{P}(L_{\mathscr{C}/S})\to\mathscr{C}$).

######  Remark 7.26 (Stabilization maps).

We could regard the formulae of Examples 7.20 and 7.22 as describing the
behavior of the derived categories under “contracting rational tails” ([Sta,
Tag 0E3G]). On the other hand, the formula of Example 7.25 describes the
behavior of the derived categories under “contracting non-reduced rational
bridges” (see [Sta, Tag 0E7M] for more about the contraction of rational
bridges). Notice that a recent result of Kuznetsov and Shinder [KS22b,
Corollary 6.8] implies that there are no semiorthogonal decompositions for the
underlying reduced scheme $\widetilde{C}^{\rm red}$ in Example 7.25 (if $C$ is
projective). These results altogether provide a comprehensive picture of the
behavior of derived categories of nodal curves under stabilization maps, since
the general stabilization maps of nodal curves are composed of operations of
contracting rational tails and rational bridges (see, for example, [Sta, Tag
0E7Q]).

We can also consider higher-dimensional variants of Example 7.20. For example:

######  Example 7.27.

Let $X=\mathbb{A}^{2}=\operatorname{Spec}\mathbb{Z}[x,y]$, and let
$\mathscr{F}$ denote the cokernel of the map

$\mathscr{O}_{\mathbb{A}^{2}}^{\oplus
2}\xrightarrow{\big{(}\begin{smallmatrix}x&0\\\ 0&y\\\
x&y\end{smallmatrix}\big{)}}\mathscr{O}_{\mathbb{A}^{2}}^{\oplus 3}.$

Then $\operatorname{rank}\mathscr{F}=1$, $\operatorname{codim}_{X}(X_{1})=1$,
$\operatorname{codim}_{X}(X_{2})=2$, so by virtue of Lemma 7.16,
$\mathbb{P}(\mathscr{F})$ is classical but
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is not. More concretely:

  * •

$\mathbb{P}(\mathscr{F})\subseteq\mathbb{A}^{2}\times\mathbb{P}^{2}=\operatorname{Spec}\mathbb{Z}[x,y]\times\operatorname{Proj}\mathbb{Z}[X,Y,Z]$
is the classical closed subscheme defined by the equations $xX+xZ=0$ and
$yY+yZ=0$. Unwinding the definition, we see that $\mathbb{P}(\mathscr{F})$ has
four irreducible components:

$\mathbb{P}(\mathscr{F})=\mathbb{A}^{2}\cup(\mathbb{A}^{1}_{x}\times\mathbb{P}^{1})\cup(\\{(0,0)\\}\times\mathbb{P}^{2})\cup(\mathbb{A}^{1}_{y}\times\mathbb{P}^{1}).$

  * •

$\mathbb{P}(\Sigma^{\vee}\mathscr{F})\subseteq\mathbb{A}^{2}\times\mathbb{P}^{1}=\operatorname{Spec}\mathbb{Z}[x,y]\times\operatorname{Proj}\mathbb{Z}[U,V]$
is the derived closed subscheme defined by the quations $xU=0$, $yV=0$, and
$xU+yV=0$. Unwinding the definition, we see that the underlying classical
scheme of $\mathbb{P}(\Sigma^{\vee}\mathscr{F})$ is the blowup of the nodal
curve defined by $(xy=0)$ along the nodal point, that is,

$\mathbb{P}(\Sigma^{\vee}\mathscr{F})_{\mathrm{cl}}=\mathbb{A}^{1}\coprod_{\rm
pt}\mathbb{P}^{1}\coprod_{\rm pt}\mathbb{A}^{1}.$

But over each closed point, $\mathbb{P}(\Sigma^{\vee}\mathscr{F})$ is equipped
with a non-trivial derived structure which is isomorphic to that of the ring
of derived dual numbers $\operatorname{Spec}\mathbb{Z}[\varepsilon]$:

$\mathbb{P}(\Sigma^{\vee}\mathscr{F})\simeq(\mathbb{A}^{1}\coprod_{\rm
pt}\mathbb{P}^{1}\coprod_{\rm
pt}\mathbb{A}^{1})\times\operatorname{Spec}\mathbb{Z}[\varepsilon].$

Theorem 7.5 implies there are semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}\big{(}\mathbb{A}^{2}\cup(\mathbb{A}^{1}\times\mathbb{P}^{1})\cup(\\{(0,0)\\}\times\mathbb{P}^{2})\cup(\mathbb{A}^{1}\times\mathbb{P}^{1})\big{)}$
$\displaystyle\qquad\qquad=\big{\langle}\operatorname{QCoh}(\mathbb{A}^{2}),~{}\operatorname{QCoh}\big{(}\mathbb{A}^{1}\coprod_{\rm
pt}\mathbb{P}^{1}\coprod_{\rm
pt}\mathbb{A}^{1}\big{)}\otimes\operatorname{QCoh}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}\mathbb{A}^{2}\cup(\mathbb{A}^{1}\times\mathbb{P}^{1})\cup(\\{(0,0)\\}\times\mathbb{P}^{2})\cup(\mathbb{A}^{1}\times\mathbb{P}^{1})\big{)}$
$\displaystyle\qquad\qquad=\big{\langle}\mathrm{Perf}(\mathbb{A}^{2}),~{}\mathrm{Perf}\big{(}\mathbb{A}^{1}\coprod_{\rm
pt}\mathbb{P}^{1}\coprod_{\rm
pt}\mathbb{A}^{1}\big{)}\otimes\mathrm{Perf}(\operatorname{Spec}\mathbb{Z}[\varepsilon])\big{\rangle}.$

####  7.3.4. Examples satisfying ($a^{\prime}$) but violating ($b$)

By virtue of Lemma 7.16, as long as the pair $(X,\mathscr{F})$ in the
situation of ($\star$ ‣ 7.3.1) satisfies condition ($a^{\prime}$) but violates
condition ($b$), $\mathbb{P}(\mathscr{F})$ is classical and irreducible, but
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ is not classical. For example:

######  Example 7.28.

Let $X=\mathbb{A}^{2}=\operatorname{Spec}\mathbb{Z}[x,y]$, and let
$\mathscr{F}$ denote the cokernel of the map

$\mathscr{O}_{\mathbb{A}^{2}}\xrightarrow{(x,y,0)^{T}}\mathscr{O}_{\mathbb{A}^{2}}^{\oplus
3}.$

Then $\mathbb{P}(\mathscr{F})$ is the quadric cone
$Q=\\{xX+yY=0\\}\subseteq\mathbb{A}^{2}\times\mathbb{P}^{2}=\operatorname{Spec}\mathbb{Z}[x,y]\times\operatorname{Proj}\mathbb{Z}[X,Y,Z]$,
which is an irreducible threefold with a generic $\mathbb{P}^{1}$-bundle
structure over $X=\mathbb{A}^{2}$ and with an $A_{1}$-singularity at the point
$((0,0),[0,0,1])\in Q$. By virtue of Proposition 4.33 again,
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})\simeq\operatorname{Spec}\mathbb{Z}[\varepsilon]$,
where
$\mathbb{Z}[\varepsilon]=\operatorname{Sym}_{\mathbb{Z}}^{*}(\Sigma\,\mathbb{Z})$
is the ring of derived dual numbers. Therefore, by virtue of Theorem 7.5 we
obtain semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}(Q)$
$\displaystyle=\big{\langle}\operatorname{QCoh}(\operatorname{Spec}\mathbb{Z}[\varepsilon]),~{}~{}\operatorname{QCoh}(\mathbb{A}^{2})\otimes\mathscr{O}_{Q}(1),\operatorname{QCoh}(\mathbb{A}^{2})\otimes\mathscr{O}_{Q}(2)\big{\rangle},$
$\displaystyle\mathrm{Perf}(Q)$
$\displaystyle=\big{\langle}\mathrm{Perf}(\operatorname{Spec}\mathbb{Z}[\varepsilon]),~{}~{}\mathrm{Perf}(\mathbb{A}^{2})\otimes\mathscr{O}_{Q}(1),\mathrm{Perf}(\mathbb{A}^{2})\otimes\mathscr{O}_{Q}(2)\big{\rangle}.$

We may replace $X=\mathbb{A}^{2}$ by any reduced surface $S$ over a field
$\kappa$ and $\mathscr{F}$ by $\mathscr{O}_{S}\oplus\mathscr{I}_{p}$, where
$\mathscr{I}_{p}$ is the ideal sheaf of a non-singular rational point $p\in
S(\kappa)$. The above semiorthogonal decompositions are compatible with
results in [Ka19, §7.1], [X21, Theorem 4.6] and [PS21, Theorem 3.6] on nodal
threefolds.

Similar to Example 7.21, we may consider the global version of Example 7.28:

######  Example 7.29.

Let $X$ be a scheme and $Y\subseteq X$ be a regularly immersed closed
subscheme of codimension $2$. Let
$\mathscr{F}=\mathscr{O}_{X}\oplus\mathscr{I}_{Y}$, where $\mathscr{I}_{Y}$ is
the ideal sheaf of $Y$.

  * •

$\mathbb{P}(\mathscr{F})$ is the classical scheme which is a
$\mathbb{P}^{1}$-bundle over $X\backslash Y$, $\mathbb{P}^{2}$-bundle over
$Y$, and has $\mathbb{A}_{1}$-singularity along a unique section of the
$\mathbb{P}^{2}$-bundle over $Y$. More concretely, let $U\subseteq X$ be any
open subscheme for which $Y\cap U\subset U$ is defined by the equation
$(f_{1}=f_{2}=0)$, where $(f_{1},f_{2})\in\mathscr{O}_{X}(U)$ is a regular
sequence. Then by virtue of Proposition 4.33,
$\mathbb{P}_{U}(\mathscr{F})\subseteq
U\times\mathbb{P}^{2}=U\times\operatorname{Proj}\mathbb{Z}[X_{0},X_{1},X_{2}]$
is an effective divisor defined by the equation $(f_{1}X_{1}+f_{2}X_{2})=0$.

  * •

By virtue of Example 4.35,
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})\simeq\mathbb{V}_{Y}(\Sigma(\det\mathcal{N}_{Y/X})^{\vee})$
which we could informally think of as the total space
$(\det\mathbb{N}_{Y/X})[-1]$ of the $(-1)$-shifted determinant bundle of
normal bundle, where $\mathcal{N}_{Y/X}$ is the normal bundle of $Y$ inside
$X$. More concretely, over the open subsets $U\subseteq X$ of the previous
bullet, $\mathbb{P}_{U}(\Sigma\mathscr{F}^{\vee})$ is equivalent to $(Y\cap
U)\times\operatorname{Spec}\mathbb{Z}[\varepsilon]$, where
$\mathbb{Z}[\varepsilon]=\operatorname{Sym}_{\mathbb{Z}}^{*}(\Sigma\mathbb{Z})$
as usual.

By virtue of Theorem 7.5 we obtain semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}(\mathbb{P}(\mathscr{F}))$
$\displaystyle=\big{\langle}\operatorname{QCoh}(\mathbb{V}_{Y}(\Sigma(\det\mathcal{N}_{Y/X})^{\vee})),~{}~{}\operatorname{QCoh}(X)\otimes\mathscr{O}(1),\operatorname{QCoh}(X)\otimes\mathscr{O}(2)\big{\rangle},$
$\displaystyle\mathrm{Perf}(\mathbb{P}(\mathscr{F}))$
$\displaystyle=\big{\langle}\mathrm{Perf}(\mathbb{V}_{Y}(\Sigma(\det\mathcal{N}_{Y/X})^{\vee})),~{}~{}\mathrm{Perf}(X)\otimes\mathscr{O}(1),\mathrm{Perf}(X)\otimes\mathscr{O}(2)\big{\rangle}.$

####  7.3.5. Examples satisfying ($b$) or even ($b^{\prime}$) but violating
($c$)

By virtue of Lemma 7.16, as long as the pair $(X,\mathscr{F})$ in the
situation of ($\star$ ‣ 7.3.1) satisfies condition ($b$) (resp.
($b^{\prime}$)) but violates condition ($c$), $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are classical (resp. classical and
irreducible) but the correspondence scheme $\widehat{Z}$ is not classical.

######  Example 7.30.

Let $X=\mathbb{A}^{2}=\operatorname{Spec}\mathbb{Z}[x,y]$, and let
$\mathscr{F}$ be the cokernel of the map

$\mathscr{O}_{X}^{\oplus 2}\xrightarrow{\big{(}\begin{smallmatrix}x&0\\\
0&y\\\ \end{smallmatrix}\big{)}}\mathscr{O}_{X}^{\oplus 2}.$

By virtue of Lemma 7.16, both $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are classical and reducible (in fact,
they are both equivalent to the blowup of $\\{xy=0\\}\subseteq\mathbb{A}^{2}$
along the origin $\mathbf{0}$), but the universal incidence space
$\widehat{Z}$ is not classical. Furthermore, Theorem 7.5 implies that the
diagram
$\mathbb{P}(\mathscr{F})\xleftarrow{r_{+}}\widehat{Z}\xrightarrow{r_{-}}\mathbb{P}(\Sigma\mathscr{F}^{\vee})$
induces equivalences of $\infty$-categories

$r_{+,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\mathbb{P}(\Sigma\,\mathscr{F}^{\vee}))\simeq\operatorname{QCoh}(\mathbb{P}(\mathscr{F}))\qquad
r_{+,*}\,r_{-}^{*}\colon\mathrm{Perf}(\mathbb{P}(\Sigma\,\mathscr{F}^{\vee}))\simeq\mathrm{Perf}(\mathbb{P}(\mathscr{F})).$

(Moreover, the same argument of [JL18, Theorem 4.1] shows that the composition
of equivalences $(r_{-,*}\,r_{+}^{*})\circ(r_{+,*}\,r_{-}^{*})$ is a non-
trivial spherical twist in the sense of [ST, AL].)

######  Example 7.31.

Let $X=\mathbb{A}^{4}=\operatorname{Spec}\mathbb{Z}[x,y,z,w]$ and let
$\mathscr{F}$ be the cokernel of the map

$\mathscr{O}_{X}^{\oplus
3}\xrightarrow{\sigma=\left(\begin{smallmatrix}x&y&w\\\ z&x&y\\\
w&z&0\end{smallmatrix}\right)}\mathscr{O}_{X}^{\oplus 3}.$

Then $\operatorname{codim}X_{1}=1$ and $\operatorname{codim}X_{2}=X_{3}=4$. By
virtue of Lemma 7.16, $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are classical, irreducible threefolds
(which are both birational equivalent to the determinantal variety
$X_{1}=\\{-xw^{2}+y^{2}w+z^{2}w-xyz=0\\}\subseteq\mathbb{A}^{4}$), and
$\widehat{Z}$ is not classical. Furthermore, the diagram
$\mathbb{P}(\mathscr{F})\xleftarrow{r_{+}}\widehat{Z}\xrightarrow{r_{-}}\mathbb{P}(\Sigma\mathscr{F}^{\vee})$
induces equivalences of $\infty$-categories

$r_{+,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\mathbb{P}(\Sigma\,\mathscr{F}^{\vee}))\simeq\operatorname{QCoh}(\mathbb{P}(\mathscr{F}))\qquad
r_{+,*}\,r_{-}^{*}\colon\mathrm{Perf}(\mathbb{P}(\Sigma\,\mathscr{F}^{\vee}))\simeq\mathrm{Perf}(\mathbb{P}(\mathscr{F})).$

(Moreover, the same argument of [JL18, Theorem 4.1] shows that the composition
of equivalences $(r_{-,*}\,r_{+}^{*})\circ(r_{+,*}\,r_{-}^{*})$ is a non-
trivial spherical twist in the sense of [ST, AL].)

There are also similar examples where $X$ is an irreducible sixfold, $\sigma$
is a $4\times 4$-matrix, and $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are classical, irreducible fivefolds
which are connected by a flop, but the universal incidence space $\widehat{Z}$
which induces the derived equivalence is not classical.

####  7.3.6. Examples satisfying ($c$) but possibly violating ($c^{\prime}$)

######  Example 7.32 (Symmetric matrices).

The examples of generic symmetric matrices of size $k\times k$, where $k\geq
3$, are typical situations where condition ($c$) is satisfied but condition
($c^{\prime}$) is not. We consider a concrete case, where
$X=\mathbb{A}^{6}=\operatorname{Spec}\mathbb{Z}[a,b,c,e,d,f]$ is the space of
symmetric $3\times 3$-matrices, and $\sigma\colon\mathscr{O}_{X}^{\oplus
3}\to\mathscr{O}_{X}^{\oplus 3}$ is the tautological map

$\sigma=\left(\begin{array}[]{ccc}a&b&c\\\ b&d&e\\\ c&e&f\end{array}\right).$

Then $\operatorname{codim}_{X}(X_{1})=1$, $\operatorname{codim}_{X}(X_{2})=3$,
and $\operatorname{codim}_{X}(X_{3})=6$. Let $Z=X_{1}$ denote the symmetric
determinantal variety. Then both $\mathbb{P}(\mathscr{F})$ and
$\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ are (classical, irreducible)
resolutions of $Z$. More concretely, they are both isomorphic to the subscheme
$\widetilde{Z}$ of
$\mathbb{A}^{6}\times\mathbb{P}^{2}=\operatorname{Spec}\mathbb{Z}[a,b,c,e,d,f]\times\operatorname{Proj}\mathbb{Z}[X,Y,Z]$
defined by the equations

$\displaystyle aX+bY+cZ=0,\qquad bX+dY+eZ=0,\qquad cX+eY+fZ=0.$

The universal incidence space $\widehat{Z}$ is classical, with two irreducible
components, and isomorphic to the classical fiber product
$\widetilde{Z}\times_{Z}^{\mathrm{cl}}\widetilde{Z}$. Moreover, by virtue of
Theorem 7.5, the diagram
$\widetilde{Z}\xleftarrow{r_{+}}\widetilde{Z}\times_{Z}^{\mathrm{cl}}\widetilde{Z}\xrightarrow{r_{-}}\widetilde{Z}$
induces auto-equivalences of $\infty$-categories

$r_{+,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\widetilde{Z})\simeq\operatorname{QCoh}(\widetilde{Z})\qquad
r_{+,*}\,r_{-}^{*}\colon\mathrm{Perf}(\widetilde{Z})\simeq\mathrm{Perf}(\widetilde{Z}).$

By virtue of [JL18, Theorem 4.1], the above auto-equivalences are non-trivial,
and the composition $(r_{-,*}\,r_{+}^{*})\circ(r_{+,*}\,r_{-}^{*})$ is a (non-
trivial) spherical twist in the sense of [ST, AL].

######  Example 7.33 (Symmetric products of curves, [Tod21, §5.5]).

Let $C$ be a complex smooth projective curve of genus $g\geq 1$ and let
$0\leq\delta\leq g-1$ be an integer. Let $C^{(g-1-\delta)}$ and
$C^{(g-1+\delta)}$ denote the $(g-1-\delta)$-th and $(g-1+\delta)$-th
symmetric products of $C$, respectively. Let $X$ denote Picard variety
$\operatorname{Pic}^{g-1+\delta}(C)$ of line bundles of degree ($g-1+\delta$)
on $C$. By virtue of [JL18, §3.1.3], there exists a quasi-coherent sheaf
$\mathscr{F}$ of perfect-amplitude contained in $[0,1]$ and rank $\delta$ over
$X$, for which there are canonical identifications

$\mathrm{pr}_{+}={\rm AJ}\colon C^{(g-1+\delta)}=\mathbb{P}(\mathscr{F})\to
X\qquad\mathrm{pr}_{-}={\rm AJ}^{\vee}\colon
C^{(g-1-\delta)}=\mathbb{P}(\Sigma^{\vee}\mathscr{F})\to X,$

where ${\rm AJ}$ is the Abel–Jacobi map which carries an effective divisor $D$
on $C$ to $\mathscr{O}(D)$, and ${\rm AJ}^{\vee}$ is the Serre-dual version of
${\rm AJ}$ which carries $D$ to $\mathscr{O}(K_{C}-D)$.

  * •

By virtue of [JL18, Lemma B.1], the condition ($c$) is always satisfied by the
pair $(X,\mathscr{F})$, that is, the universal incidence space $\widehat{Z}$
is classical, and is isomorphic to the classical fiber product of
$C^{(g-1-\delta)}$ and $C^{(g-1+\delta)}$ over $X$.

  * •

Suppose $g\geq 3$. Then by virtue of the proof of [J19, Corollary 5.1], we see
that the condition ($c^{\prime}$) is satisfied (that is, $\widehat{Z}$ is
irreducible) if $C$ is not hyperelliptic.

The converse of the second assertion is almost true in the following sense:

  * •

Suppose $g\geq 3$, then as long as $C$ is hyperelliptic and $0\leq\delta\leq
g-3$, the condition ($c^{\prime}$) is violated, that is, $\widehat{Z}$ is
reducible. (To prove this assertion, observe that any complete linear series
$g_{g-1-\delta}^{1}$ on the hyperelliptic curve $C$ is of the form
$g_{2}^{1}+p_{1}+\cdots+p_{g-\delta-3}$, where $p_{i}\in C(\mathbb{C})$.
Therefore the Brill–Noether locus $W_{g-1-\delta}^{1}(C)\neq\emptyset$ and
$\dim(W_{g-1-\delta}^{1}(C))=g-\delta-3$, that is, the degeneracy locus
$X_{2}\neq\emptyset$ satisfies
$\operatorname{codim}_{X}(X_{2})=\delta+3<\delta+2\cdot 2$, which violates
($c^{\prime}$). See [ACGH, Chapter IV, §3] for the involved notations.)

Since the condition ($c$) is satisfied by all curves $C$, Theorem 7.5 in this
case reproduces Toda’s result [Tod21, Corollary 5.11] (see also [JL18,
Corollary 3.10], [BK19]):

$\displaystyle\operatorname{QCoh}(C^{(g-1+\delta)})$
$\displaystyle=\big{\langle}\Phi(\operatorname{QCoh}(C^{(g-1-\delta)})),~{}\operatorname{QCoh}(\operatorname{Pic}^{g-1+\delta}(C))(1),\cdots,\operatorname{QCoh}(\operatorname{Pic}^{g-1+\delta}(C))(\delta)\big{\rangle},$
$\displaystyle\mathrm{Perf}(C^{(g-1+\delta)})$
$\displaystyle=\big{\langle}\Phi(\mathrm{Perf}(C^{(g-1-\delta)})),~{}\mathrm{Perf}(\operatorname{Pic}^{g-1+\delta}(C))(1),\cdots,\mathrm{Perf}(\operatorname{Pic}^{g-1+\delta}(C)(\delta)\big{\rangle}$

where $\Phi=r_{+,*}\,r_{-}^{*}$ is the functor induced by the diagram
$C^{(g-1+\delta)}\xleftarrow{r_{+}}\widehat{Z}\xrightarrow{r_{-}}C^{(g-1-\delta)}$
of the classical fiber product. Therefore, to understand the $\infty$-category
$\operatorname{QCoh}(C^{(g-1+\delta)})$ is equivalent to understand the
$\infty$-category $\operatorname{QCoh}(C^{(g-1-\delta)})$, the morphism
algebra of the relative exceptional sequence
$\\{\mathscr{O}_{C^{(g-1+\delta)}}(i)\\}_{1\leq i\leq\delta}$, and the
morphism spaces from $\operatorname{QCoh}(C^{(g-1-\delta)})$ to
$\mathscr{O}_{C^{(g-1+\delta)}}(i)$. By a result of Lin [L21], the
$\infty$-category $\operatorname{QCoh}(C^{(g-1-\delta)})$ is indecomposable.
On the other hand, the results on Beilinson’s relations (Corollary 6.21)
completely describe the morphism algebra of the part
$\\{\mathscr{O}_{C^{(g-1+\delta)}}(i)\\}_{1\leq i\leq\delta}$: for all $1\leq
i,j\leq\delta$,

$\underline{\mathrm{Map}}_{\operatorname{Pic}^{g-1+\delta}(C)}(\mathscr{O}_{C^{(g-1+\delta)}}(i),\mathscr{O}_{C^{(g-1+\delta)}}(j))\simeq\operatorname{Sym}^{j-i}_{\operatorname{Pic}^{g-1+\delta}(C)}(\mathscr{F})\simeq
S^{j-i}_{\operatorname{Pic}^{g-1+\delta}(C)}(\pi_{0}(\mathscr{F}))$

(the last equivalence follows from Lemma 7.18). Moreover, we can describe the
dual relative exceptional sequences in terms of the sheaf of relative
differentials $\Omega_{{\rm AJ}}^{i}$ (Proposition 6.19).

######  Remark 7.34.

Notice that in the situation of ($\star$ ‣ 7.3.1), as long as the pair
$(X,\mathscr{F})$ satisfies condition ($c$), all the three spaces
$\mathbb{P}(\mathscr{F})$, $\mathbb{P}(\Sigma\mathscr{F}^{\vee})$ and
$\widehat{Z}$ are classical. Therefore, the classical results in [JL18,
Theorem 3.4], [J21, Theorem 6.16] are already applicable to these cases.
Moreover, ($c$) (or even ($c^{\prime}$)) is satisfied by a generic map
$\sigma\colon\mathscr{W}\to\mathscr{V}$ if
$\mathscr{W}^{\vee}\otimes\mathscr{V}$ is globally generated. The new
information we get in these cases from the current paper is the generalized
Serre’s theorem 5.6, and Beilison’s relations (Proposition 6.19, Corollary
6.21). Furthermore, by virtue of Lemmata 7.16 and 7.18, we can formulate the
output of these results in terms of classical algebraic geometry (although the
proofs require the machinery of derived algebraic geometry).

##  8\. Hecke correspondence moduli as derived projectivizations

Many moduli problems naturally fit into the framework of derived
projectivizations. We will consider one such example in this section: the
Hecke correspondence moduli. To motivate our discussion, we first consider the
following two examples:

######  Example 8.1 (Hilbert schemes of points on surfaces).

Let $X$ be a smooth complex surface, and for an integer $n\geq 0$, let
$\mathrm{Hilb}_{n}$ denote the moduli space of Hilbert scheme of points on
$X$, that is, $\mathrm{Hilb}_{n}$ parametrizes ideal sheaves
$\mathscr{I}_{n}\subseteq\mathscr{O}_{X}$ for which
$\mathscr{O}_{X}/\mathscr{I}_{n}$ is a zero dimensional sheaf of length $n$.
We let $\mathrm{Hilb}_{n,n+1}$ denote the nested Hilbert scheme which
parametrizes the pairs $(\mathscr{I}_{n}\supseteq\mathscr{I}_{n+1})$, where
$\mathscr{I}_{n}\in\mathrm{Hilb}_{n}$,
$\mathscr{I}_{n+1}\in\mathrm{Hilb}_{n+1}$, for which there is an exact
sequence $0\to\mathscr{I}_{n+1}\to\mathscr{I}_{n}\to\mathbb{C}_{x}\to 0$ for
some closed point $x\in X(\mathbb{C})$. There is a diagram:

${\mathrm{Hilb}_{n,n+1}}$${\mathrm{Hilb}_{n+1}}$${X}$${\mathrm{Hilb}_{n}}$$\scriptstyle{p_{+}}$$\scriptstyle{p_{X}}$$\scriptstyle{p_{-}}$

where $p_{\pm}$ are the forgetful maps which carry
$(\mathscr{I}_{n}\supseteq\mathscr{I}_{n+1})$ to $\mathscr{I}_{n+1}$ and
$\mathscr{I}_{n}$, respectively, and $p_{X}$ is the map which carries
$(\mathscr{I}_{n}\supseteq\mathscr{I}_{n+1})$ to $x\in X$, where
$\mathscr{I}_{n}/\mathscr{I}_{n+1}\simeq\mathbb{C}_{x}$. We let
$Z_{n}\subseteq\mathrm{Hilb}_{n}\times X$ denote the universal subscheme. From
the work of [ES], we know that

  * •

The projection $\pi_{-}=(p_{-}\times
p_{X})\colon\mathrm{Hilb}_{n,n+1}\to\mathrm{Hilb}_{n}\times X$ identifies
$\mathrm{Hilb}_{n,n+1}$ as the projectivization
$\mathbb{P}(\mathscr{I}_{Z_{n}})$, where $\mathscr{I}_{Z_{n}}$ is the ideal
sheaf of $Z_{n}$.

  * •

The projection $\pi_{+}=(p_{+}\times
p_{X})\colon\mathrm{Hilb}_{n-1,n}\to\mathrm{Hilb}_{n}\times X$ identifies
$\mathrm{Hilb}_{n-1,n}$ as the projectivization
$\mathbb{P}(\mathscr{E}\kern-1.0ptxt^{1}(\mathscr{I}_{Z_{n}},\mathscr{O}_{\mathrm{Hilb}_{n}\times
X})\simeq\mathbb{P}(\omega^{\mathrm{cl}}_{Z_{n}})$, where
$\omega^{\mathrm{cl}}_{Z_{n}}$ is the classical dualizing sheaf of the
Cohem–Macaulay scheme $Z_{n}$.

In particular, the situation fits into the picture of the classical version
([JL18, Theorem 3.1]) of the projectivization formula Theorem 7.5, and there
is a semiorthogonal decomposition:

$\mathrm{D}_{\mathrm{qc}}(\mathrm{Hilb}_{n,n+1})=\langle\mathrm{D}_{\mathrm{qc}}(\mathrm{Hilb}_{n-1,n}),~{}\mathrm{D}_{\mathrm{qc}}(\mathrm{Hilb}_{n}\times
X)\otimes\mathscr{O}(1)\rangle.$

See [BK19] (or [JL18, §3.1.4], [J19, §5.2]) for more details.

######  Example 8.2 (One-point modifications of Gieseker stable sheaves on
surfaces [Ne19, Ne18]).

Let $X$ be a smooth complex projective surface with an ample divisor
$H\subseteq X$. We fix a pair of invariants $(r,c_{1})\in\mathbb{N}\times
H^{2}(X,\mathbb{Z})$ such that $r>0$ and $\gcd(r,c_{1}\cdot H)=1$, and let
$\mathcal{M}$ be the disjoint union
$\bigsqcup_{c_{2}\geq\frac{r-1}{2r}c_{1}^{2}}\mathcal{M}_{r,c_{1},c_{2}}(X)$,
where $\mathcal{M}_{r,c_{1},c_{2}}(X)$ is the moduli space of Gieseker stable
(discrete) sheaves with numerical invariants $(r,c_{1},c_{2})$. We consider
the moduli space $\mathfrak{Z}$ which parametrizes pairs of Gieseker stable
sheaves $(\mathscr{F}^{\prime}\subset\mathscr{F})$ for which
$\mathscr{F},\mathscr{F}^{\prime}\in\mathcal{M}$ and
$\mathscr{F}/\mathscr{F}^{\prime}\simeq\mathbb{C}_{x}$, where $x\in
X(\mathbb{C})$ is a closed point and $\mathbb{C}_{x}$ is the skyscraper sheaf
supported at $x$. Then the moduli $\mathfrak{Z}$ fits into a similar diagram
as in Example 8.1,

${\mathfrak{Z}}$${\mathcal{M}}$${X}$${\mathcal{M},}$$\scriptstyle{p_{+}}$$\scriptstyle{p_{X}}$$\scriptstyle{p_{-}}$

where $p_{+},p_{-}$ and $p_{X}$ are the forgetful maps which carry
$(\mathscr{F}^{\prime}\subset\mathscr{F})$ to $\mathscr{F}^{\prime}$,
$\mathscr{F}$ and $x$, respectively. Similar to Example 8.1, the forgetful
maps $\pi_{\pm}=(p_{\pm}\times_{S}p_{X})$ are (derived) projectivizations of
sheaves of perfect-amplitude in $[0,1]$. See [Ne19, Ne18] for more details.

This section considers a simultaneous generalization of the above two
examples. More concretely, let $X\to S$ be a family of derived schemes over a
fixed prestack $S$, and let $\mathfrak{M}$ denote the moduli prestack of
connective quasi-coherent sheaves on $X$ over $S$.

In §8.1, we will define a prestack $\mathfrak{Z}$ of “one-point modifications
of connective sheaves on $X$ over $S$”, which parametrizes exact triangles of
the form

$\mathscr{F}^{\prime}\to\mathscr{F}\to x_{*}\mathscr{L}$

where $\mathscr{F}^{\prime},\mathscr{F}\in\mathfrak{M}$, $\mathscr{L}$ is a
line bundle, and $x\colon S\to X$ is a section of $X\to S$. We show that under
mild assumptions ($X\to S$ is separated), the forgetful map
$\pi_{-}\colon\mathfrak{Z}\to\mathfrak{M}\times_{S}X$ is always a derived
projectivization (Proposition 8.5). Moreover, if we restrict our attention to
the moduli of “one-point modifications of perfect objects on $X$ over $S$ with
a specified Tor-amplitude”, and suppose that $X\to S$ is smooth and separated,
then both the restrictions of the forgetful functors
$\pi_{\pm}\colon\mathfrak{Z}\to\mathfrak{M}\times_{S}X$ are derived
projectivizations (Proposition 8.7).

In §8.2, we consider the moduli $\mathfrak{Z}_{\lambda}$ of general “flags of
one-point modifications of sheaves on $X$ over $S$”, where $\lambda$ is a set
partition which encodes the combinatorial information of the flags. We show
that the moduli of general flags also fit into similar derived
projectivization pictures as above (Propositions 8.11 and 8.13).

In §8.3, we focus on the case where $X\to S$ is a family of smooth, separated
derived surfaces. In this case, the above forgetful functors are derived
projectivizations of sheaves of perfect-amplitude in $[0,1]$. In particular,
we could apply all the results of this paper to this situation; see Corollary
8.15 for more detail.

######  Remark 8.3.

This section’s exposition and notations closely follow Neguţ’s works [Ne17,
Ne18, Ne19] which study the shuffle algebra structures and Hecke
correspondences for the moduli of Gieseker stable sheaves on surfaces; see
also [PZ20, Z21] for more results in this direction. We expect the results of
this paper to be helpful for the study of similar algebraic structures for the
general prestacks of perfect objects of Tor-amplitude $[0,1]$ on a surface $X$
or a family $X\to S$ of surfaces. Furthermore, the general framework of this
section allows us to consider the similar structures on the moduli spaces with
other types of stability conditions. For example, it allows us to study
algebraic structures on the moduli spaces $\mathfrak{M}_{\sigma}(X/S)$ of
Bridgeland (semi-)stable objects on a family $X\to S$ of surfaces, where $X\to
S$ is equipped with (a family of) Bridgeland stability conditions $\sigma$ in
the sense of [Bri07, Bri08, AB12, BM11, BM14, BLM+]. We wish to explore these
aspects in the future.

###  8.1. Hecke correspondences: the case of one-step flags

Let $X$ be a prestack over the fixed base prestack $S$. In this subsection, we
construct moduli functors $\mathfrak{Z}$ (resp.
$\mathfrak{Z}^{\mathrm{perf}}$) of “one-step flag of one-point modifications
of connective quasi-coherent sheaves (resp. perfect coherent sheaves)” on $X$
and study their basic properties.

######  Construction 8.4.

Let $X$ be a prestack over $S$. Consider the following constructions:

  1. (1)

We let $\mathfrak{M}$ denote the functor
$\operatorname{CAlg}^{\Delta}\to\mathcal{S}$ over $S$ which carries each map
$\eta\colon T=\operatorname{Spec}R\to S$ to the space
$\mathfrak{M}(\eta)=\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn},\,\simeq}$.
We let
$\mathscr{U}\in\operatorname{QCoh}(X\times_{S}\mathfrak{M})^{\mathrm{cn}}$
denote the universal connective quasi-coherent sheaf on
$X\times_{S}\mathfrak{M}$.

  2. (2)

We let $\mathfrak{Z}\colon\operatorname{CAlg}^{\Delta}\to\mathcal{S}$ denote
the moduli functor over $S$ which carries each map $\eta\colon
T=\operatorname{Spec}R\to S$ to the full subspace of

$\mathrm{Fun}(\Delta^{1},\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn}})^{\simeq}\times\mathrm{Map}_{/T}(T,X\times_{S}T)\times\operatorname{Pic}(T)^{\simeq}$

spanned by those elements

$((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});\,x\colon T\to
X\times_{S}T;\,\mathscr{L}\in\operatorname{Pic}(T))$

for which there is a fiber sequence

$\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F}\to x_{*}\mathscr{L}$

in $\operatorname{QCoh}(X\times_{S}T)$. (Here,
$\mathrm{Map}_{/T}(T,X\times_{S}T)$ denotes the space of sections $x\colon
T\to X\times_{S}T$ of the natural projection $X\times_{S}T\to T$.) We could
informally think of $\mathfrak{Z}$ as the “moduli of one-step flag of one-
point modifications of connective quasi-coherent sheaves on $X$ over $S$”. We
denote the universal element of $\mathfrak{Z}$ by

$((\mathscr{U}^{\prime}\xrightarrow{\phi_{\rm univ}}\mathscr{U});\,x_{\rm
univ};\,\mathscr{L}_{\rm univ}),$

where $\phi_{\rm univ}\colon\mathscr{U}^{\prime}\to\mathscr{U}$ is a morphism
in $\operatorname{QCoh}(X\times_{S}\mathfrak{Z})^{\mathrm{cn}}$, $x_{\rm
univ}\colon\mathfrak{Z}\to X\times_{S}\mathfrak{Z}$ is a section of the
projection $X\times_{S}\mathfrak{Z}\to\mathfrak{Z}$, $\mathscr{L}_{\rm
univ}\in\operatorname{Pic}(X\times_{S}\mathfrak{Z})$ is a line bundle, and for
which there is a universal fiber sequence in
$\operatorname{QCoh}(X\times_{S}\mathfrak{Z})$:

$\mathscr{U}^{\prime}\xrightarrow{\phi_{\rm univ}}\mathscr{U}\to(x_{\rm
univ})_{*}\mathscr{L}_{\rm univ}.$

  3. (3)

We consider the following natural forgetful maps

${\mathfrak{Z}}$${\mathfrak{M}}$${X}$${\mathfrak{M},}$$\scriptstyle{p_{+}}$$\scriptstyle{p_{X}}$$\scriptstyle{p_{-}}$

where $p_{+}$, $p_{-}$, and $p_{X}$ are the respective forgetful maps which
carry each element

$\zeta=((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});\,x\colon T\to
X\times_{S}T;\,\mathscr{L}\in\operatorname{Pic}(T))\in\mathfrak{Z}(\eta),$

where $\eta\colon T=\operatorname{Spec}R\to S$, to the elements
$p_{+}(\zeta)=\mathscr{F}^{\prime}\in\mathfrak{M}(\eta)$,
$p_{-}(\zeta)=\mathscr{F}\in\mathfrak{M}(\eta)$, and the composition
$p_{X}(\zeta)=(T\xrightarrow{x}X\times_{S}T\xrightarrow{\mathrm{pr}_{X}}X)\in
X(\eta)$, respectively.

  4. (4)

We consider the following maps:

$\displaystyle\pi_{+}=(p_{+}\times_{S}p_{X})\colon\mathfrak{Z}\to\mathfrak{M}\times_{S}X,\qquad\pi_{-}=(p_{-}\times_{S}p_{X})\colon\mathfrak{Z}\to\mathfrak{M}\times_{S}X.$

(Therefore $\pi_{+}$ is the map which carries the element
$\zeta=((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});x;\mathscr{L})$ to
$(\mathscr{F}^{\prime},\mathrm{pr}_{X}\circ x)$, and $\pi_{+}$ is the map
which carries
$\zeta=((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});\,x;\,\mathscr{L})$
to $(\mathscr{F},\mathrm{pr}_{X}\circ x)$.)

The following result relates the functors $\mathfrak{Z}$ with derived
projectivizations.

######  Proposition 8.5 (Compare with [Ne19, Definition 2.5, Proposition
2.8]).

In the situation of Construction 8.4, suppose that the map $X\to S$ is
separated and a relative derived scheme, then the projection map
$\pi_{-}\colon\mathfrak{Z}\to\mathfrak{M}\times_{S}X$ identifies
$\mathfrak{Z}$ with the derived projectivization
$\mathbb{P}_{\mathfrak{M}\times_{S}X}(\mathscr{U})$ and for which there is a
canonical identification $\mathscr{O}(1)\simeq\mathscr{L}_{\rm univ}$.

###### Proof.

For any map $\eta\colon T=\operatorname{Spec}R\to S$, where
$R\in\operatorname{CAlg}^{\Delta}$, and a map
$\widetilde{\eta}=(f\times_{S}g)\colon T\to\mathfrak{M}\times_{S}X$ over
$\eta$, we set
$\mathscr{F}=(\operatorname{id}_{X}\times_{S}f)^{*}(\mathscr{U})\in\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn}}$
and $x=(g\times_{S}\operatorname{id}_{T})\colon T\to X\times_{S}T$, so that
$\widetilde{\eta}^{*}\,\mathscr{U}\simeq x^{*}\mathscr{F}$. Then the space of
liftings $T\to\mathfrak{Z}$ of $\widetilde{\eta}$ is the space of pairs
$((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F})\in\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn}}_{/\mathscr{F}},\mathscr{L}\in\operatorname{Pic}(T))$
for which there is a fiber sequence
$\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F}\to x_{*}\mathscr{L}$. Such
a space is homotopy equivalent to the space of pairs
$(\mathscr{L}\in\operatorname{Pic}(T),(\mathscr{F}\xrightarrow{\varphi}x_{*}\mathscr{L})\in\operatorname{QCoh}(X\times_{S}T)_{\mathscr{F}/})$
for which $\varphi$ is surjective on $\pi_{0}$ (and a homotopy inverse map is
given by setting $\mathscr{F}^{\prime}$ to be the fiber of $\varphi$). We
claim that the above space of pairs $(\mathscr{L},\varphi)$ is homotopy
equivalent to the space of pairs
$(\mathscr{L}\in\operatorname{Pic}(T),x^{*}\mathscr{F}\xrightarrow{\rho}\mathscr{L})$
for which $\rho$ is surjective on $\pi_{0}$, where $\rho$ is the map induced
by $\varphi$ from the adjunction

$\mathrm{Map}_{\operatorname{QCoh}(X\times_{S}T)}(\mathscr{F},x_{*}\mathscr{L})\simeq\mathrm{Map}_{\operatorname{QCoh}(T)}(x^{*}\mathscr{F},\mathscr{L}).$

In other words, the space of liftings $T\to\mathfrak{Z}$ of $\widetilde{\eta}$
along $\pi_{-}$ is canonical homotopy equivalent to the space
$\mathbb{P}_{\mathfrak{M}\times_{S}X}(\mathscr{U})(\widetilde{\eta})$ over
$\widetilde{\eta}$, where $\mathbb{P}_{\mathfrak{M}\times_{S}X}(\mathscr{U})$
is the derived projectivization (§4.2), which will prove the desired
assertion.

Let $\rho\colon x^{*}\mathscr{F}\to\mathscr{L}$ be a morphism which
corresponds to $\varphi\colon\mathscr{F}\to x_{*}\mathscr{L}$ through the
above adjunction. Then to prove the above claim, it suffices to show the
following:

  1. $(enumi)$

$\rho$ is surjective on $\pi_{0}$ if and only if $\varphi$ is surjective on
$\pi_{0}$.

We now prove ($enumi$). First, if $\varphi$ is surjective on $\pi_{0}$, then
the counit map $x^{*}x_{*}\mathscr{L}\to\mathscr{L}$ is surjective on
$\pi_{0}$ (since $x$ is an affine morphism), and $x^{*}\varphi$ is surjective
on $\pi_{0}$ (since the property “surjection on $\pi_{0}$” is stable under
pullback). Therefore, we obtain that the composition $\rho\colon
x^{*}\mathscr{F}\xrightarrow{x^{*}\varphi}x^{*}x_{*}\mathscr{L}\to\mathscr{L}$
is surjective on $\pi_{0}$. On the other hand, if $\rho$ is surjective on
$\pi_{0}$, then $x_{*}\rho$ is surjective on $\pi_{0}$ (since $x$ is affine).
To show $\varphi\colon\mathscr{F}\to
x_{*}x^{*}\mathscr{F}\xrightarrow{x_{*}\rho}x_{*}\mathscr{L}$ is surjective on
$\pi_{0}$, it suffices to show that the unit map $\mathscr{F}\to
x_{*}x^{*}\mathscr{F}$ is surjective on $\pi_{0}$ for all closed immersions
$x$ and connective quasi-coherent sheaves $\mathscr{F}$. As this assertion is
local, it suffices to show that if $A\to B$ is map of simplicial commutative
rings which induces a surjective map of discrete commutative rings
$\pi_{0}(A)\twoheadrightarrow\pi_{0}(B)$, and $M$ is a connective $A$-module,
then the natural map $M\to M\otimes_{A}B$ induces an epimorphism
$\pi_{0}(M)\twoheadrightarrow\pi_{0}(M\otimes_{A}B)$. But this is evident by
virtue of
$\pi_{0}(M\otimes_{A}B)\simeq\operatorname{Tor}_{0}^{\pi_{0}(A)}(\pi_{0}(M),\pi_{0}(B))$
(see [HA, Corollary 7.2.1.23]). ∎

Next, we restrict our attention to the “one-step flag of one-point
modifications” of quasi-coherent sheaves with a specified range of perfect-
amplitude.

######  Notation 8.6.

Suppose that the map of prestacks $X\to S$ is smooth, separated and a relative
derived scheme of constant relative dimension $d\geq 1$. Then the smooth
morphism $X\to S$ is equipped with a relative canonical line bundle given by
the formula $\mathscr{K}_{X/S}=\bigwedge^{d}L_{X/S}$, where $L_{X/S}$ is the
relative cotangent complex (Definition 4.28). In this situation:

  1. (1)

For any prestack $Y$, we let $\mathrm{Perf}(Y)_{[0,d-1]}$ denote the full
subcategory of $\operatorname{QCoh}(Y)$ spanned by objects of perfect-
amplitude contained in $[0,d-1]$ (that is, spanned by connective perfect
objects of Tor-amplitude $\leq d-1$).

  2. (2)

We let $\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}$ denote the subfunctor of
$\mathfrak{M}$ such that for each map $\eta\colon T=\operatorname{Spec}R\to
S$,
$\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}(\eta)=\mathrm{Perf}(X\times_{S}T)_{[0,d-1]}^{\simeq}$
is the full subcategory of
$\mathfrak{M}(\eta)=\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn},\,\simeq}$
spanned by objects of perfect-amplitude contained in $[0,d-1]$. By abuse of
notation, we also let
$\mathscr{U}\in\mathrm{Perf}(X\times_{S}\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]})_{[0,d-1]}$
denote the universal perfect object.

  3. (3)

Similarly, we let $\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}$ denote the
subfunctor of $\mathfrak{Z}$ such that for each map $\eta\colon
T=\operatorname{Spec}R\to S$, $\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}(\eta)$
is the full subcategory of $\mathfrak{Z}(\eta)$ spanned by those

$\zeta=((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});\,x\colon T\to
X\times_{S}T;\,\mathscr{L}\in\operatorname{Pic}(T))\in\mathfrak{Z}(\eta)$

for which
$\mathscr{F},\mathscr{F}^{\prime}\in\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}(T)$
(that is, $\mathscr{F}$ and $\mathscr{F}^{\prime}$ are of perfect-amplitude in
$[0,d-1]$). By abuse of notation, we use the same notation as Construction
8.4,

$((\mathscr{U}^{\prime}\xrightarrow{\phi_{\rm univ}}\mathscr{U});\,x_{\rm
univ};\,\mathscr{L}_{\rm univ}),$

to denote the universal element of $\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}$,
where
$\mathscr{U}^{\prime},\mathscr{U}\in\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X)_{[0,d-1]}$.

The following result shows that when restricted to perfect objects with a
specified range of Tor-amplitudes, the projection $\pi_{+}$ also fits in the
picture of derived projectivizations.

######  Proposition 8.7 (Compare with [Ne19, Proposition 2.8]).

In the situation of Notation 8.6 and Construction 8.4, the following is true:

  1. (1)

The restriction
$\pi_{-}|_{\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}\to\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$
of the projection map $\pi_{-}$ identifies
$\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}$ with the derived projectivization
$\mathbb{P}_{\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X}(\mathscr{U})$,
and for which there is a canonical identification of line bundles
$\mathscr{O}(1)\simeq\mathscr{L}_{\rm univ}$.

  2. (2)

The restriction
$\pi_{+}|_{\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}\to\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$
of the projection map $\pi_{+}$ identifies
$\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}$ with the derived projectivization
$\mathbb{P}_{\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X}(\Sigma^{d-1}\,\mathscr{U}^{\vee}\otimes\mathscr{K}_{X/S})$,
and for which there is a canonical identification of line bundles
$\mathscr{O}(1)\simeq\mathscr{L}_{\rm univ}^{\vee}$.

Notation. In the above assertion (2) and for the rest of this section, by
abuse of notations, we shall use the same notation $\mathscr{K}_{X/S}$ to
denote the line bundle $f^{*}\mathscr{K}_{X/S}$ for any given map of prestacks
$f\colon Y\to X$. For example, in the expression of (2),
$\mathscr{U}^{\vee}\otimes\mathscr{K}_{X/S}$ denotes the object
$\mathscr{U}^{\vee}\otimes\mathrm{pr}_{2}^{*}\mathscr{K}_{X/S}$, where
$\mathrm{pr}_{2}\colon\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X\to X$
is the natural projection.

###### Proof.

For any map $\eta\colon T=\operatorname{Spec}R\to S$,
$R\in\operatorname{CAlg}^{\Delta}$, since $X\to S$ is a smooth, separated
relative derived scheme, any section $x\colon T\to X\times_{S}T$ of the
projection $X\times_{S}T\to T$ is a quasi-smooth closed immersion. In
particular, $x_{*}\colon\operatorname{QCoh}(T)\to\operatorname{QCoh}(X\times
T)$ carries perfect complexes to perfect complexes (Theorem 3.7 (3)). Let
$x_{!}$ denote the left adjoint of the pullback functor $x^{*}$, then there is
a canonical equivalence
$x_{!}(\mathscr{E})\simeq\Sigma^{-d}x_{*}(\mathscr{E}\otimes\mathscr{K}_{X/S}^{\vee})$
for all $\mathscr{E}\in\operatorname{QCoh}(X\times_{S}T)$ (Theorem 3.7 (22
_g_)). In particular, for any $\mathscr{L}\in\operatorname{Pic}(T)$, we have
canonical equivalences $(x_{*}\mathscr{L})^{\vee}\simeq
x_{!}(\mathscr{L}^{\vee})\simeq\Sigma^{-d}x_{*}(\mathscr{L}^{\vee}\otimes\mathscr{K}_{X/S}^{\vee})$
(Theorem 3.7 (22 _d_)), so that $x_{*}\mathscr{L}$ and
$\Sigma^{d}\,(x_{*}\mathscr{L})^{\vee}$ both have perfect-amplitude contained
in $[0,d]$.

We first prove assertion (1). By virtue of Proposition 8.5, we only need to
show for any $\mathscr{F}\in\mathrm{Perf}(X\times_{S}T)_{[0,d-1]}$,
$\mathscr{L}\in\operatorname{Pic}(T)$, and a map $\varphi\colon\mathscr{F}\to
x_{*}\mathscr{L}$ which is surjective on $\pi_{0}$, the fiber of $\varphi$ is
of perfect-amplitude in $[0,d-1]$. Let
$\mathscr{F}^{\prime}=\operatorname{fib}(\varphi)$, then
$\mathscr{F}^{\prime}$ is clearly a connective perfect object. On the other
hand, from the fiber sequence
$\Sigma^{d-1}\mathscr{F}^{\vee}\to\Sigma^{d-1}\mathscr{F}^{\prime\vee}\to\Sigma^{d}(x_{*}\mathscr{L})^{\vee}$
we obtain that $\Sigma^{d-1}\mathscr{F}^{\prime\vee}$ is a connective perfect
object. Therefore $\mathscr{F}^{\prime}$ is of perfect-amplitude contained in
$[0,d-1]$. This proves assertion (1).

We next prove assertion (2). For any given map
$\widetilde{\eta}=(f^{\prime}\times_{S}g)\colon
T\to\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$, we set
$\mathscr{F}^{\prime}=(\operatorname{id}_{X}\times_{S}f^{\prime})^{*}(\mathscr{U})\in\mathrm{Perf}(X\times_{S}T)^{\mathrm{cn}}_{[0,d-1]}$
and $x=(g\times_{S}\operatorname{id}_{T})\colon T\to X\times_{S}T$, so that
$\widetilde{\eta}^{*}\,\mathscr{U}\simeq x^{*}\mathscr{F}^{\prime}$. Then the
space of liftings of $\widetilde{\eta}$ along
$\pi_{+}|_{\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}}$ is canonically identified
with the space of pairs

$((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F})\in(\mathrm{Perf}(X\times_{S}T)_{[0,d-1]})_{\mathscr{F}^{\prime}/},\mathscr{L}\in\operatorname{Pic}(T))$

for which there is a fiber sequence
$\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F}\to x_{*}\mathscr{L}$. This
space is homotopy equivalent to the space of pairs
$(\mathscr{L}\in\operatorname{Pic}(T),x_{*}\mathscr{L}\xrightarrow{\psi}\Sigma\mathscr{F}^{\prime})$
for which ${\operatorname{fib}}(\psi)$ is of perfect-amplitude in $[0,d-1]$
(and a homotopy inverse of the above map is given by setting $\mathscr{F}={\rm
fib}(\psi)$). There are canonical homotopy equivalences of spaces

$\displaystyle\mathrm{Map}_{\operatorname{QCoh}(X\times_{S}T)}(x_{*}\mathscr{L},\Sigma\mathscr{F}^{\prime})$
$\displaystyle\simeq\mathrm{Map}_{\operatorname{QCoh}(X\times_{S}T)}(\Sigma^{-1}\,\mathscr{F}^{\prime\vee},(x_{*}\mathscr{L})^{\vee})$
$\displaystyle\simeq\mathrm{Map}_{\operatorname{QCoh}(X\times_{S}T)}(\Sigma^{-1}\,\mathscr{F}^{\prime\vee},x_{!}(\mathscr{L}^{\vee}))$
$\displaystyle\simeq\mathrm{Map}_{\operatorname{QCoh}(X\times_{S}T)}(\Sigma^{-1}\,\mathscr{F}^{\prime\vee},\Sigma^{-d}x_{*}(\mathscr{L}^{\vee}\otimes\mathscr{K}_{X/S}^{\vee}))$
$\displaystyle\simeq\mathrm{Map}_{\operatorname{QCoh}(T)}(x^{*}(\Sigma^{d-1}\,\mathscr{F}^{\prime\vee}\otimes\mathscr{K}_{X/S}),\mathscr{L}^{\vee}).$

Let $\rho\colon
x^{*}(\Sigma^{d-1}\,\mathscr{F}^{\prime\vee}\otimes\mathscr{K}_{X/S})\to\mathscr{L}^{\vee}$
denote the map which corresponds to $\psi\colon
x_{*}\mathscr{L}\to\Sigma\,\mathscr{F}^{\prime}$ via the above equivalences.
To prove the assertion (2), it remains to show

  1. $(enumi^{\prime})$

$\rho$ is surjective on $\pi_{0}$ if and only if $\operatorname{fib}(\psi)$ is
of perfect-amplitude in $[0,d-1]$.

On the one hand, the assertion ($enumi$) of Proposition 8.5 implies that

  1. $(enumienumi^{\prime})$

$\rho$ is surjective on $\pi_{0}$ if and only if
$\Sigma^{d}\psi^{\vee}\colon\Sigma^{d-1}\mathscr{F}^{\prime\vee}\to\Sigma^{d}(x_{*}\mathscr{L})^{\vee}$
is surjective on $\pi_{0}$.

We now deduce assertion ($enumi^{\prime}$) from ($enumienumi^{\prime}$) as
follows. If we assume $\mathscr{F}:=\operatorname{fib}(\psi)$ is of perfect-
amplitude in $[0,d-1]$, then
$\operatorname{fib}(\Sigma^{d}\psi^{\vee})\simeq\Sigma^{d-1}\mathscr{F}^{\vee}$
is connective, which implies that $\rho$ is surjective on $\pi_{0}$, by virtue
of assertion ($enumienumi^{\prime}$). On the other hand, if $\rho$ is
surjective on $\pi_{0}$, then
$\mathscr{M}:=\operatorname{fib}(\Sigma^{d}\psi^{\vee})$ is connective by
virtue of assertion ($enumienumi^{\prime}$). Let
$\mathscr{F}:=\operatorname{fib}(\psi)$, then from the fiber sequence
$\mathscr{F}^{\prime}\to\mathscr{F}\to x_{*}\mathscr{L}$ we obtain that
$\mathscr{F}$ is connective and perfect. By virtue of the equivalence
$\mathscr{M}\simeq\Sigma^{d-1}\mathscr{F}^{\vee}$ again and the connectivity
of $\mathscr{M}$, we obtain $\mathscr{F}$ is of perfect-amplitude contained in
$[0,d-1]$. This proves assertion ($enumi^{\prime}$). ∎

###  8.2. Hecke correspondences: the case of general flags

Let $X\to S$ be a map of prestacks. In this section, we generalize the
constructions and results of the preceding §8.1 to the case of general “flags
of one-point modifications” of quasi-coherent sheaves on $X$ over $S$.

######  Construction 8.8 (Compare with [Ne18, Definition 2.13]).

Let $X$ be a prestack over $S$. A set partition $\lambda$ of size $n$ is an
equivalence relation on the set $\\{1,2,\ldots,n\\}$. Let $\lambda$ a set
partition of size $n\geq 1$, and we let
$\mathfrak{Z}_{\lambda}\colon\operatorname{CAlg}^{\Delta}\to\mathcal{S}$
denote the functor which carries each map $\eta\colon
T=\operatorname{Spec}R\to S$ to the full subspace of

$\mathrm{Fun}(\Delta^{n},\operatorname{QCoh}(X\times_{S}T)^{\mathrm{cn}})^{\simeq}\times\mathrm{Map}_{/T}(T,X\times_{S}T)^{\times
n}\times(\operatorname{Pic}(T)^{\simeq})^{\times n}$

spanned by those elements

$\zeta=((\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\xrightarrow{\phi_{2}}\cdots\xrightarrow{\phi_{n}}\mathscr{F}_{n}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{1\leq i\leq n},\\{\mathscr{L}_{i}\\}_{1\leq i\leq n})$

for which the following conditions are satisfied:

  * •

The sections satisfy $x_{i}=x_{j}$ if $i\sim j$ in $\lambda$;

  * •

There are equivalences ${\rm
cofib}(\mathscr{F}_{i-1}\xrightarrow{\phi_{i}}\mathscr{F}_{i})\simeq(x_{i})_{*}\mathscr{L}_{i}$.

We denote the universal element of $\mathfrak{Z}_{\lambda}$ by

$\zeta_{\rm univ}=((\mathscr{U}_{0}\xrightarrow{\phi_{1,\rm
univ}}\mathscr{U}_{1}\xrightarrow{\phi_{2,\rm
univ}}\cdots\xrightarrow{\phi_{n,\rm univ}}\mathscr{U}_{n}),\\{x_{i,\rm
univ}\\}_{1\leq i\leq n},\\{\mathscr{L}_{i,\rm univ}\\}_{1\leq i\leq n}),$

where
$\mathscr{U}_{i}\in\operatorname{QCoh}(X\times_{S}\mathfrak{Z}_{\lambda})^{\mathrm{cn}}$,
$x_{i,\rm
univ}\in\mathrm{Map}_{/\mathfrak{Z}_{\lambda}}(\mathfrak{Z}_{\lambda},X\times_{S}\mathfrak{Z}_{\lambda})$,
$\mathscr{L}_{i,\rm univ}\in\operatorname{Pic}(\mathfrak{Z}_{\lambda})$.

Notice that if $n=1$, then $\mathfrak{Z}_{\lambda}=\mathfrak{Z}$. By
convention, we set $\mathfrak{Z}_{\lambda}=\mathfrak{M}$ if $n=0$.

######  Variant 8.9.

In the situation of Construction 8.8, we let
$\widetilde{\mathfrak{Z}}_{\lambda}$ be the functor which carries each map
$\eta\colon T=\operatorname{Spec}R\to S$ to the space of elements

$\widetilde{\zeta}=(\zeta,\\{\varphi_{i}\\}_{1\leq i\leq n})$

where

$\zeta=((\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\xrightarrow{\phi_{2}}\cdots\to\mathscr{F}_{n}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{1\leq i\leq n},\\{\mathscr{L}_{i}\\}_{1\leq i\leq
n})\in\mathfrak{Z}_{\lambda}(\eta),$
$(\varphi_{i}\colon\mathscr{F}_{i}\to(x_{i})_{*}\mathscr{L}_{i})\in\mathrm{Fun}(\Delta^{1},\operatorname{QCoh}(X\times_{S}T))^{\simeq},$

and for which the following conditions are satisfied:

  * •

The sections satisfy $x_{i}=x_{j}$ if $i\sim j$ in $\lambda$;

  * •

The sequences
$\mathscr{F}_{i-1}\xrightarrow{\phi_{i}}\mathscr{F}_{i}\xrightarrow{\varphi_{i}}(x_{i})_{*}\mathscr{L}_{i}$
are fiber sequences for all $1\leq i\leq n$.

Then by virtue of [HA, Remark 1.1.1.7], the natural forgetful map
$\widetilde{\mathfrak{Z}}_{\lambda}(\eta)\to\mathfrak{Z}_{\lambda}(\eta)$
which carries $\widetilde{\zeta}=(\zeta,\\{\varphi_{i}\\}_{1\leq i\leq n})$ to
$\zeta$ is a trivial Kan fibration. Therefore, we obtain that the forgetful
map $\widetilde{\mathfrak{Z}}_{\lambda}\to\mathfrak{Z}_{\lambda}$ is a natural
isomorphism of functors. From now on, we will fix an inverse of the
isomorphism $\widetilde{\mathfrak{Z}}_{\lambda}\to\mathfrak{Z}_{\lambda}$.
Unwinding the definitions, this means that we will fix a choice of fiber
sequences of the universal quasi-coherent sheaves

$\mathscr{U}_{i-1}\xrightarrow{\phi_{i,\rm
univ}}\mathscr{U}_{i}\xrightarrow{\varphi_{i,\rm univ}}(x_{i,\rm
univ})_{*}(\mathscr{L}_{i,\rm univ})$

in $\operatorname{QCoh}(X\times_{S}\mathfrak{Z}_{\lambda})$ for all $1\leq
i\leq n$.

We also consider the natural forgetful maps between the functors of the form
$\mathfrak{Z}_{\lambda}$:

######  Notation 8.10 (Compare with [Ne18, §2.14]).

In the situation of Construction 8.8:

  1. (1)

Let $\lambda$ be a set partition of size $n\geq 1$. We let $\lambda|$ denote
the set partition of $\\{1,\ldots,n-1\\}$ which is obtained by dropping the
last element of $\lambda$, and let $|\lambda$ denote the set partition of
$\\{2,\ldots,n\\}$ which is obtained by dropping the first element of
$\lambda$.

  2. (2)

We consider the following maps:

${\mathfrak{Z}_{\lambda}}$${\mathfrak{Z}_{\lambda|}}$${X}$${\mathfrak{Z}_{|\lambda},}$$\scriptstyle{p_{+}}$$\scriptstyle{p_{X,i}}$$\scriptstyle{p_{-}}$

where $p_{+}$, $p_{-}$, and $p_{X}$ are the forgetful maps which carry each
element

$\zeta=((\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\xrightarrow{\phi_{2}}\cdots\xrightarrow{\phi_{n}}\mathscr{F}_{n}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{1\leq i\leq n},\\{\mathscr{L}_{i}\\}_{1\leq i\leq
n})\in\mathfrak{Z}_{\lambda}(\eta),$

where $\eta\colon T=\operatorname{Spec}R\to S$ is any given map, to the
elements

$\displaystyle p_{+}(\zeta)$
$\displaystyle=((\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\to\cdots\xrightarrow{\phi_{n-1}}\mathscr{F}_{n-1}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{1\leq i\leq n-1},\\{\mathscr{L}_{i}\\}_{1\leq i\leq
n-1})\in\mathfrak{Z}_{\lambda|}(\eta),$ $\displaystyle p_{-}(\zeta)$
$\displaystyle=((\mathscr{F}_{1}\xrightarrow{\phi_{2}}\mathscr{F}_{2}\to\cdots\xrightarrow{\phi_{n}}\mathscr{F}_{n}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{2\leq i\leq n},\\{\mathscr{L}_{i}\\}_{2\leq i\leq
n})\in\mathfrak{Z}_{|\lambda}(\eta),$

and the composition
$p_{X,i}(\zeta)=(T\xrightarrow{x_{i}}T\times_{S}X\xrightarrow{\mathrm{pr}_{X}}X)\in
X(\eta)$, respectively.

  3. (3)

We define a forgetful map $\pi_{-}$ with source $\mathfrak{Z}_{\lambda}$ by
the following rules: we let

     * •

$\pi_{-}=(p_{-}\times_{S}p_{X,1})\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{|\lambda}\times_{S}X$,
if $1$ is not equivalent to any $i\in\\{2,\ldots,n\\}$ in $\lambda$.

     * •

$\pi_{-}=p_{-}\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{|\lambda}$, if $1$
is equivalent to some $i\in\\{2,\ldots,n\\}$ in $\lambda$.

Similarly, we define a forgetful map $\pi_{+}$ with source
$\mathfrak{Z}_{\lambda}$ by the following rules: we let

     * •

$\pi_{+}=(p_{+}\times_{S}p_{X,n})\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{\lambda|}\times_{S}X$,
if $n$ is not equivalent to any $i\in\\{1,\ldots,n-1\\}$ in $\lambda$.

     * •

$\pi_{+}=p_{+}\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{\lambda|}$, if $1$
is equivalent to some $i\in\\{1,\ldots,n-1\\}$ in $\lambda$.

The following result is a direct generalization of Proposition 8.5:

######  Proposition 8.11 (Compare with [Ne18, Proposition 2.16])).

In the situation of Construction 8.8 and Notation 8.10, and suppose that the
map of prestacks $X\to S$ is separated and a relative derived scheme.

  1. (1)

If $1$ is not equivalent to any $i\in\\{2,\ldots,n\\}$ in $\lambda$, then the
map $\pi_{-}\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{|\lambda}\times_{S}X$
identifies $\mathfrak{Z}_{\lambda}$ with the derived projectivization
$\mathbb{P}_{\mathfrak{Z}_{|\lambda}\times_{S}X}(\mathscr{U}_{1})$, and for
which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{1,\rm univ}$.

  2. (2)

If $1$ is equivalent to some $i\in\\{2,\ldots,n\\}$ in $\lambda$, then the map
$\pi_{-}\colon\mathfrak{Z}_{\lambda}\to\mathfrak{Z}_{|\lambda}$ identifies
$\mathfrak{Z}_{\lambda}$ with the derived projectivization
$\mathbb{P}_{\mathfrak{Z}_{|\lambda}}(x_{i,\mathrm{univ}}^{*}(\mathscr{U}_{1}))$,
and for which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{1,\rm univ}$.

###### Proof.

The assertion (1) can be proved in the same way as Proposition 8.5. To prove
assertion (2), assume that $1\sim i$ in $\lambda$, where $2\leq i\leq n$, so
that $x_{1}=x_{i}$, then for any $\eta\colon T=\operatorname{Spec}R\to S$, the
fiber of $\pi_{-}$ over a point

$\zeta=((\mathscr{F}_{1}\xrightarrow{\phi_{2}}\cdots\xrightarrow{\phi_{n}}\mathscr{F}_{n}),\\{x_{2}\colon
T\to X\times_{S}T\\}_{2\leq i\leq n},\\{\mathscr{L}_{2}\\}_{1\leq 2\leq
n})\in\mathfrak{Z}_{|\lambda}(\eta)$

is canonically identified with the space of pairs $(\mathscr{F}_{1}\to
x_{1*}\,\mathscr{L}_{1},\mathscr{L}_{1}\in\operatorname{Pic}(T))$ for which
the map $\mathscr{F}_{1}\to x_{1*}\,\mathscr{L}_{1}$ is surjective on
$\pi_{0}$. By virtue of the adjunction

$\underline{\mathrm{Map}}_{\operatorname{QCoh}(X\times_{S}T)}(\mathscr{F}_{1},x_{1\,*}\mathscr{L}_{1})\simeq\underline{\mathrm{Map}}_{\operatorname{QCoh}(T)}(x_{i}^{*}\mathscr{F}_{1},\mathscr{L}_{1})$

and the same argument of the proof of Proposition 8.5, the fiber of $\pi_{-}$
over $\zeta$ is canonically homotopy equivalent to
$\mathbb{P}_{\mathfrak{Z}_{|\lambda}}(x_{i}^{*}\mathscr{U}_{1})(\eta)$, which
proves the desired assertion. ∎

Next, we focus our attention to general “flags of one-point modifications” of
quasi-coherent sheaves with a specified range of perfect-amplitude.

######  Notation 8.12.

In the situation of Construction 8.8 and Notation 8.6, we let
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ denote the subfunctor of
$\mathfrak{Z}_{\lambda}$ such that, for each $\eta\colon
T=\operatorname{Spec}R\to S$,
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}(\eta)$ is the full
subcategory of $\mathfrak{Z}_{\lambda}(\eta)$ spanned by those elements

$\zeta=((\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\xrightarrow{\phi_{2}}\cdots\xrightarrow{\phi_{n}}\mathscr{F}_{n}),\\{x_{i}\colon
T\to X\times_{S}T\\}_{1\leq i\leq n},\\{\mathscr{L}_{i}\\}_{1\leq i\leq
n})\in\mathfrak{Z}_{\lambda}(\eta)$

for which
$\mathscr{F}_{i}\in\mathfrak{M}^{\mathrm{perf}}_{[0,d-1]}(\eta)=\mathrm{Perf}(X\times_{S}T)_{[0,d-1]}^{\simeq}$,
$0\leq i\leq n$. By abuse of notation, we denote the universal element of
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ by the same notation as
Construction 8.8:

$\zeta_{\rm univ}=((\mathscr{U}_{0}\xrightarrow{\phi_{1,\rm
univ}}\mathscr{U}_{1}\xrightarrow{\phi_{2,\rm
univ}}\cdots\xrightarrow{\phi_{n,\rm univ}}\mathscr{U}_{n}),\\{x_{i,\rm
univ}\\}_{1\leq i\leq n},\\{\mathscr{L}_{i,\rm univ}\\}_{1\leq i\leq n}),$

where
$\mathscr{U}_{i}\in\mathrm{Perf}(X\times_{S}(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]})_{[0,d-1]}$,
$x_{i,\rm univ}$ are sections of the projection
$X\times_{S}(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\to(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$,
$\mathscr{L}_{i,\rm
univ}\in\operatorname{Pic}((\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]})$.
Then there are fiber sequences on
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$:

$\mathscr{U}_{i-1}\xrightarrow{\phi_{i,\rm
univ}}\mathscr{U}_{i}\xrightarrow{\varphi_{i,\rm univ}}(x_{i,\rm
univ})_{*}\mathscr{L}_{i,\rm univ}.$

The following is a generalization of Proposition 8.7 and [Ne18, Proposition
2.16]:

######  Proposition 8.13.

Suppose that $X\to S$ is map of prestacks which is smooth, separated, and a
relative derived scheme of constant relative dimension $d\geq 1$. We let
$\mathscr{K}_{X/S}$ denote the relative canonical line bundle of $X\to S$ and
let $\lambda$ be a set partition of size $n\geq 2$. Consider the moduli
functors and maps defined in Notations 8.10 and 8.12. Then:

  1. (1)

The restriction map
$\pi_{-}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}}$ is a derived
projectivization of a quasi-coherent sheaf of perfect-amplitude contained in
$[0,d-1]$. More concretely:

    1. (i)

If $1$ is not equivalent to any element $i\in\\{2,\ldots,n\\}$ in $\lambda$,
then the restriction

$\pi_{-}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}}\colon(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\to(\mathfrak{Z}_{|\lambda})^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$

identifies $(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ with the
derived projectivization
$\mathbb{P}_{(\mathfrak{Z}_{|\lambda})^{\mathrm{perf}}_{[0,d-1]}\times_{S}X}(\mathscr{U}_{1})$,
and for which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{1,\rm univ}$.

    2. (ii)

If $1$ is equivalent to some $i\in\\{2,\ldots,n\\}$ in $\lambda$, then the
restriction

$\pi_{-}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}}\colon(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\to(\mathfrak{Z}_{|\lambda})^{\mathrm{perf}}_{[0,d-1]}$

identifies $(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ with the
derived projectivization
$\mathbb{P}_{(\mathfrak{Z}_{|\lambda})^{\mathrm{perf}}_{[0,d-1]}}(x_{i,\mathrm{univ}}^{*}(\mathscr{U}_{1}))$,
and for which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{1,\rm univ}$.

  2. (2)

The restriction map $\pi_{+}|_{\mathfrak{Z}^{\mathrm{perf}}_{[0,d-1]}}$ is a
derived projectivization of a quasi-coherent sheaf of perfect-amplitude
contained in $[0,d-1]$. More concretely:

    1. (i)

If $1$ is not equivalent to any element $i\in\\{2,\ldots,n\\}$ in $\lambda$,
then the restriction

$\pi_{+}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}}\colon(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\to(\mathfrak{Z}_{\lambda|})^{\mathrm{perf}}_{[0,d-1]}\times_{S}X$

identifies $(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ with the
derived projectivization
$\mathbb{P}_{(\mathfrak{Z}_{\lambda|})^{\mathrm{perf}}_{[0,d-1]}\times_{S}X}(\Sigma^{d-1}\,\mathscr{U}_{1}^{\vee}\otimes\mathscr{K}_{X/S})$,
and for which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{n,\rm univ}^{\vee}$;

    2. (ii)

If $1$ is equivalent to some $i\in\\{2,\ldots,n\\}$ in $\lambda$, then the
restriction

$\pi_{+}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}}\colon(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}\to(\mathfrak{Z}_{\lambda|})^{\mathrm{perf}}_{[0,d-1]}$

identifies $(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,d-1]}$ with the
derived projectivization
$\mathbb{P}_{(\mathfrak{Z}_{\lambda|})^{\mathrm{perf}}_{[0,d-1]}}(\Sigma^{d-1}x_{i,\mathrm{univ}}^{*}(\mathscr{U}_{1}^{\vee})\otimes\mathscr{K}_{X/S})$,
and for which there is a canonical identification
$\mathscr{O}(1)\simeq\mathscr{L}_{n,\rm univ}^{\vee}$.

###### Proof.

The assertions (1i) and (2i) follow from the same argument of Proposition 8.7;
The assertions (1ii) and (2ii) are proved similarly, in view of the proof of
Proposition 8.11 (2). ∎

###  8.3. Hecke correspondences for surfaces

In the situation of Proposition 8.13, if we set $d=2$, that is, if $X\to S$ is
a family of smooth, separated derived surfaces, then both the projections
$\pi_{\pm}|_{(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,1]}}$ are derived
projectivizations of sheaves of perfect-amplitude contained in $[0,1]$. In
particular, we could apply the generalized Serre’s Theorem 5.6, the results on
Beilinson’s relations (Proposition 6.19, Corollary 6.21), and the
projectivization formula (Theorem 7.5) to these situations. For simplicity of
exposition, we only include here part of all these results for the “one-step
flags” $\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$; The generalizations of these
results to the “general flags”
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,1]}$ are straightforward, by
virtue of Proposition 8.13.

######  Notation 8.14.

Let $X\to S$ be map of prestacks which is smooth, separated, and a relative
derived surface, with relative canonical line bundle
$\mathscr{K}_{X/S}=\bigwedge^{2}L_{X/S}$, where $L_{X/S}$ is the relative
cotangent complex (Definition 4.28). Let the functors $\mathfrak{Z}$,
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$, $\mathfrak{M}$,
$\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$, $\mathfrak{Z}_{\lambda}$,
$\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,1]}$ be defined as in
Constructions 8.4 and 8.8, and Notations 8.6 and 8.12.

  1. (1)

We let $\mathfrak{Z}_{2}^{\bullet}$ (resp.
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$) denote the prestack
$\mathfrak{Z}_{\lambda}$ (resp.
$(\mathfrak{Z}_{\lambda})^{\mathrm{perf}}_{[0,1]}$) in the case where
$\lambda$ is the set partition of $\\{1,2\\}$ for which $1\sim 2$. We let

$((\mathscr{U}_{0}\xrightarrow{\phi_{1,\rm
univ}}\mathscr{U}_{1}\xrightarrow{\phi_{2,\rm univ}}\mathscr{U}_{2});\,x_{\rm
univ};\,(\mathscr{L}_{1,\rm univ},\mathscr{L}_{2,\rm univ}))$

denote the universal element of
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$, where
$\mathscr{U}_{i}\in\mathrm{Perf}(X\times_{S}(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]})_{[0,1]}$,
$x_{\rm univ}\colon(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}\to
X\times_{S}(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ is a section
of the natural projection
$X\times_{S}(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}\to(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$,
and $\mathscr{L}_{i,\rm
univ}\in\operatorname{Pic}((\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]})$
are line bundles, and for which there are universal fiber sequences of quasi-
coherent sheaves on
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}\times_{S}X$:

$\mathscr{U}_{0}\xrightarrow{\phi_{1,\rm univ}}\mathscr{U}_{1}\to(x_{\rm
univ})_{*}\mathscr{L}_{1,\rm
univ},\qquad\mathscr{U}_{1}\xrightarrow{\phi_{2,\rm
univ}}\mathscr{U}_{2}\to(x_{\rm univ})_{*}\mathscr{L}_{2,\rm univ}.$

These fiber sequences induce canonical maps

$\rho_{1,\rm univ}\colon x_{\rm
univ}^{*}(\mathscr{U}_{1})\to\mathscr{L}_{1,\rm univ},\qquad\rho_{2,\rm
univ}\colon\Sigma\,x_{\rm
univ}^{*}(\mathscr{U}_{1}^{\vee})\to\mathscr{L}_{2,\rm
univ}^{\vee}\otimes\mathscr{K}_{X/S}^{\vee}.$

  2. (2)

We let $(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ denote
the universal incidence locus (Notation 7.4) for the derived projectivizations
$\pi_{-}|\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$ and
$\pi_{+}|\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$ of Proposition 8.13 over
$\mathfrak{M}^{\mathrm{perf}}_{[0,1]}\times_{S}X$. Unwinding the definitions,
$(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ is the closed
subfunctor of $(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ defined
by the pullback square

${(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}$${(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}$${(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}$${\mathbb{V}_{(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}(\Sigma\,(\mathscr{L}_{1,\rm
univ}^{\vee}\boxtimes\mathscr{L}_{2,\rm
univ})\otimes\mathscr{K}_{X/S}).}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$$\scriptstyle{i_{\mathbf{1}}}$$\scriptstyle{i_{\mathbf{0}}}$

Here, $i_{\mathbf{0}}$ is the inclusion of the zero section, and
$i_{\mathbf{1}}$ is the section of the natural projection
$\mathbb{V}_{(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}(\Sigma(\mathscr{L}_{1,\rm
univ}^{\vee}\boxtimes\mathscr{L}_{2,\rm
univ})\otimes\mathscr{K}_{X/S})\to(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$
that classifies the composite map

$\Sigma\,\mathscr{L}_{1,\rm univ}^{\vee}\boxtimes(\mathscr{L}_{2,\rm
univ}\otimes\mathscr{K}_{X/S})\xrightarrow{\Sigma\,\rho_{1,\rm
univ}^{\vee}\boxtimes\rho_{2,\rm univ}^{\vee}}x_{\rm
univ}^{*}(\mathscr{U}_{1})\otimes x_{\rm
univ}^{*}(\mathscr{U}_{1}^{\vee})\xrightarrow{\rm
ev}\mathscr{O}_{(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}}.$

  3. (3)

Let $\delta$ be an integer. For a prestack $Y$, we let
$\mathrm{Perf}(Y)_{[0,1],\operatorname{rk}=\delta}$ denote the full
subcategory of $\mathrm{Perf}(Y)$ spanned by perfect objects of Tor-amplitude
in $[0,1]$ and rank $\delta$. Similarly, we let
$\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$,
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$,
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$
and
$(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$
denote the subfunctors of $\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$,
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$,
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ and
$(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ such that
their spaces of $T$-point over a map $\eta\colon T\to S$ are respectively
spanned by those elements for which the components $\mathscr{F}_{i}$ all have
rank $\delta$, that is,
$\mathscr{F}_{i}\in\mathrm{Perf}(X\times_{S}T)_{[0,1],\operatorname{rk}=\delta}$.
(These notations are justified by the following fact: if $x$ is a quasi-smooth
closed immersion and $\mathscr{L}$ is a line bundle, then $x_{*}\mathscr{L}$
is perfect of rank $0$. Therefore, if a pair of perfect quasi-coherent sheaves
$(\mathscr{F}_{i-1},\mathscr{F}_{i})$ fits into a fiber sequence
$\mathscr{F}_{i-1}\to\mathscr{F}_{i}\to x_{*}\mathscr{L}$, then
$\operatorname{rk}\mathscr{F}_{i-1}=\operatorname{rk}\mathscr{F}_{i}$.)

  4. (4)

In particular, for each $\delta\geq 0$, we have a pullback square and a
commutative square:

${(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\times_{S}X}$$\scriptstyle{r_{+}^{\prime}}$$\scriptstyle{r_{-}^{\prime}}$$\scriptstyle{\mathrm{pr}_{-}}$$\scriptstyle{\mathrm{pr}_{+}}$
and
${(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}$${\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\times_{S}X,}$$\scriptstyle{r_{+}}$$\scriptstyle{r_{-}}$$\scriptstyle{\mathrm{pr}_{-}}$$\scriptstyle{\mathrm{pr}_{+}}$

respectively, where
$\mathrm{pr}_{+}=\pi_{-}|\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$
and
$\mathrm{pr}_{-}=\pi_{+}|\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$
are the forgetful maps (notice the change of signs due to the different
conventions) and the maps $r_{\pm}$ are given by the compositions
$r_{\pm}^{\prime}\circ\iota$.

######  Corollary 8.15.

Let $X\to S$ be map of prestacks which is smooth, separated, and a relative
derived surface, with relative canonical line bundle $\mathscr{K}_{X/S}$.
Consider the moduli functors and the maps between them defined in Notation
8.14. For simplicity of expressions, we introduce the following notations:

$\mathfrak{X}:=\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\times_{S}X,\qquad\mathscr{W}:=\Sigma\,\mathscr{U}^{\vee}\otimes\mathscr{K}_{X/S}\in\mathrm{Perf}(\mathfrak{X}).$

In particular, $\mathscr{L}_{\rm univ}$ denotes the universal line bundle on
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$,
$\mathscr{U}\in\mathrm{Perf}(\mathfrak{X})$ denotes the universal perfect
object over
$\mathfrak{X}=\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\times_{S}X$,
and the natural projections

$\mathrm{pr}_{\pm}=\pi_{\mp}|_{\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\to\mathfrak{X}$

are the maps which, for any given $\eta\colon T=\operatorname{Spec}R\to S$,
$R\in\operatorname{CAlg}^{\Delta}$, carry the element

$\zeta=((\mathscr{F}^{\prime}\xrightarrow{\phi}\mathscr{F});\,x\colon T\to
X\times_{S}T;\,\mathscr{L}\in\operatorname{Pic}(T))\in\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}(\eta)$

to the respective elements

$\displaystyle\mathrm{pr}_{+}(\zeta)$
$\displaystyle=(\mathscr{F},(T\xrightarrow{x}X\times_{S}T\xrightarrow{\mathrm{pr}_{X}}X))\in\mathfrak{X}(\eta),$
$\displaystyle\mathrm{pr}_{-}(\zeta)$
$\displaystyle=(\mathscr{F}^{\prime},(T\xrightarrow{x}X\times_{S}T\xrightarrow{\mathrm{pr}_{X}}X))\in\mathfrak{X}(\eta).$

  1. (1)

If the rank $\delta\geq 1$. Then:

    1. (1i)

For any $d\in\mathbb{Z}$, there are canonical equivalences

$(\mathrm{pr}_{+})_{*}(\mathscr{L}_{\rm univ}^{\otimes
d})\simeq\begin{cases}\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{U})&\text{if~{}}d\geq
0;\\\ 0&\text{if ~{}}-\delta+1\leq d\leq-1;\\\
\Sigma^{1-\delta}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{-d-\delta}(\mathscr{U})\otimes_{\mathscr{O}_{\mathfrak{X}}}\det\mathscr{U}\big{)}^{\vee}&\text{if~{}}d\leq-\delta\end{cases}$

and canonical fiber sequences

$\displaystyle\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{W})\to(\mathrm{pr}_{-})_{*}(\mathscr{L}_{\rm
univ}^{\otimes-d})\to\Sigma^{1+\delta}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{\delta-d}(\mathscr{W})\otimes\det\mathscr{W}\big{)}^{\vee}.$

    2. (1ii)

For any perfect stack $T$ (see Definition 6.3; for example, for any quasi-
compact, quasi-separated derived scheme $T$) and any morphism
$T\to\mathfrak{X}$, we let $\mathrm{pr}_{+,T}\colon\mathfrak{Z}_{T}\to T$
denote the base-change of
$\mathrm{pr}_{+}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\to\mathfrak{X}$
along $T\to\mathfrak{X}$, let $\mathscr{L}_{T}$ denote the base-change of
$\mathscr{L}_{\rm univ}$, and let $\mathscr{F}_{T}$, $x_{T}$ be similarly
defined. Then the sequence of line bundles
$\mathscr{O}_{\mathfrak{Z}_{T}}=\mathscr{L}_{T}^{0},\mathscr{L}_{T},\cdots,\mathscr{L}_{T}^{\otimes\delta-1}$
is a relative exceptional sequence on $\mathfrak{Z}_{T}$ over $T$, whose left
dual relative exceptional sequence is given by
$\Sigma^{\delta-1}\bigwedge^{\delta-1}(L_{\mathrm{pr}_{+,T}}\otimes\mathscr{L}_{T}),\cdots,\Sigma(L_{\mathrm{pr}_{+,T}}\otimes\mathscr{L}_{T}),\mathscr{O}_{\mathfrak{Z}_{T}}$.
(Notice that, by virtue of Euler’s fiber sequence Theorem 4.27,
$L_{\mathrm{pr}_{+,T}}\otimes\mathscr{L}_{T}$ is canonically equivalent to
$\operatorname{fib}(x_{T}^{*}(\mathscr{F}_{T})\to\mathscr{L}_{T})$.) Moreover,
these two relative exceptional sequences satisfy Beilinson’s relations
Corollary 6.21.

    3. (1iii)

The functors

$\Phi=r_{+\,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$

and the functors

$\Psi_{i}=\mathrm{pr}_{+}^{*}(\underline{\hphantom{A}})\otimes\mathscr{L}_{\rm
univ}^{\otimes
i}\colon\operatorname{QCoh}(\mathfrak{X})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}),$

for all $i\in\mathbb{Z}$, and their restrictions to the subcategories of
perfect objects, are all fully faithful. Furthermore, there are an induced
semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$
$\displaystyle=\big{\langle}\Phi(\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})),~{}\Psi_{1}(\operatorname{QCoh}(\mathfrak{X})),\cdots,\Psi_{\delta}(\operatorname{QCoh}(\mathfrak{X}))\big{\rangle}.$
$\displaystyle\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$
$\displaystyle=\big{\langle}\Phi(\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})),~{}\Psi_{1}(\mathrm{Perf}(\mathfrak{X})),\cdots,\Psi_{\delta}(\mathrm{Perf}(\mathfrak{X}))\big{\rangle}.$

  2. (2)

If the rank $\delta=0$, then for each $d\in\mathbb{Z}$, there are canonical
equivalences

$\displaystyle(\mathrm{pr}_{+})_{*}(\mathscr{L}_{\rm univ}^{\otimes d})$
$\displaystyle\simeq\operatorname{cofib}\Big{(}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{-d}(\mathscr{U})\otimes_{\mathscr{O}_{\mathfrak{X}}}\det\mathscr{U}\big{)}^{\vee}\to\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{U})\Big{)}.$
$\displaystyle(\mathrm{pr}_{-})_{*}(\mathscr{L}_{\rm univ}^{\otimes-d})$
$\displaystyle\simeq\operatorname{cofib}\Big{(}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{-d}(\mathscr{W})\otimes_{\mathscr{O}_{\mathfrak{X}}}\det\mathscr{W})^{\vee}\to\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{W})\Big{)}.$

Moreover, the functors

$r_{+\,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\qquad
r_{+\,*}\,r_{-}^{*}\colon\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\to\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})$
$r_{-\,*}\,r_{+}^{*}\colon\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\qquad
r_{-\,*}\,r_{+}^{*}\colon\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})\to\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=0})$

are exact equivalences of stable $\infty$-categories.

  3. (3)

If the rank $\delta\leq-1$. Then:

    1. (3i)

For all $d\in\mathbb{Z}$, there are canonical equivalences

$(\mathrm{pr}_{-})_{*}(\mathscr{L}_{\rm
univ}^{\otimes-d})\simeq\begin{cases}\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{W})&\text{if~{}}d\geq
0;\\\ 0&\text{if ~{}}\delta+1\leq d\leq-1;\\\
\Sigma^{1+\delta}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{-d+\delta}(\mathscr{W})\otimes_{\mathscr{O}_{\mathfrak{X}}}\det\mathscr{W}\big{)}^{\vee}&\text{if~{}}d\leq\delta,\end{cases}$

and canonical fiber sequences

$\displaystyle\operatorname{Sym}_{\mathfrak{X}}^{d}(\mathscr{U})\to(\mathrm{pr}_{+})_{*}(\mathscr{L}_{\rm
univ}^{\otimes
d})\to\Sigma^{1-\delta}\big{(}\operatorname{Sym}_{\mathfrak{X}}^{-\delta-d}(\mathscr{U})\otimes_{\mathscr{O}_{\mathfrak{X}}}\det\mathscr{U}\big{)}^{\vee}.$

    2. (3ii)

For any perfect stack $T$ (for example, for any quasi-compact, quasi-separated
derived scheme $T$) and any morphism $T\to\mathfrak{X}$, we let
$\mathrm{pr}_{-,T}\colon\mathfrak{Z}_{T}\to T$ denote the base-change of
$\mathrm{pr}_{-}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}\to\mathfrak{X}$
along $T\to\mathfrak{X}$, let $\mathscr{L}_{T}$ denote the base-change of
$\mathscr{L}_{\rm univ}$, and let $\mathscr{F}^{\prime}_{T}$, $x_{T}$ be
similarly defined. Then the sequence of line bundles
$\mathscr{O}_{\mathfrak{Z}_{T}}=\mathscr{L}_{T}^{0},\mathscr{L}_{T}^{\vee},\cdots,\mathscr{L}_{T}^{\otimes-\delta-1}$
is a relative exceptional sequence on $\mathfrak{Z}_{T}$ over $T$, whose left
dual relative exceptional sequence is given by
$\Sigma^{\delta-1}\bigwedge^{\delta-1}(L_{\mathrm{pr}_{-,T}}\otimes\mathscr{L}_{T}^{\vee}),\cdots,\Sigma(L_{\mathrm{pr}_{-,T}}\otimes\mathscr{L}_{T}^{\vee}),\mathscr{O}_{\mathfrak{Z}_{T}}$.
(Notice that, by virtue of Euler fiber sequence Theorem 4.27,
$L_{\mathrm{pr}_{-,T}}\otimes\mathscr{L}_{T}^{\vee}$ is canonically equivalent
to
$\operatorname{fib}(x_{T}^{*}(\Sigma\,\mathscr{F}_{T}^{\prime\vee}\otimes\mathscr{K}_{X_{T}/T})\to\mathscr{L}_{T}^{\vee})$.)
Moreover, these two relative exceptional sequences satisfy Beilinson’s
relations Corollary 6.21.

    3. (3iii)

The functors

$\Phi^{\prime}=r_{-\,*}\,r_{+}^{*}\colon\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$

and the functor

$\Psi_{i}^{\prime}=\mathrm{pr}_{-}^{*}(\underline{\hphantom{A}})\otimes\mathscr{L}_{\rm
univ}^{\otimes-i}\colon\operatorname{QCoh}(\mathfrak{X})\to\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}),$

for all $i\in\mathbb{Z}$, and their restrictions to the subcategories of
perfect objects, are all fully faithful. Furthermore, there are an induced
semiorthogonal decompositions

$\displaystyle\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$
$\displaystyle=\big{\langle}\Phi^{\prime}(\operatorname{QCoh}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})),~{}\Psi_{1}^{\prime}(\operatorname{QCoh}(\mathfrak{X})),\cdots,\Psi_{-\delta}^{\prime}(\operatorname{QCoh}(\mathfrak{X}))\big{\rangle}.$
$\displaystyle\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})$
$\displaystyle=\big{\langle}\Phi^{\prime}(\mathrm{Perf}(\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta})),~{}\Psi_{1}^{\prime}(\mathrm{Perf}(\mathfrak{X})),\cdots,\Psi_{-\delta}^{\prime}(\mathrm{Perf}(\mathfrak{X}))\big{\rangle}.$

###### Proof.

In view of Proposition 8.7, the desired results follow from the Generalized
Serre’s Theorem 5.6, the results on Beilinson’s relations (Proposition 6.19,
Corollary 6.21), and the projectivization formula (Theorem 7.5). ∎

######  Remark 8.16.

In the situation of assertion 2, the same argument of [JL18, Theorem 4.1]
shows that the equivalence functors are non-trivial, and the composition of
equivalences $(r_{-,*}\,r_{+}^{*})\circ(r_{+,*}\,r_{-}^{*})$ is a spherical
twist in the sense of [ST, AL].

######  Remark 8.17.

As a consequence, we could “decategorify” all the formulae of Corollary 8.15
and obtain the corresponding formulae on the level of any additive invariant
$\mathbb{E}$ in the sense of [BGT, Definition 6.1]; see Remark 7.7. For
example, we could take $\mathbb{E}$ to be algebraic K-theory, topological
Hochschild homology, or topological cyclic homology.

##  Appendix A Simplicial Models for Symmetric and Koszul Algebras

In this appendix, we provide the concrete simplicial models for symmetric
algebras and Koszul algebras, especially in the case of two-term complexes of
locally free modules. These constructions are useful for computations in
concrete examples. (However, they do not play a role in the proofs of this
paper, so we might delete them in the final version.)

###  A.1. Dold–Puppe and Quillen’s construction

In this subsection, we show that Lurie’s framework of non-abelian derived
functors recovers the construction of Dold–Puppe [DP] and Quillen [Qu] using
simplicial resolutions. Let $R$ be an ordinary commutative ring, and suppose
that
$F\colon\operatorname{Mod}_{R}^{\mathrm{cn}}\to\operatorname{Mod}_{R}^{\mathrm{cn}}$
is a functor which preserves sifted colimits and projective $R$-modules (for
example, if $F$ is one of the functors $\operatorname{Sym}^{n}_{R}$,
$\bigwedge^{n}_{R}$, or $\Gamma_{R}^{n}$ of Construction 2.10). We let
$f=F|\operatorname{Mod}_{R}^{\mathrm{proj}}$ be the restriction of $F$ to
projective $R$-modules. From the canonical equivalence
$\operatorname{Mod}_{R}^{\mathrm{cn}}\simeq\operatorname{N}_{\bullet}^{\rm
hc}(\mathrm{Fun}(\operatorname{{\bf\Delta}},\operatorname{Mod}_{R}^{\heartsuit})^{\circ})$,
we obtain that each $R$-module $M\in\operatorname{Mod}_{R}^{\mathrm{cn}}$
corresponds to a simplicial $R$-module $P_{\bullet}$ for which each $P_{i}$ is
a projective $R$-module. We could regard $P_{\bullet}$ as a simplicial object
of $\operatorname{Mod}^{\mathrm{cn}}_{R}$ via the map
$\operatorname{{\bf\Delta}}^{\mathrm{op}}\to\operatorname{Mod}_{R}^{\mathrm{proj}}\subseteq\operatorname{Mod}^{\mathrm{cn}}_{R}$.
By virtue of [HTT, Theorem 4.2.4.1] and [HTT, Proposition 5.5.9.14], we can
identify $M$ with the geometric realization $|P_{\bullet}|$ of $P_{\bullet}$
(in other words, $P_{\bullet}$ is a simplicial resolution of $M$). Since $F$
preserves sifted colimits, we obtain canonical equivalences

$F(M)\simeq F(|P_{\bullet}|)\simeq|f(P_{\bullet})|,$

where $f(P_{\bullet})$ is the simplicial object obtained by applying $f$
level-wisely to $P_{\bullet}$, that is,

$f(P_{\bullet})\colon\operatorname{{\bf\Delta}}^{\mathrm{op}}\xrightarrow{P_{\bullet}}\operatorname{Mod}_{R}^{\mathrm{proj}}\xrightarrow{f=F|\operatorname{Mod}_{R}^{\mathrm{proj}}}\operatorname{Mod}_{R}^{\mathrm{proj}}\subseteq\operatorname{Mod}^{\mathrm{cn}}_{R}.$

Similarly, the symmetric algebra functor
$\operatorname{Sym}_{R}^{*}\colon\operatorname{Mod}_{R}^{\mathrm{cn}}\to\operatorname{CAlg}^{\Delta}_{R}$
could also be computed by choosing a simplicial resolution $P_{\bullet}$,
where $P_{n}$ are projective $R$-modules. We could summarize the above
discussion by saying that Lurie’s construction of non-abelian derived functors
extends the construction of Dold–Puppe [DP] and Quillen [Qu] using simplicial
resolutions.

###  A.2. Simplicial symmetric algebras

By virtue of the preceding subsection, in the case where $R$ is a discrete
commutative ring and $N\in\operatorname{Mod}_{R}^{\mathrm{cn}}$ is a
connective $R$-module which is represented by a complex $P_{*}$ of discrete
projective $R$-modules, we could compute the derived symmetric algebra
$\operatorname{Sym}_{R}^{*}(N)$ by the Dold–Kan construction ${\rm
DK}_{\bullet}(P_{*})$ ([HA, Construction 1.2.3.5]). In this subsection, we
spell out this construction in the case where $N$ is represented by a cofiber
of a map $\rho\colon M^{\prime}\to M$ between finite locally free $R$-modules.

######  Construction A.1.

Let $R$ be a discrete commutative ring, and let $\rho\colon M^{\prime}\to M$
be a morphism of finitely generated locally free modules. The simplicial
symmetric algebras of $\rho\colon M^{\prime}\to M$ over $R$ is a simplicial
object of discrete commutative rings

$\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M^{\prime}\to
M)\in\mathrm{Fun}(\operatorname{{\bf\Delta}}^{\mathrm{op}},\operatorname{CAlg}_{R}^{\heartsuit})$

defined as follows: The terms of $\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon
M^{\prime}\to M)$ are given by for all $n\geq 0$,

$(\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M^{\prime}\to
M))_{n}=S^{*}_{R}(M)\otimes S_{R}^{*}(M^{\prime})^{\otimes n};$

The face maps $d_{i}\colon(\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon
M^{\prime}\to M))_{n}\to(\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon
M^{\prime}\to M))_{n-1}$, $0\leq i\leq n$, and the degeneracy maps
$s_{i}\colon(\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M^{\prime}\to
M))_{n}\to(\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M^{\prime}\to
M))_{n+1}$, $0\leq i\leq n$, are respectively given by the formulae: for $f\in
S_{R}^{*}(M)$ and $g_{i}\in S_{R}^{*}(M^{\prime})$,

$\displaystyle d_{i}(f\otimes g_{1}\otimes\cdots\otimes
g_{n})=\begin{cases}(f\cdot(S_{R}^{*}(\rho)(g_{1})))\otimes
g_{2}\otimes\cdots\otimes g_{n}&\text{if~{}}i=0;\\\ f\otimes
g_{1}\otimes\cdots\otimes g_{i-1}\otimes(g_{i}\cdot g_{i+1})\otimes
g_{i+1}\otimes\cdots\otimes g_{n}&\text{if~{}}1\leq i\leq n-1;\\\
(f\cdot(S_{R}^{*}(0)(g_{n})))\otimes g_{1}\otimes\cdots\otimes
g_{n-1}&\text{if~{}}i=n.\end{cases}$ $\displaystyle s_{i}(f\otimes
g_{1}\otimes\cdots\otimes g_{n})=f\otimes g_{1}\otimes\cdots g_{i}\otimes
1\otimes g_{i+1}\otimes\cdots\otimes g_{n}.$

Here $S_{R}^{*}(\rho),S_{R}^{*}(0)\colon S_{R}^{*}(M^{\prime})\to
S_{R}^{*}(M)$ denote the canonical morphisms of algebras induced by the
morphisms $\rho\colon M^{\prime}\to M$ and $0\colon M^{\prime}\to M$,
respectively. Notice $S_{R}^{*}(0)$ by definition factorizes as a composition
$S_{R}^{*}(M^{\prime})\twoheadrightarrow R\hookrightarrow S_{R}^{*}(M)$. There
is a natural grading of $\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon
M^{\prime}\to M)$,

$\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M^{\prime}\to
M)=\bigoplus_{d\geq 0}\operatorname{Sym}_{\Delta}^{d}(R,\rho\colon
M^{\prime}\to M)$

where for each $d\geq 0$, $\operatorname{Sym}_{\Delta}^{d}(R,\rho\colon
M^{\prime}\to M)$ is the simplicial module that has terms

$(\operatorname{Sym}_{\Delta}^{d}(R,\rho\colon M^{\prime}\to
M))_{n}=\bigoplus_{d_{0}+d_{1}+\cdots+d_{n}=d,d_{i}\geq
0}S_{R}^{d_{0}}M\otimes_{R}S_{R}^{d_{1}}M^{\prime}\otimes_{R}\cdots\otimes_{R}S_{R}^{d_{n}}M^{\prime}.$

The face maps and degeneracy maps are given by the same formulae.

###  A.3. Simplicial Koszul algebras

The concrete models of the preceding subsection provides simplicial models for
derived zero loci of a cosection. Let $R$ be a discrete commutative ring, and
let $\rho\colon M\to R$ be a cosection of a connective $R$-module $M$. By
virtue of Remark 4.15 and §A.1, we could compute the simplicial commutative
ring $R\otimes_{{\rm ev}_{0},\operatorname{Sym}_{R}^{*}(M),{\rm ev}_{\rho}}R$
(whose spectrum is the derived zero locus of $\rho$) using either the
simplicial model
$\operatorname{Sym}_{\Delta}^{*}(R,\operatorname{id}_{M})\otimes_{\operatorname{Sym}_{\Delta}^{*}(M),{\rm
ev}_{\rho}}R$ or the simplicial model
$\operatorname{Sym}_{\Delta}^{*}(R,\rho\colon M\to R)\otimes_{R[T],T\mapsto
1}R$.

In this subsection, we spell out the above simplicial model for
$R\otimes_{{\rm ev}_{0},\operatorname{Sym}_{R}^{*}(M),{\rm ev}_{\rho}}R$ in
the case where $M$ is a finite free $R$-module. Let $R$ be a discrete
commutative ring, and let $\mathbf{r}=(r_{1},\cdots,r_{m})$ be a sequence of
elements $r_{i}\in R$, which we may regard as a section $\mathbf{r}\colon R\to
M^{\vee}$, where $M=R^{m}$, or dually, a cosection map
$\mathbf{r}^{\vee}\colon M=R^{m}\to R$. Therefore, by adjunction, the
cosection maps $\mathbf{r}^{\vee}$ and $\mathbf{0}^{\vee}$ induce maps of
discrete commutative algebras $\mathrm{ev}_{\mathbf{r}}\colon S_{R}^{*}(M)\to
R$ and $\mathrm{ev}_{\mathbf{0}}\colon S_{R}^{*}(M)\to R$, respectively.

######  Construction A.2.

Let $R$ be a discrete commutative ring, and let
$\mathbf{r}=(r_{1},\cdots,r_{m})$ be a sequence of elements $r_{i}\in R$. Then
simplicial Koszul algebras of the sequence $\mathbf{r}$, which we shall denote
by $\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})$, is the following simplicial
object of the ordinary category of discrete commutative rings

$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})=\operatorname{Sym}_{\Delta}^{*}(R,\operatorname{id}_{M}\colon
M\to
M)\otimes_{S_{R}^{*}(M)}R\in\mathrm{Fun}(\operatorname{{\bf\Delta}}^{\mathrm{op}},\operatorname{CAlg}_{R}^{\heartsuit}).$

Here, the $S_{R}^{*}(M)$-algebra structure of $R$ is given by the map
$\mathrm{ev}_{\mathbf{r}}\colon S_{R}^{*}(M)\to R$. Unwinding the definition,
the simplicial Koszul algebra takes the form:

$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})_{n}=S^{*}_{R}(M)^{\otimes
n}=S_{R}^{*}(M)\otimes_{R}S_{R}^{*}(M)\otimes_{R}\cdots\otimes_{R}S_{R}^{*}(M).$

The face maps

$d_{i}\colon S^{*}_{R}(M)^{\otimes n}\to S^{*}_{R}(M)^{\otimes(n-1)}$

and the degeneracy maps

$s_{i}\colon S^{*}_{R}(M)^{\otimes n}\to S^{*}_{R}(M)^{\otimes(n+1)}$

are respectively given by the formulae: for all $g_{i}\in S_{R}^{*}(M)$,

$\displaystyle d_{i}(g_{1}\otimes\cdots\otimes
g_{n})=\begin{cases}\mathrm{ev}_{\mathbf{r}}(g_{1})\cdot(g_{2}\otimes\cdots\otimes
g_{n})&\text{if~{}}i=0;\\\ g_{1}\otimes\cdots\otimes g_{i-1}\otimes(g_{i}\cdot
g_{i+1})\otimes g_{i+1}\otimes\cdots\otimes g_{n}&\text{if~{}}1\leq i\leq
n-1;\\\ \mathrm{ev}_{\mathbf{0}}(g_{n})\cdot(g_{1}\otimes\cdots\otimes
g_{n-1})&\text{if~{}}i=n.\end{cases}$ $\displaystyle
s_{i}(g_{1}\otimes\cdots\otimes g_{n})=g_{1}\otimes\cdots g_{i}\otimes
1\otimes g_{i+1}\otimes\cdots\otimes g_{n}.$

In particular, the face maps
$d_{0},d_{1}\colon\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})_{1}=S_{R}^{*}(M)\to\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})_{0}=R$
are respectively given by $d_{0}(g)=\mathrm{ev}_{\mathbf{r}}(g)$ and
$d_{1}(g)=\mathrm{ev}_{\mathbf{0}}(g).$ Hence, there is an augmentation map
$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})_{0}\to
R/(\mathbf{r}-\mathbf{0})R=R/\mathbf{r}R$.

######  Remark A.3 (Bar convention).

Since $S_{R}^{*}(M)=R[X_{1},\ldots,X_{m}]$, therefore we can write

$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})_{n}=R[X_{1},\ldots,X_{m}]^{\otimes
n}.$

The morphisms $\mathrm{ev}_{\mathbf{r}}\colon R[X_{1},\ldots,X_{m}]\to R$ and
$\mathrm{ev}_{\mathbf{0}}\colon R[X_{1},\ldots,X_{m}]\to R$ are respectively
given by the formulae: $\mathrm{ev}_{\mathbf{r}}\colon X_{i}\mapsto r_{i}$,
and $\mathrm{ev}_{\mathbf{0}}\colon X_{i}\mapsto 0$. If we use the “bar
convention” and write elements of $R[X_{1},\ldots,X_{m}]^{\otimes n}$ in the
form $f[g_{1}|g_{2}|\cdots|g_{n}]$, where $f\in R$ and $g_{i}\in
R[X_{1},\ldots,X_{m}]$, then we can rewrite the formulae of face and
degeneracy maps as

$\displaystyle
d_{i}[g_{1}|\cdots|g_{n}]=\begin{cases}\mathrm{ev}_{\mathbf{r}}(g_{1})[g_{2}|\cdots|g_{n}]&\text{if~{}}i=0;\\\
[g_{1}|\cdots|g_{i-1}|g_{i}\cdot
g_{i+1}|g_{i+2}|\cdots|g_{n}]&\text{if~{}}1\leq i\leq n-1;\\\
\mathrm{ev}_{\mathbf{0}}(g_{n})[g_{1}|\cdots|g_{n-1}]&\text{if~{}}i=n.\end{cases}$
$\displaystyle
s_{i}[g_{1}|\cdots|g_{n}]=[g_{1}|\cdots|g_{i}|\,1\,|g_{i+1}|\cdots|g_{n}].$

In the situation of Construction A.2, by virtue of Remark 4.15 and §A.1, we
obtain that the geometric realization of the simplicial Koszul algebra
$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})$ (when regarded as a simplicial
diagram in $\operatorname{CAlg}^{\Delta}_{R}$) is canonically equivalent to
the simplicial commutative ring $R\otimes_{{\rm
ev}_{0},\operatorname{Sym}_{R}^{*}(M),{\rm ev}_{\mathbf{r}^{\vee}}}R$. Next,
we compare it with another common description:

######  Lemma A.4.

Let $R$ be a discrete commutative ring, and $\mathbf{r}=(r_{1},\ldots,r_{m})$
be a sequence of elements of $R$. Then $\mathbf{r}$ determines a canonical
ring homomorphism $\mathbb{Z}[x_{1},\ldots,x_{m}]\to R$ that carries $x_{i}$
to $r_{i}$. If we let $\psi$ denote the canonical equivalence

$\psi\colon\mathrm{N}_{\bullet}^{\rm
hc}(\mathrm{Fun}(\operatorname{{\bf\Delta}}^{\mathrm{op}},\operatorname{CAlg}_{R}^{\heartsuit})^{\circ})\simeq\operatorname{CAlg}^{\Delta}_{R},$

then there is a canonical equivalence of simplicial commutative $R$-algebras:

$\psi(\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r}))\simeq
R\otimes_{\mathbb{Z}[x_{1},\ldots,x_{m}]}\frac{\mathbb{Z}[x_{1},\ldots,x_{m}]}{(x_{1},\ldots,x_{m})}.$

###### Proof.

Since the formation of $\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})$ commutes with
base change of discrete commutative rings $R^{\prime}\to R$, we may reduce to
the case where $R=\mathbb{Z}[x_{1},\ldots,x_{m}]$, $r_{i}=x_{i}$. It follows
from Illusie’s result Corollary 2.30 that
$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{r})$ is quasi-isomorphism to the Koszul
complex $\mathrm{Kos}^{*}(R,\mathbf{x})$; On the other hand, we know the
latter complex resolves the module
$\mathbb{Z}[x_{1},\ldots,x_{m}]/(x_{1},\ldots,x_{m})$. Hence the augmentation
map
$\mathrm{Kos}_{\Delta}^{*}(R,\mathbf{x})\to\mathbb{Z}[\mathbf{x}]/(\mathbf{x})$
is a homotopy equivalence of simplicial commutative rings. ∎

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