# Derived Projectivizations of complexes

Qingyuan Jiang  School of Mathematics, University of Edinburgh, James Clerk
Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the counterpart of Grothendieck’s projectivization
construction in the context of derived algebraic geometry. Our main results
are as follows: First, we define the derived projectivization of a connective
complex, study its fundamental properties such as finiteness properties and
functorial behaviors, and provide explicit descriptions of their relative
cotangent complexes. We then focus on the derived projectivizations of
complexes of perfect-amplitude contained in $[0,1]$. In this case, we prove a
generalized Serre’s theorem, a derived version of Beilinson’s relations, and
establish semiorthogonal decompositions for their derived categories. Finally,
we show that many moduli problems fit into the framework of derived
projectivizations, such as moduli spaces that arise in Hecke correspondences.
We apply our results to these situations.

###### Contents

  1. 1 Introduction
    1. 1.1 Derived projectivizations
    2. 1.2 Derived projectivizations of sheaves of perfect-amplitude in $[0,1]$
    3. 1.3 Applications to Classical Situations
      1. 1.3.1 Derived categories of reducible schemes; see Examples 7.21 and 7.23
      2. 1.3.2 Contracting rational tails and bridges for curves; see Examples 7.22 and 7.25
      3. 1.3.3 Threefolds with rational double points; see Example 7.28
      4. 1.3.4 Derived equivalences of classical schemes induced by non-trivially derived schemes
    4. 1.4 Applications to Hecke correspondences moduli
    5. 1.5 The framework of derived algebraic geometry
    6. 1.6 Related works
    7. 1.7 Notations and conventions
  2. 2 Derived symmetric algebras
    1. 2.1 Classical symmetric, exterior and divided power algebras
    2. 2.2 Non-abelian derived functors
    3. 2.3 Derived symmetric, exterior, and divided powers
    4. 2.4 Derived symmetric algebras
    5. 2.5 Universal fiber sequences associated to symmetric powers
    6. 2.6 Filtrations of symmetric powers
  3. 3 Quasi-coherent sheaves and algebras on prestacks
    1. 3.1 $\infty$-categories of sheaves of objects on prestacks
    2. 3.2 Quasi-coherent sheaves on prestacks
    3. 3.3 Quasi-coherent algebras on prestacks and relative spectra
    4. 3.4 Derived symmetric powers and universal fiber sequences on prestacks
    5. 3.5 Derived symmetric algebras over prestacks
  4. 4 Derived projectivizations
    1. 4.1 Affine cones
    2. 4.2 Derived projectivizations
  5. 5 Generalized Serre’s theorem
    1. 5.1 Serre’s theorem for projective bundles
    2. 5.2 Generalized Serre’s theorem
  6. 6 Beilinson’s relations
    1. 6.1 Exceptional sequences and mutation functors
    2. 6.2 Beilinson’s relations
  7. 7 Semiorthogonal decompositions
    1. 7.1 The Fourier–Mukai functors and semiorthogonal decompositions
    2. 7.2 The local situation
    3. 7.3 Applications to classical examples
      1. 7.3.1 Criteria for derived projectivizations and symmetric powers being classical
      2. 7.3.2 Examples violating ($a$)
      3. 7.3.3 Examples satisfying ($a$) but violating ($a^{\prime}$) and ($b$)
      4. 7.3.4 Examples satisfying ($a^{\prime}$) but violating ($b$)
      5. 7.3.5 Examples satisfying ($b$) or even ($b^{\prime}$) but violating ($c$)
      6. 7.3.6 Examples satisfying ($c$) but possibly violating ($c^{\prime}$)
  8. 8 Hecke correspondence moduli as derived projectivizations
    1. 8.1 Hecke correspondences: the case of one-step flags
    2. 8.2 Hecke correspondences: the case of general flags
    3. 8.3 Hecke correspondences for surfaces
  9. A Simplicial Models for Symmetric and Koszul Algebras
    1. A.1 Dold–Puppe and Quillen’s construction
    2. A.2 Simplicial symmetric algebras
    3. A.3 Simplicial Koszul algebras

##  1\. Introduction

Grothendieck’s construction of projectivizations plays a vital role in
classical algebraic geometry. For example, projectivizations and their
properties are crucial for defining projective schemes and morphisms and
studying their properties. In this paper, we study the counterpart of
Grothendieck’s construction in the framework of derived algebraic geometry.
Our main results are as follows:

  1. (1)

For a derived scheme $X$ (or more generally, a prestack $X$) and a connective
quasi-coherent sheaf $\mathscr{E}$ on $X$:

    1. (i)

we will define a (relative) derived scheme $\mathbb{P}(\mathscr{E})$ over $X$,
called the derived projectivization of $\mathscr{E}$ over $X$ (§4.2), whose
underlying classical scheme agrees with Grothendieck’s construction for the
classical truncation (Proposition 4.21).

    2. (ii)

We study the fundamental properties of the derived projectivizations,
including the finiteness properties (Proposition 4.26) and their functorial
behaviors under base change, tensoring with line bundles (Proposition 4.25),
and quotient maps (Corollary 4.32, Proposition 4.33). We also explicitly
describe their relative cotangent complexes by proving a generalized Euler
sequence (Theorem 4.27).

  2. (2)

In the case where $\mathscr{E}$ has perfect-amplitude contained in $[0,1]$,
the derived projectivization $\mathbb{P}(\mathscr{E})$ enjoys special pleasant
features. In this case:

    1. (i)

We will prove a generalized Serre’s theorem for $\mathbb{P}(\mathscr{E})$
(Theorem 5.6).

    2. (ii)

We show that $\mathbb{P}(\mathscr{E})$ has a natural pair of dual relative
exceptional sequences which satisfies a derived version of Beilinson’s
relations (Proposition 6.19).

    3. (iii)

We establish a semiorthogonal decomposition of the $\infty$-category
$\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$ of quasi-coherent sheaves on
$\mathbb{P}(\mathscr{E})$ (Theorem 7.5).

  3. (3)

We apply the our results to various situations:

    1. (i)

(Classical situations) For example, we could apply these results and obtain:

       * •

Semiorthogonal decompositions of the derived categories of schemes of the form
$X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}$, where $D\subset X$ is any effective
Cartier divisor, or the reducible schemes appearing as the central fiber of
deformation to normal cones; see §1.3.1.

       * •

Descriptions of derived categories under stabilization maps of nodal curves
(§1.3.2).

       * •

Semiorthogonal decompositions of derived categories of certain threefolds with
rational double points (§1.3.3).

       * •

Examples of derived equivalences of classical schemes induced by non-trivially
derived schemes (§1.3.4).

    2. (ii)

(Moduli problems) Many moduli spaces have natural derived enhancements which
fit into the framework of derived projectivizations, such as moduli spaces
that arise in Hecke correspondences and moduli of extensions.

       * •

We study the situations of Hecke correspondences for schemes, and apply our
construction and results to this situation (§8).

       * •

In [J23b], we apply the results of this paper to study the derived Abel maps
for integral curves to their compactified Jacobians, and provide
semiorthogonal decompositions of the derived Hilbert schemes of points on
integral curves.

######  Remark 1.1 (Conventions).

This paper works in the framework of derived algebraic geometry and uses
Lurie’s terminology (§1.5, §1.7). Most results of this paper require no
conditions on the base $X$: it could be arbitrarily stacky or derived (or
singular). However, since the projectivization is a “relative phenomenon”, the
readers might feel free to assume $X$ to be a nice classical (underived)
scheme and still get the essence of most of the results in this introduction.
If one wishes to assume that $X$ is a classical Noetherian scheme, beware the
following distinctions of conventions between derived and classical algebraic
geometry:

  * •

A quasi-coherent sheaf $\mathscr{E}$ on $X$ in the derived context (as in this
paper) corresponds to an unbounded complex
$\mathscr{E}_{*}=[\,\cdots\to\mathscr{E}_{1}\to\mathscr{E}_{0}\to\mathscr{E}_{-1}\to\cdots\,]$
of quasi-coherent sheaves on $X$ in the classical sense. The $i$th homotopy
group $\pi_{i}(\mathscr{E})$ corresponds to the sheaf (co)homology
$\mathscr{H}_{i}(\mathscr{E}_{*})=\mathscr{H}^{-i}(\mathscr{E}_{*})$. We say
$\mathscr{E}$ is connective if satisfies
$\pi_{i}(\mathscr{E})=\mathscr{H}_{i}(\mathscr{E}_{*})\simeq 0$ for all $i<0$,
and we say $\mathscr{E}$ is discrete if
$\pi_{i}(\mathscr{E})=\mathscr{H}_{i}(\mathscr{E}_{*})\simeq 0$ for all $i\neq
0$. In particular, discrete quasi-coherent sheaves on $X$ correspond to the
usual quasi-coherent sheaves on $X$ in the classical sense.

  * •

The collection of all quasi-coherent sheaves on $X$ is organized into a stable
$\infty$-category $\operatorname{QCoh}(X)$, whose homotopy category is the
usual derived category $\mathrm{D}_{\mathrm{qc}}(X)$ of quasi-coherent
complexes on $X$. The subcategory of discrete quasi-coherent sheaves, denoted
by $\operatorname{QCoh}(X)^{\heartsuit}$, is equivalent to the (nerve of the)
usual abelian category ${\rm qcoh}(X)$ in the classical setting.

  * •

$\mathrm{Perf}(X)$ denotes the stable $\infty$-category of perfect objects of
$\operatorname{QCoh}(X)$, whose homotopy category is the full triangulated
subcategory $\operatorname{D}^{\mathrm{perf}}(X)$ of
$\mathrm{D}_{\mathrm{qc}}(X)$ spanned by perfect complexes.

  * •

A quasi-coherent sheaf $\mathscr{E}\in\operatorname{QCoh}(X)$ is said to have
perfect-amplitude (contained) in $[0,1]$ if it is represented by a perfect
complex $\mathscr{E}_{*}$ on $X$ of homological Tor-amplitude in $[0,1]$ (or
equivalently, of cohomological Tor-amplitude in $[-1,0]$) in the classical
sense.

###  1.1. Derived projectivizations

Let $X$ be a classical scheme, and let $\mathscr{E}$ be a discrete quasi-
coherent sheaf on $X$. Then Grothendieck’s projectivization of $\mathscr{E}$
over $X$, which we shall denote as
$\mathbb{P}_{\mathrm{cl}}(\mathscr{E})=\operatorname{Proj}\operatorname{Sym}_{\mathrm{cl}}^{*}\mathscr{E}$,
is the scheme over $X$ which represents the following functor: for any
morphism of schemes $\eta\colon T\to X$, the set of $T$-points
$\mathbb{P}_{\mathrm{cl}}(\mathscr{E})(\eta)$ over $\eta$ is the set of
equivalence classes of rank-one locally free quotients of $\mathscr{E}$.

Let $X$ be a derived scheme (or more generally, a prestack), and let
$\mathscr{E}$ be a connective quasi-coherent sheaf on $X$. We define the
derived projectivization $\mathbb{P}(\mathscr{E})$ of $\mathscr{E}$ over $X$
via a similar manner as above, that is, it is the following functor:

  * •

For every map $\eta\colon T\to X$ from a derived scheme $T$, the space of
$T$-points $\mathbb{P}(\mathscr{E})(\eta)$ over $\eta$ is the space of maps of
quasi-coherent sheaves $\eta^{*}\mathscr{E}\to\mathscr{L}$ which induce a
surjection
$\pi_{0}(\eta^{*}\mathscr{E})\twoheadrightarrow\pi_{0}(\mathscr{L})$ , where
$\mathscr{L}$ is a line bundle on $T$.

(See §4.2 for more details.) More informally, we could think of
$\mathbb{P}(\mathscr{E})$ as the derived moduli space which parametrizes rank-
one locally free quotients of $\mathscr{E}$.

The derived projectivization theory extends its classical counterpart in two
ways:

  1. (a)

We allow $X$ to be any derived scheme (or more generally, any prestack).

  2. (b)

We allow $\mathscr{E}$ to be any connective quasi-coherent sheaf, that is, it
needs not be discrete.

We show that the derived projectivization $\mathbb{P}(\mathscr{E})$ enjoys the
following features:

  1. $(0)$

(Underlying classical schemes) The underlying classical scheme of
$\mathbb{P}(\mathscr{E})$ is canonically equivalent to Grothendieck’s
projectivization construction applied to its classical truncation, that is,
there is a canonical equivalence
$(\mathbb{P}_{X}(\mathscr{E}))_{\mathrm{cl}}\simeq\mathbb{P}_{X_{\mathrm{cl}},\mathrm{cl}}(\pi_{0}(\mathscr{E}))$
(Proposition 4.21).

For example, suppose $X$ is a classical scheme and $\mathscr{E}$ is
represented by a complex $\mathscr{E}_{*}$ of discrete quasi-coherent sheaves
on $X$. In this case, the derived projectivization $\mathbb{P}(\mathscr{E})$
has the same underlying classical scheme as Grothendieck’s construction
$\mathbb{P}_{\mathrm{cl}}(\mathscr{H}_{0}(\mathscr{E}_{*}))$, but usually with
a non-trivial derived structure induced from the “hidden higher information”
of $\mathscr{E}$.

######  Warning 1.2.

Let $X$ be classical and let $\mathscr{E}$ be represented by a complex
$\mathscr{E}_{*}$. The “hidden higher information” of $\mathscr{E}$ which
contributes to the derived structure of $\mathbb{P}(\mathscr{E})$ includes not
only all the higher homologies $\\{\mathscr{H}_{i}(\mathscr{E}_{*})\\}_{i>0}$
of $\mathscr{E}_{*}$ but also the higher homologies
$\\{\mathscr{H}_{i}(\operatorname{Sym}^{d}(\mathscr{E}_{*}))\\}_{i>0,d>0}$ of
the derived symmetric powers of $\mathscr{E}_{*}$. For example, even in the
simple case where $\mathscr{E}=\mathscr{O}_{p}^{\oplus 2}$ on
$\mathbb{A}^{1}$, where $p\in\mathbb{A}^{1}$ is the origin, the derived
projectivization $\mathbb{P}(\mathscr{O}_{p}^{\oplus 2})$ is not isomorphic to
Grothendieck’s projectivization
$\mathbb{P}_{\mathrm{cl}}(\mathscr{O}_{p}^{\oplus 2})\simeq\mathbb{P}^{1}$ but
is equipped with a non-trivial derived structure (Example 7.19). If $X$ is an
integral Cohen–Macaulay scheme and $\mathscr{E}$ has homological dimension
one, we will provide criterion for $\mathbb{P}(\mathscr{E})$ to be classical;
see Lemma 7.16.

  1. $(1)$

(Representability) The functor $\mathbb{P}(\mathscr{E})$ is representable by a
relative derived scheme over $X$ (Proposition 4.18).

  2. $(2)$

(Functoriality) The formation of derived projectivizations commutes with
arbitrary base change and tensoring with line bundles (Proposition 4.25).

  3. $(3)$

(Finiteness properties) If $\mathscr{E}$ is perfect to order $n$ (resp. almost
perfect, perfect), then the projection
$\mathrm{pr}\colon\mathbb{P}(\mathscr{E})\to X$ is locally of finite
presentation to order $n$ (resp. locally almost of finite presentation,
locally of ﬁnite presentation); see Proposition 4.26.

  4. $(4)$

(Closed immersions) For any map $\varphi\colon\mathscr{F}\to\mathscr{E}$ of
quasi-coherent sheaves on $X$ which is surjective on $\pi_{0}$, there is a
canonical closed immersion
$\iota\colon\mathbb{P}(\mathscr{E})\hookrightarrow\mathbb{P}(\mathscr{F})$
which is compatible with the formation of universal line bundles and
tautological quotients. Moreover, we have an explicit description of the
closed immersion $\iota$ and its relative cotangent complex in terms of the
morphism $\varphi$ (Corollary 4.32 and Proposition 4.33).

  5. $(5)$

(Euler fiber sequences) The projection
$\mathrm{pr}\colon\mathbb{P}(\mathscr{E})\to X$ admits a connective relative
cotangent complex $L_{\mathbb{P}(\mathscr{E})/X}$ which fits into a canonical
exact triangle

$L_{\mathbb{P}(\mathscr{E})/X}\otimes\mathscr{O}(1)\to\mathrm{pr}^{*}\mathscr{E}\xrightarrow{\rho}\mathscr{O}(1)$

(here $\rho$ denotes the tautological morphism from
$\mathrm{pr}^{*}\mathscr{E}$ to $\mathscr{O}(1)$); see Theorem 4.27. We shall
refer to the above sequence as the Euler fiber sequence.

######  Remark 1.3.

The property $(5)$ seems to provide a perfect example where an object in
derived algebraic geometry behaves manifestly better than its counterpart in
classical algebraic geometry. The above Euler sequence provides an explicit
description of the relative cotangent complexes for all derived
projectivizations $\mathbb{P}(\mathscr{E})$. On the other hand, as far as the
author is aware, there seems to be no general explicit description of the
relative cotangent complexes for classical projectivizations (except in the
special cases where $\mathscr{E}$ is a vector bundle or a “generic” discrete
sheaf of homological dimension one).

The construction of derived projectivization is closely related to affine
cones $\mathbb{V}(\mathscr{E})$, which we will define and study in §4.1. Our
proof of the properties of derived projectivizations and affine cones depends
heavily on the properties of the derived symmetric algebras, which we will
explore in detail in §2 (the local situation) and §3 (the global situation).

###  1.2. Derived projectivizations of sheaves of perfect-amplitude in $[0,1]$

The derived projectivization $\mathbb{P}(\mathscr{E})$ enjoys further pleasant
properties if $\mathscr{E}$ has perfect-amplitude contained in $[0,1]$. For
example, in this case, (as an immediate consequence of the above properties
$(3)$ and $(5)$) the projection $\mathrm{pr}\colon\mathbb{P}(\mathscr{E})\to
X$ is a quasi-smooth proper morphism.

Our main results regarding derived projectivizations of quasi-coherent sheaves
of perfect-amplitude in $[0,1]$ are as follows:

  1. (1)

(Generalized Serre’s theorem) Let $X$ be a derived scheme (or more generally,
a prestack), and let $\mathscr{E}\in\operatorname{QCoh}(X)$ be of perfect-
amplitude contained in $[0,1]$ and rank $\delta$. If $\mathscr{E}$ has rank
$\delta\geq 1$, then for all integers $d$, there are canonical equivalences

$\mathrm{pr}_{*}(\mathscr{O}(d))\simeq\begin{cases}\operatorname{Sym}_{X}^{d}\mathscr{E}&\qquad\text{if~{}}d\geq
0;\\\ 0&\qquad\text{if ~{}}-\delta+1\leq d\leq-1;\\\
(\operatorname{Sym}_{X}^{-d-\delta}\mathscr{E}\otimes_{\mathscr{O}_{X}}\det\mathscr{E})^{\vee}[1-\delta]&\qquad\text{if~{}}d\leq-\delta,\end{cases}$

where $\operatorname{Sym}_{X}^{d}\mathscr{E}$ denotes the derived symmetric
powers of $\mathscr{E}$ over $X$ (see §2.2, §3.4). If $\mathscr{E}$ has rank
$\delta\leq 0$, then there is a canonical exact triangle

$\operatorname{Sym}_{X}^{d}(\mathscr{E})\to\mathrm{pr}_{*}(\mathscr{O}(d))\to(\operatorname{Sym}_{X}^{-d-\delta}\mathscr{E}\otimes_{\mathscr{O}_{X}}\det\mathscr{E})^{\vee}[1-\delta]$

which generalizes the classical Eagon–Northcott complex; see Theorem 5.6.

  2. (2)

(Beilinson’s relations) Let $X$ be a quasi-compact, quasi-separated derived
scheme (or more generally, a perfect stack (Definition 6.3)), and let
$\mathscr{E}\in\operatorname{QCoh}(X)$ be of perfect-amplitude in $[0,1]$ and
rank $\delta\geq 1$. Let $\mathscr{O}(1)$ denote the universal line bundle on
$\mathbb{P}(\mathscr{E})$ and let $\mathscr{O}(i)$ denote
$\mathscr{O}(1)^{\otimes i}$, where $i\geq 0$ is an integer. Then the sequence
of line bundles

$\mathscr{O},\mathscr{O}(1),\cdots,\mathscr{O}(\delta-1)$

forms a relative exceptional sequence (§6.1) of
$\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$ over $X$, whose left dual
relative exceptional sequence is given by

$\operatorname{Sym}_{\mathbb{P}(\mathscr{E})}^{\delta-1}(L_{\mathbb{P}(\mathscr{E})/X}(1)[1]),\cdots,L_{\mathbb{P}(\mathscr{E})/X}(1)[1],\mathscr{O},$

where $L_{\mathbb{P}(\mathscr{E})/X}$ is the relative cotangent complex for
the projection $\mathrm{pr}\colon\mathbb{P}(\mathscr{E})\to X$,
$L_{\mathbb{P}(\mathscr{E})/X}(1)$ denotes
$L_{\mathbb{P}(\mathscr{E})/X}\otimes\mathscr{O}(1)$, and $[1]$ is the degree
shift; see Proposition 6.19. By virtue of Illusie’s result (Proposition 2.11
(4)), there are canonical equivalences

$\operatorname{Sym}_{\mathbb{P}(\mathscr{E})}^{i}\big{(}L_{\mathbb{P}(\mathscr{E})/X}(1)[1]\big{)}\simeq\big{(}\bigwedge\nolimits_{\mathbb{P}(\mathscr{E})}^{i}L_{\mathbb{P}(\mathscr{E})/X}\big{)}(i)[i],$

where $\operatorname{Sym}_{\mathbb{P}(\mathscr{E})}^{i}$ and
$\bigwedge_{\mathbb{P}(\mathscr{E})}^{i}$ denote the $i$th derived symmetric
powers and $i$th derived exterior powers over ${\mathbb{P}(\mathscr{E})}$,
respectively (see §2.2, §3.4). Therefore we could equally write the above left
dual relative exceptional sequence as

$\big{(}\bigwedge\nolimits_{\mathbb{P}(\mathscr{E})}^{\delta-1}L_{\mathbb{P}(\mathscr{E})/X}\big{)}(\delta-1)[\delta-1],\cdots,L_{\mathbb{P}(\mathscr{E})/X}(1)[1],\mathscr{O}.$

Furthermore, for all integers $i,j\in[0,\delta-1]$, we have the following
derived version of Beilinson’s relations ([Be], where similar results for
projective spaces are obtained):

$\left\\{\begin{split}&\underline{\mathrm{Map}}_{\mathbb{P}(\mathscr{E})/X}(\mathscr{O}(i),\mathscr{O}(j))\simeq\operatorname{Sym}_{X}^{j-i}(\mathscr{E}),\\\
&\underline{\mathrm{Map}}_{\mathbb{P}(\mathscr{E})/X}\Big{(}\mathscr{O}(i),\operatorname{Sym}_{\mathbb{P}(\mathscr{E})}^{j}\big{(}L_{\mathbb{P}(\mathscr{E})/X}(1)[1]\big{)}\Big{)}\simeq\delta_{i,j}\cdot\mathscr{O}_{X},\\\
&\underline{\mathrm{Map}}_{\mathbb{P}(\mathscr{E})/X}\Big{(}\big{(}\bigwedge\nolimits_{\mathbb{P}(\mathscr{E})}^{i}L_{\mathbb{P}(\mathscr{E})/X}\big{)}(i),\big{(}\bigwedge\nolimits_{\mathbb{P}(\mathscr{E})}^{j}L_{\mathbb{P}(\mathscr{E})/X}\big{)}(j)\Big{)}\simeq(\bigwedge\nolimits_{X}^{i-j}\mathscr{E})^{\vee},\end{split}\right.$

where
$\underline{\mathrm{Map}}_{\mathbb{P}(\mathscr{E})/X}(\mathscr{F},\mathscr{G})\in\operatorname{QCoh}(X)$
denotes the mapping object of a pair of objects
$\mathscr{F}\in\mathrm{Perf}(\mathbb{P}(\mathscr{E}))$ and
$\mathscr{G}\in\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$ (Construction
6.6), and $\mathscr{H}^{\vee}$ denotes the dual in $\operatorname{QCoh}(X)$ of
an object $\mathscr{H}\in\mathrm{Perf}(X)$; see Corollary 6.21.

  3. (3)

(Semiorthogonal decompositions) Let $X$ be a derived scheme (or more
generally, a prestack) and $\mathscr{E}$ a quasi-coherent sheaf on $X$ with
perfect-amplitude in $[0,1]$ and rank $\delta\geq 0$. Then the “shifted
derived dual” $\mathscr{E}^{\vee}[1]$ also has perfect-amplitude contained in
$[0,1]$. We let $\widehat{Z}$ be the universal incidence locus of
$\mathbb{P}(\mathscr{E})$ and $\mathbb{P}(\mathscr{E}^{\vee}[1])$ over $X$,
which is defined as the derived closed subscheme of the (derived) fiber
product $\mathbb{P}(\mathscr{E})\times_{X}\mathbb{P}(\mathscr{E}^{\vee}[1])$
which classifies paths between the composite map

$\mathscr{O}_{\mathbb{P}(\mathscr{E})}(-1)\boxtimes\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee}[1])}(-1)[1]\to\mathscr{E}^{\vee}\boxtimes\mathscr{E}\xrightarrow{\rm
ev}\mathscr{O}_{\mathbb{P}(\mathscr{E})\times_{X}\mathbb{P}(\mathscr{E}^{\vee}[1])}$

and the zero map; see Notation 7.4. Therefore, we have a commutative diagram

${\widehat{Z}}$${\mathbb{P}(\mathscr{E})}$${\mathbb{P}(\mathscr{E}^{\vee}[1])}$${X.}$$\scriptstyle{\widehat{\mathrm{pr}}}$$\scriptstyle{r_{+}}$$\scriptstyle{r_{-}}$$\scriptstyle{\mathrm{pr}_{+}}$$\scriptstyle{\mathrm{pr}_{-}}$

We will show that the functor induced by the above correspondence
$\widehat{Z}$,

$\Phi=r_{+\,*}\circ
r_{-}^{*}\colon\operatorname{QCoh}(\mathbb{P}(\mathscr{E}^{\vee}[1]))\to\operatorname{QCoh}(\mathbb{P}(\mathscr{E})),$

is fully faithful, and if $\delta\geq 1$, then for all integers
$i\in\mathbb{Z}$, the functors

$\mathrm{pr}_{+}^{*}(\underline{\hphantom{A}})\otimes\mathscr{O}(i)\colon\operatorname{QCoh}(X)\to\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$

are fully faithful. Furthermore, there are induced semiorthogonal
decomposition

$\displaystyle\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$
$\displaystyle=\big{\langle}\Phi\big{(}\operatorname{QCoh}(\mathbb{P}(\mathscr{E}^{\vee}[1]))\big{)},~{}\mathrm{pr}_{+}^{*}(\operatorname{QCoh}(X))\otimes\mathscr{O}(1),\ldots,\mathrm{pr}_{+}^{*}(\operatorname{QCoh}(X))\otimes\mathscr{O}(\delta)\big{\rangle},$
$\displaystyle\mathrm{Perf}(\mathbb{P}(\mathscr{E}))$
$\displaystyle=\big{\langle}\Phi\big{(}\mathrm{Perf}(\mathbb{P}(\mathscr{E}^{\vee}[1]))\big{)},~{}\mathrm{pr}_{+}^{*}(\mathrm{Perf}(X))\otimes\mathscr{O}(1),\ldots,\mathrm{pr}_{+}^{*}(\mathrm{Perf}(X))\otimes\mathscr{O}(\delta)\big{\rangle};$

see Theorem 7.5 for more details.

######  Remark 1.4.

Combining these semiorthogonal decompositions with the above result $(2)$, we
could obtain a whole series of semiorthogonal decompositions of
$\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))$ (or
$\mathrm{Perf}(\mathbb{P}(\mathscr{E}))$) by mutating the relative exceptional
sequence $\mathscr{O}(1),\mathscr{O}(2),\ldots,\mathscr{O}(\delta)$.

######  Remark 1.5.

In the context of classical algebraic geometry, the above semiorthogonal
decompositions are referred to as the projectivization formula in [JL18,
Theorem 3.4]; see also [Ku07, Theorem 5.5], [Tod19, Theorem 4.6.11], and [J21,
Theorem 6.16].

###  1.3. Applications to classical situations

The results of the preceding sections have many interesting applications even
in the classical situation. For this purpose, we provide in §7.3 criteria for
the derived projectivizations $\mathbb{P}(\mathscr{E})$ and
$\mathbb{P}(\mathscr{E}^{\vee}[1])$, the universal incidence loci
$\widehat{Z}$, and derived symmetric products
$\operatorname{Sym}^{d}(\mathscr{E})$ and
$\operatorname{Sym}^{d}(\mathscr{E}^{\vee}[1])$ to be classical (Lemmata 7.16
and 7.18). For example, assume that $X$ is an integral Cohen–Macaulay scheme,
and $\mathscr{E}$ is cofiber of a generically injective map between vector
bundles. If $\operatorname{rank}\mathscr{E}>0$, then under the above
assumption, there is a hierarchy of conditions on the codimensions of
degeneracy loci of $(X,\mathscr{E})$,

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

which is in one-to-one correspondence to the following sequence of conditions:

$\displaystyle\text{$\mathbb{P}(\mathscr{E})$ is
classical}\impliedby\text{$\mathbb{P}(\mathscr{E})$ is
irreducible}\impliedby\text{$\mathbb{P}(\mathscr{E}^{\vee}[1])$ is classical}$
$\displaystyle\impliedby\text{$\mathbb{P}(\mathscr{E}^{\vee}[1])$ is
irreducible}\impliedby\text{$\widehat{Z}$ is
classical}\impliedby\text{$\widehat{Z}$ is irreducible}.$

For example, $\eqref{eqn:condition:PF}$, $\eqref{eqn:condition:irr:PF}$, and
$\eqref{eqn:condition:PK}$ are the conditions
$\operatorname{codim}_{X}(X_{i})\geq i$, $\operatorname{codim}_{X}(X_{i})\geq
i+1$, and
$\operatorname{codim}_{X}(X_{i})\geq\operatorname{rank}\mathscr{E}+i$, for all
degeneracy loci $X_{i}\subseteq X$, respectively. There is also a similar
description if $\operatorname{rank}\mathscr{E}=0$; see Lemma 7.16 for more
details. In particular, there are numerous counter-examples for each of the
inverse of the above implications “$\impliedby$”, and the results of the
preceding subsection have interesting consequences in these situations.

We will only mention some of these applications in this introduction, and
refer the readers to §7.3 in the main text for more examples.

####  1.3.1. Derived categories of reducible schemes; see Examples 7.21 and
7.23

Let $X$ be any scheme, let $D\subseteq X$ be any effective Cartier divisor.
Let $\mathscr{E}=\mathscr{O}_{X}\oplus\mathscr{O}_{D}$. Then (locally)
condition ($a$) is satisfied but ($a^{\prime}$) is not. More concretely,
$\mathbb{P}(\mathscr{E})=X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}$ is a
classical reducible scheme, which is the union of $X$ and
$\mathbb{P}^{1}_{D}=D\times\mathbb{P}^{1}$ along $D$; and
$\mathbb{P}(\mathscr{E}^{\vee}[1])=\mathbb{V}_{D}(\mathscr{O}_{D}(-D)[1])$ is
the total space ${\rm Tot}(\mathcal{N}_{D/X}[-1])$ of “$(-1)$-shifted normal
bundle of $D$ in $X$” in the sense [ACH, PTVV], which has the same underlying
classical scheme as $D$ but equipped with non-trivial derived structure
everywhere. Then Theorem 7.5 implies that there are semiorthogonal
decompositions

$\displaystyle\operatorname{QCoh}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}\big{)}$
$\displaystyle=\big{\langle}\operatorname{QCoh}(X),~{}\operatorname{QCoh}({\rm
Tot}(\mathcal{N}_{D/X}[-1]))\big{\rangle},$
$\displaystyle\mathrm{Perf}\big{(}X\coprod\nolimits_{D}\mathbb{P}_{D}^{1}\big{)}$
$\displaystyle=\big{\langle}\mathrm{Perf}(X),~{}\mathrm{Perf}({\rm
Tot}(\mathcal{N}_{D/X}[-1]))\big{\rangle}.$

See Examples 7.21 for more details.

On the other hand, if we let
$\mathscr{E}^{\prime}=\mathscr{O}_{X}\oplus\mathscr{O}_{D}(D)$, then
$\mathbb{P}(\mathscr{E}^{\prime})=X\coprod_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})$,
where $\mathcal{N}=\mathcal{N}_{D/X}$ is the normal bundle, and
$\mathbb{P}(\mathscr{E}^{\prime\vee}[1])=D\times\operatorname{Spec}\mathbb{Z}[\varepsilon]$.
Therefore we obtain

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

Here, the reducible scheme
$X\coprod_{D}\mathbb{P}_{D}(\mathcal{N}\oplus\mathscr{O}_{D})$ is precisely
the special fiber appearing in the construction of deformation to normal cones
of [Ful, §5.1]; see Variant 7.23.

We could regard the semiorthogonal decompositions of the above two examples as
blowup formulae, where $\mathbb{P}(\mathscr{E})$ and
$\mathbb{P}(\mathscr{E}^{\prime})$ are the derived blowups in the sense of
[KR18, He21] of $X$ along the virtual codimension two closed derived
subschemes ${\rm Tot}(\mathcal{N}_{D/X}[-1])\subseteq X$ and
$D\times\operatorname{Spec}\mathbb{Z}[\varepsilon]\subseteq X$, respectively;
see Remark 7.24.

####  1.3.2. Contracting rational tails and bridges for curves; see Examples
7.22 and 7.25

In the preceding example, if we let $X=C$ be a reduced curve over a field
$\kappa$, let $p\in C(\kappa)$ be a non-singular $\kappa$-point, and let
$\mathscr{E}=\mathscr{O}_{C}\oplus\mathscr{O}_{p}$, 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 $\kappa[\varepsilon]=\operatorname{Sym}^{*}_{\kappa}(\kappa[1])$ is the
ring of derived dual numbers over $\kappa$ (where we might informally think of
$\varepsilon$ as a generator in homological degree $1$). See 7.22 for more
details. This result agrees with Kuznetsov and Shinder’s in [Ku21, Example
5.2, Theorem 5.12].

We might also repeat the above process by setting $C_{0}=C$, $p_{0}=p$, and
letting $C_{i}=C_{i-1}\coprod_{p_{i-1}}\mathbb{P}^{1}$, where $p_{i-1}\in
C_{i-1}(\kappa)$ is any nonsingular $\kappa$-point. Then for any $n\geq 1$, we
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}.$

The morphism spaces of the latter components depend (only) on the
configurations of the trees of $\mathbb{P}^{1}$’s on $C_{n}$. For example, in
the case where $C_{n}$ is the union of $C$ and a chain $\Gamma_{n}$ of
$\mathbb{P}^{1}$’s (and $\kappa$ has characteristic zero), the endomorphism
algebra (of a choice of tilting bundle on $\Gamma_{n}$) of the latter
component is described explicitly in [Bur, Theorem 2.1]. The cases of where
$C_{n}$ is union of $C$ and trees of $\mathbb{P}^{1}$’s can be described
similarly, in view of [KPS21, Theorem 4.9].

On the other hand, let $C=\\{xy=0\\}\subset\mathbb{A}_{\kappa}^{2}$, where
$\kappa$ is a field, and let $o=(0,0)$ denote the node of $C$. Then
$\widetilde{C}:=\mathbb{P}(\Omega_{C/\kappa}^{1})$ is the total transform of
$C$ along the usual blowup
$\operatorname{Bl}_{o}(\mathbb{A}^{2})\to\mathbb{A}^{2}$, and the map
$\pi\colon\widetilde{C}\to C$ is the contraction of the (non-reduced) rational
bridge $E=2\mathbb{P}_{\kappa}^{1}$. We obtain from Theorem 7.5 semiorthogonal
decompositions

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

where $\kappa[\delta]=\kappa[\delta]/(\delta^{2})$ denote the ring of usual
dual numbers; see Example 7.25 for more details. In this example, one could
replace $C$ by any nodal curve $C$. Notice that a recent result [KS22b,
Corollary 6.8] implies that there are no semiorthogonal decompositions for the
reduced scheme $\widetilde{C}^{\rm red}$ when $C$ is projective.

These results altogether provide a comprehensive picture of the behavior of
derived categories of nodal curves under stabilization maps, since the
stabilization maps of nodal curves are compositions of contracting rational
tails and rational bridges ([Sta, Tag 0E7Q]).

####  1.3.3. Threefolds with rational double points; see Example 7.28

Let $X=\mathbb{A}^{2}=\operatorname{Spec}\mathbb{Z}[x,y,z]$, 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 ($a^{\prime}$) is satisfied but ($b$) is not. In this case,
$\mathbb{P}(\mathscr{E})=Q\subseteq\mathbb{A}^{2}\times\mathbb{P}^{2}=\mathbb{A}^{2}\times\operatorname{Proj}\mathbb{Z}[X,Y,Z]$
is the quadric cone defined by the equation $xX+yY=0$, which is an irreducible
threefold with a unique $A_{1}$-singularity at the cone point
$o=((0,0),[0,0,1])\in Q$. Theorem 7.5 implies that there are semiorthogonal
decompositions:

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

where
$\mathbb{Z}[\varepsilon]=\operatorname{Sym}_{\mathbb{Z}}^{*}(\mathbb{Z}[1])$
is the ring of derived dual numbers. 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 $p\in S(\kappa)$ is a non-
singular rational point. The above semiorthogonal decompositions are locally
compatible with Kawamata’s results [Ka19, §7.1] on quadric cones in
$\mathbb{P}^{4}$, the results of Xie [X21, Theorem 4.6] and Pavic and Shinder
[PS21, Theorem 3.6] on nodal del Pezzo threefolds. We may also consider the
global version of this example by letting $X$ be any scheme, and $\mathscr{F}$
by $\mathscr{O}_{X}\oplus\mathscr{I}_{Y}$, where $Y\subseteq X$ is a regularly
immersed closed subscheme of codimension two; see Example 7.29.

####  1.3.4. Derived equivalences of classical schemes induced by non-
trivially derived schemes

If the condition ($b^{\prime}$) is satisfied but ($c$) is not, then the
schemes in the semiorthogonal decompositions are classical, but the fully
faithful embedding is induced by a derived scheme. In the case where
$\mathscr{E}$ has rank zero, we could obtain various examples where
$\mathbb{P}(\mathscr{E})$ and $\mathbb{P}(\mathscr{E}^{\vee}[1])$ are
birationally equivalent classical, irreducible threefolds (or fivefolds, or
even higher dimensional varieties), but the universal incidence locus
$\widehat{Z}$ which induces the (non-trivial) equivalences of
$\infty$-categories

$r_{+,*}\,r_{-}^{*}\colon\operatorname{QCoh}(\mathbb{P}(\mathscr{E}^{\vee}[1]))\simeq\operatorname{QCoh}(\mathbb{P}(\mathscr{E}))\qquad
r_{+,*}\,r_{-}^{*}\colon\mathrm{Perf}(\mathbb{P}(\mathscr{E}^{\vee}[1]))\simeq\mathrm{Perf}(\mathbb{P}(\mathscr{E}))$

is not classical (where
$\mathbb{P}(\mathscr{E})\xleftarrow{r_{+}}\widehat{Z}\xrightarrow{r_{-}}\mathbb{P}(\mathscr{E}^{\vee}[1])$
is the incidence diagram). See Example 7.31.

We refer the readers to §7.3 for more examples, where we only list lower
dimensional cases in each situation, but all these results have direct
analogues in higher dimensions.

###  1.4. Applications to Hecke correspondences moduli

Many moduli problems which have the nature of parametrizing extensions
naturally fit into the framework of derived projectivizations. In §8, we will
consider one such example, the Hecke correspondence moduli which parametrizes
“one-point modifications of complexes”. The geometry of these moduli spaces
are extensively studied in Neguţ’s works [Ne17, Ne18, Ne19] in the classical
context.

Let $X\to S$ be a family of smooth, separated surfaces, and let
$\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$ denote the prestack of perfect
complexes on $X$ relative to $S$ of Tor-amplitude contained in $[0,1]$. Let
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$ denote the moduli of “one-point
modifications of objects of $\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$”, that is,
$\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$ is the prestack which parametrizes
exact triangles of the form

$\zeta=(\mathscr{F}^{\prime}\to\mathscr{F}\to x_{*}\mathscr{L}),$

where
$\mathscr{F}^{\prime},\mathscr{F}\in\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$,
$\mathscr{L}$ is a line bundle on $S$, and $x\colon S\to X$ is a section of
the projection $X\to S$. Then there are natural forgetful maps

${\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}}$${\mathfrak{M}^{\mathrm{perf}}_{[0,1]}}$${X}$${\mathfrak{M}^{\mathrm{perf}}_{[0,1]},}$$\scriptstyle{p_{+}}$$\scriptstyle{p_{X}}$$\scriptstyle{p_{-}}$

where $p_{+}$, $p_{-}$, and $p_{X}$ are the maps which carry the triangle
$\zeta$ to $\mathscr{F}^{\prime}\in\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$,
$\mathscr{F}\in\mathfrak{M}^{\mathrm{perf}}_{[0,1]}$, and $x\in X$,
respectively. We show that the natural projections

$\pi_{\pm}=p_{\pm}\times_{S}p_{X}\colon\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}\to\mathfrak{M}^{\mathrm{perf}}_{[0,1]}\times_{S}X$

are derived projectivizations of perfect-complexes of Tor-amplitude in $[0,1]$
(Proposition 8.7).

######  Remark 1.6.

In fact, we will show that, for any family $X\to S$ of (smooth separated)
derived schemes of any given dimension $d$, the Hecke correspondence moduli
$\mathfrak{Z}$ fits into the framework of derived projectivizations
(Propositions 8.5 and 8.7); and the results are even true for any type of flag
of “one-point modifications” $\mathfrak{Z}_{\lambda}$ (Propositions 8.11 and
8.13).

In particular, we could apply all of the above-mentioned results (the
generalized Serre’s Theorem 5.6, Beilinson’s relations Proposition 6.19,
Corollary 6.21, and Semiorthogonal Decompositions Theorem 7.5) to this
situation. For example, if we let
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ denote the moduli of
“two-step flags of one-point modifications with support on one point”, that
is, the prestack which parametrizes the pair of exact triangles

$\mathscr{F}_{0}\xrightarrow{\phi_{1}}\mathscr{F}_{1}\to
x_{*}(\mathscr{L}_{1})\qquad\mathscr{F}_{1}\xrightarrow{\phi_{2}}\mathscr{F}_{2}\to
x_{*}(\mathscr{L}_{2}),$

where $\mathscr{L}_{1},\mathscr{L}_{2}$ are line bundles on $S$ and $x\colon
S\to X$ is a section of $X\to S$. We let
$(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ denote the
universal incidence locus for $\pi_{\pm}$, that is, the locus on
$(\mathfrak{Z}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}$ where the composition

$\mathscr{L}_{1}^{\vee}\boxtimes(\mathscr{L}_{2}\otimes\mathscr{K}_{X/S})[1]\to
x^{*}\mathscr{F}_{1}\otimes x^{*}(\mathscr{F}_{1}^{\vee})\xrightarrow{\rm
ev}\mathscr{O}$

is nullhomotopic. Therefore, we have a correspondence diagram

$\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}\xleftarrow{r_{+}}(\widehat{\mathfrak{Z}}_{2}^{\bullet})^{\mathrm{perf}}_{[0,1]}\xrightarrow{r_{-}}\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}.$

We let $\mathfrak{Z}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$ denote
the subfunctor of $\mathfrak{Z}^{\mathrm{perf}}_{[0,1]}$ for which
$\mathscr{F}$ and $\mathscr{F}^{\prime}$ have rank $\delta$, and let
$\mathfrak{M}^{\mathrm{perf}}_{[0,1],\operatorname{rk}=\delta}$ be defined
similarly. Theorem 7.5 implies that, if $\delta>0$, the functor

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

and the functors, for all $i\in\mathbb{Z}$,

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

are fully faithful, and there are induced semiorthogonal decompositions

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

Similarly for other cases; see Corollary 8.15 for more details.

######  Remark 1.7.

The general framework of §8 not only unifies various situations such as the
Hilbert schemes of points (Example 8.1) and moduli of discrete Gieseker stable
sheaves (Example 8.2), but also allows us to consider moduli spaces under
various types of stability conditions. For example, we expect the results of
§8 to be helpful for the study of the similar algebraic structures for the
moduli spaces $\mathfrak{M}_{\sigma}(X)$ of Bridgeland (semi-)stable objects
on a surface $X$ (or a family $X\to S$ of surfaces), where the surface $X$ (or
the family $X\to S$ of surfaces) is equipped with (a family of) Bridgeland
stability conditions $\sigma$ in the sense of [Bri07, Bri08, AB12, BM11, BM14]
(the absolute cases) and [BLM+] (the family case).

###  1.5. The framework of derived algebraic geometry

We work in the context of derived algebraic geometry, as developed by Lurie
[DAG, HTT, DAG-V, HA, SAG, Lu-Ke], Toën and Vezzosi [TVe05, TVe08, Toë14] and
many others [BFN, GR, TVa07, PTVV].

There are various models for the framework of derived algebraic geometry. For
definiteness and completeness, we choose to work within Lurie’s framework, and
we will use Lurie’s works as the only references for all results of derived
algebraic geometry in the proofs of this paper. However, we expect our
arguments to work in the other models of derived algebraic geometry.

From a technical perspective, the framework of derived algebraic geometry
plays an essential role in this paper from at least the following aspects:

  * •

Non-abelian derived functors. Historically, symmetric algebras of complexes
are studied either using concrete Koszul models [Ei, ABW] or simplicial
resolutions [DP, Qu]. However, for this paper’s goal, either approach has its
advantages and certain drawbacks:

    * –

The definition using Koszul models is practical for computations but is a
priori only defined for complexes of vector bundles (and might not be our
desired definition if the characteristic is not zero). For example, even the
fact that the symmetric algebras depend only on the quasi-isomorphism classes
of complexes is a theorem rather than an immediate consequence of this
definition.

    * –

On the other hand, the approach using simplicial resolutions is general and
functorial, but might not be effective for concrete computations.

    * –

Illusie’s results in [Ill] bridge the above two approaches, but the “bridging
process” is slightly complicated and lacks sufficient functoriality for this
paper’s goal.

Lurie’s theory of non-abelian derived functors [HTT, §5.5.8] using the
framework of $\infty$-categories in a way unifies these two perspectives, and
enjoys the best features of both approaches. See §2.4, §2.5.

  * •

Functorial exact triangles (fiber sequences). In the classical context of
triangulated categories, the construction of cones is known to be non-
functorial. Therefore, it is hard to show that two non-trivial exact triangles
obtained from different ways are equivalent. The framework of derived
algebraic geometry (see, for example, [HTT, HA, SAG]) not only solves the
problem of non-functorial triangles but also explains why certain triangles
should be equivalent, by “remembering their origins”. This perspective plays a
vital role in many proofs of this paper; see, for example, the proofs of
generalized Serre’s theorem (Theorem 5.6) and Beilinson’s relations
(Proposition 6.19 and Corollary 6.21).

  * •

Translating problems of algebraic geometry to problems of algebraic topology.
The framework of derived algebraic geometry (for example, the theory of
general derived higher moduli (pre)stacks, as developed by Lurie in [DAG,
SAG], and Toën and Vezzosi in [TVe05, TVe08]) allows us to reduce certain
problems of algebraic geometry to that of algebraic topology, where the
$\infty$-amount of homological information that is hard to manipulate in
algebraic geometry is effectively encoded by topological spaces that are
easier to handle in algebraic topology. This perspective plays a vital role in
proving the Euler fiber sequence (Theorem 4.27).

  * •

Phenomena only visible in the derived context. Let $V$ be a finite-dimensional
vector space (or a vector bundle over a base scheme). The classical universal
incidence quadric
$\mathcal{H}\subseteq\mathbb{P}(V)\times\mathbb{P}(V^{\vee})$, defined as the
zero locus of the composite map

$\mathscr{O}(-1,-1)\to V^{\vee}\otimes V\xrightarrow{\rm ev}\mathscr{O},$

plays a crucial role in classical and homological projective duality [Ku07].
Suppose that $V=[V_{1}\to V_{0}]$ is a complex of vector spaces (or vector
bundles), and $V^{\vee}$ is the dual of $V$. Then the role of $\mathcal{H}$ is
now played by the universal incidence locus
$\widehat{Z}\subseteq\mathbb{P}(V)\times\mathbb{P}(V^{\vee}[1])$ (Notation
7.4), defined as the zero locus of the composite map

$\mathscr{O}(-1,-1)[1]\to V^{\vee}\otimes V\xrightarrow{\rm ev}\mathscr{O}.$

However, since the zero loci of non-zero degree shifts of line bundles only
affect the derived structure and have no effects on the underlying classical
schemes, the semiorthogonal decompositions of Theorem 7.5 induced by
$\widehat{Z}$ are best understood in the derived context, even in the cases
where $\mathbb{P}(V)$ and $\mathbb{P}(V^{\vee}[1])$ are classical.

###  1.6. Related works

The classical projectivization formula is studied in [Ku07, Theorem 5.5],
[Tod21, Tod19], [JL18, Theorem 3.4], [J21, Theorem 6.16], and the Chow-
theoretical counterpart is studied in [J19]. We expect the current work to be
helpful to extend the results of [J19] to the context of bivariant and motivic
theories for derived schemes.

The projectivization formula is also related to the theory of homological
projective duality developed by Kuznetsov [Ku07]; see also [JLX17, P19].

Our construction of affine cones in §4.1 is closely related to the
construction of Jiang and Thomas in [JT17] in the context of perfect
obstruction theory [BF, LT] and $(-1)$-shifted symplectic structure [PTVV]. It
would be interesting to explore the further connections between the results of
this paper and the framework of perfect obstruction theory.

In [Kh21], Khan defined and studied the derived symmetric algebras (§3.5) on
derived Artin stacks and applied them to obtain an excess intersection formula
for the algebraic $K$-theory.

In [He21], Hekking constructs projective spectra for all graded simplicial
commutative algebras over a derived scheme. In the case where the graded
algebra $\mathcal{B}_{*}$ is given by a derived symmetric algebra
$\operatorname{Sym}_{X}^{*}(\mathscr{E})$, our construction of
$\mathbb{P}(\mathscr{E})$ in §4.2 is equivalent to his
$\operatorname{Proj}(\mathcal{B}_{*})$.

In [Z20], Zhao also established a version of the generalized Serre’s theorem
5.6.

The derived projectivization is closely related to the theory of
categorification of wall-crossing of Donalson–Thomas moduli spaces and the
d-critical birational geometry, developed by Toda in [Tod18, Tod21, Tod19,
Tod21a, Tod21b, Tod21c].

The applications in §7.3 are related to the study of derived categories of
nodal curves in [Bur, KPS21] and nodal threefolds in [Ku21, Ka19, X21, PS21].

The study of Hecke correspondence in §8 is closely related to Neguţ’s works
[Ne17, Ne18, Ne19]; see also [KV19, PS19, PZ20, Z20, Z21].

In [J22b, J23a], we extend the constructions and results presented in this
paper to the context of Grassmannians in characteristic zero. In this
generalization, the derived symmetric and exterior products are replaced by
more general derived Schur functors.

###  1.7. Notations and conventions

We made an effort to keep our notations and terminologies as close to Lurie’s
in [DAG, HTT, HA, SAG, Lu-Ke] as possible, so that readers of Lurie’s works
will not need to make a significant extra effort to adjust to those of this
paper.

For the readers’ convenience, here is a list of notations and terminologies
that we will frequently use in this paper:

  * •

We let $\operatorname{{\bf\Delta}}$ denote the simplex category (that is, the
category of nonempty finite linearly ordered sets, with morphisms given by
nondecreasing maps). We let
$\operatorname{Set}_{\Delta}=\mathrm{Fun}(\operatorname{{\bf\Delta}}^{\mathrm{op}},\operatorname{Set})$
denote the simplicial category of simplicial sets. For each $n\geq 0$, we let
$\Delta^{n}\in\operatorname{Set}_{\Delta}$ denote the simplicial set that
represents the functor
$\operatorname{Hom}_{\operatorname{{\bf\Delta}}}(\underline{\hphantom{A}},[n])$.

  * •

We let $\mathcal{S}$ denote the $\infty$-category of spaces, that is, the
homotopy-coherent nerve of simplicial category of Kan complexes.

  * •

We let $\mathrm{Sp}$ denote the $\infty$-category of spectra ([HA, Definition
1.4.3.1]).

  * •

We let $\mathcal{C}\mathrm{at}_{\infty}$ denote the $\infty$-category of small
$\infty$-categories, and $\widehat{\mathcal{C}\mathrm{at}}_{\infty}$ the
$\infty$-category of not necessarily small $\infty$-categories ([HTT, Chapter
3]). For an $\infty$-category $\mathcal{C}$ and objects $C,D\in\mathcal{C}$,
$\mathrm{Map}_{\mathcal{C}}(C,D)\in\mathcal{S}$ denotes the mapping space. For
a pair of $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$, we let
$\mathrm{Fun}(\mathcal{C},\mathcal{D})$ denote the $\infty$-category of
functors from $\mathcal{C}$ to $\mathcal{D}$.

  * •

For an $\infty$-category $\mathcal{C}$, we let $\mathcal{C}^{\simeq}$ denote
the largest Kan complex contained in $\mathcal{C}$, that is, the
$\infty$-category obtained from $\mathcal{C}$ by discarding all non-invertible
morphisms.

  * •

If $\mathcal{C}$ is a pointed $\infty$-category which admits cofibers (resp.
fibers), we let $\Sigma=\Sigma_{\mathcal{C}}$ (resp.
$\Omega=\Omega_{\mathcal{C}}$) denote the suspension (resp. loop) functor. If
a pointed $\infty$-category $\mathcal{C}$ admits both cofibers and fibers, we
will define a translation functor $[i]$ for each $i\in\mathbb{Z}$ by the
following rule: let $[i]=\Sigma^{\circ i}$ if $i>0$, $[i]=\Omega^{\circ-i}$ if
$i<0$, and $[i]=\operatorname{id}_{\mathcal{C}}$ if $i=0$.

  * •

We let $\operatorname{CAlg}^{\Delta}$ denote the $\infty$-category of
simplicial commutative rings ([SAG, Definition 25.1.1.1]), and we will refer
to the elements of $\operatorname{CAlg}^{\Delta}$ as simplicial commutative
rings 111These are also referred to as animated commutative rings; see [CS19,
§5.1], [BL22, Appendix A].. The $\infty$-category
$\operatorname{CAlg}^{\Delta}$ can be obtained from ordinary category ${\rm
Poly}$ of finite type polynomial rings by freely adjoining sifted colimits
(that is, $\operatorname{CAlg}^{\Delta}=\mathcal{P}_{\Sigma}({\rm Poly})$),
and we will review this construction in §2.2. For a simplicial commutative
ring $R\in\operatorname{CAlg}^{\Delta}$, we let
$\operatorname{CAlg}^{\Delta}_{R}$ denote the coslice category
$(\operatorname{CAlg}^{\Delta})_{/R}$ ([SAG, Notation 25.1.4.4]).

  * •

Let $\mathcal{C}^{\otimes}\to{\rm Comm}^{\otimes}$ be a symmetric monoidal
$\infty$-category ([HA, Definition 2.0.0.7]) with underlying $\infty$-category
$\mathcal{C}$. We let
$\operatorname{CAlg}(\mathcal{C})=\mathrm{Alg}_{/\mathrm{Comm}}(\mathcal{C})$
denote $\infty$-category of commutative algebra objects of $\mathcal{C}$ ([HA,
Definition 2.1.3.1]). For $A\in\operatorname{CAlg}(\mathcal{C})$, we let
$\operatorname{Mod}_{A}(\mathcal{C})$ denote the $\infty$-category of
$A$-module objects of $\mathcal{C}$ ([HA, Definition 3.3.3.8]).

  * •

We let $\operatorname{CAlg}=\operatorname{CAlg}(\mathrm{Sp})$ denote the
$\infty$-category of $\mathbb{E}_{\infty}$-rings ([HA, Definition 7.1.0.1]).
There is a forgetful functor
$\Theta\colon\operatorname{CAlg}^{\Delta}\to\operatorname{CAlg}$. For a
simplicial commutative ring $A$, we will denote the image $\Theta(A)$ as
$A^{\circ}$ and refer to it as the underlying $\mathbb{E}_{\infty}$-ring of
$A$ ([SAG, Construction 25.1.2.1]).

  * •

For any $A\in\operatorname{CAlg}^{\Delta}$, we will let
$\operatorname{Mod}_{A}$ denote the $\infty$-category
$\operatorname{Mod}_{A^{\circ}}(\mathrm{Sp})$, and refer to the elements of
$\operatorname{Mod}_{A}$ as $A$-modules. For a simplicial commutative ring $A$
and an $A$-module $M$, we let $\pi_{i}(A)=\pi_{i}(A^{\circ})$ (resp.
$\pi_{i}(M)$) denotes the $i$th homotopy group of the underlying spectrum of
$A$ (resp. $M$). The action map $R\otimes M\to M$ endows
$\pi_{*}(M)=\bigoplus_{m\in\mathbb{Z}}\pi_{m}(M)$ with the structure of a
graded $\pi_{*}(R)$-module. In particular, $\pi_{0}(M)$ is a module over the
ordinary commutative ring $\pi_{0}(R)$. We will say that $M$ is connective if
$\pi_{i}(M)=0$ for all $i<0$. We let $\operatorname{Mod}_{A}^{\mathrm{cn}}$
denote the full subcategory of $\operatorname{Mod}_{A}$ spanned by connective
$A$-modules. See [HA, §7.1.1] for more details. We say a simplicial
commutative ring $A$ (resp. an $A$-module $M$) is discrete if $\pi_{i}(A)=0$
for $i\neq 0$ (resp. $\pi_{i}(M)=0$ for $i\neq 0$). We let
$\operatorname{CAlg}^{\heartsuit}$ and $\operatorname{Mod}_{A}^{\heartsuit}$
denote the full subcategory of $\operatorname{CAlg}^{\Delta}$ and
$\operatorname{Mod}_{A}$ spanned by discrete objects, respectively.

  * •

By a derived scheme we mean a pair $X=(|X|,\mathscr{O}_{X})$ where $|X|$ is a
topological space, and $\mathscr{O}_{X}$ is an
$\operatorname{CAlg}^{\Delta}$-valued sheaf on $|X|$, such that the following
conditions are satisfied: (i) The underlying ringed space
$X_{\mathrm{cl}}=(|X|,\pi_{0}\mathscr{O}_{X})$ is a (classical) scheme, which
is called the underlying classical scheme of $X$. (ii) Each of the sheaves
$\pi_{n}(\mathscr{O}_{X})$ is a quasi-coherent
$\pi_{0}(\mathscr{O}_{X})$-module on $X_{\mathrm{cl}}$. (iii) The structure
sheaf $\mathscr{O}_{X}$ is hypercomplete when regarded as an object of the
$\infty$-topos $\mathcal{S}\mathrm{hv}_{\mathcal{S}}(X)$ in the sense of [HTT,
§6.5.2].

  * •

By a prestack we mean an element of the $\infty$-category
$\mathrm{Fun}(\operatorname{CAlg}^{\Delta},\mathcal{S})$, that is, a functor
$X\colon\operatorname{CAlg}^{\Delta}\to\mathcal{S}$. A map $f\colon X\to Y$
between prestacks is a natural transformation of the functors
$X,Y\colon\operatorname{CAlg}^{\Delta}\to\mathcal{S}$. For a prestack $X$, we
let $\operatorname{QCoh}(X)$ denote the $\infty$-category of quasi-coherent
sheaves on $X$ (which we will review in §3.2), and let $\mathrm{Perf}(X)$
denote the full subcategory of $\operatorname{QCoh}(X)$ spanned by perfect
objects.

Here is a list of notations and terminologies that either we introduce in this
paper or for which our usage is slightly different from (some of) Lurie’s
works:

  * •

We use $\operatorname{Sym}_{A}^{n}(M)$ to denote the derived $n$th symmetric
product of a connective $A$-module $M$ over $A$ (§2.3), and use
$\operatorname{Sym}_{X}^{n}(\mathscr{M})$ to denote the global version, that
is, the derived $n$th symmetric power of a connective quasi-coherent sheaf
$\mathscr{M}$ over a prestack $X$ (§3.4). This notation agrees with Lurie’s in
[DAG], but is different from his in [HA, SAG]. 222In [HA, SAG], Lurie uses
$\operatorname{Sym}_{A}^{n}(M)$ to denote the $A$-module obtained by taking
homotopy coinvariants for the action of the symmetric group $\Sigma_{n}$ on
the $n$-fold relative tensor product $M\otimes_{A}\cdots\otimes_{A}M$, which
is generally different from the derived $n$th symmetric power unless $A$ has
characteristic zero; We will not use this construction in this paper. Our
notation $\operatorname{Sym}_{A}^{n}(M)$ corresponds to Lurie’s notation ${\rm
LSym}_{A}^{n}(M)$ in [SAG, §25.2].

  * •

For an ordinary commutative ring $R$ and a discrete $R$-module $M$,
$S_{R}^{d}(M)=\operatorname{Sym}_{\mathrm{cl},R}^{d}(M)$ denote the classical
symmetric products, reviewed in §2.1. Let $\rho\colon M\to M^{\prime}$ be a
map of discrete $R$-modules, $\mathbf{S}^{d}(R,\rho)$ denotes the Koszul
complex defined in Construction 2.19.

  * •

$\operatorname{QCAlg}^{\Delta}(X)$ denotes the $\infty$-category of quasi-
coherent algebras on a prestack $X$; see §3.3. For a morphism of prestacks
$f\colon X\to Y$, we let $f_{\rm
alg}^{*}\colon\operatorname{QCAlg}^{\Delta}(Y)\to\operatorname{QCAlg}^{\Delta}(X)$
denote the pullback functor, and $f_{*}^{\rm alg}$ the right adjoint of
$f_{\rm alg}^{*}$ (if exists).

  * •

$\operatorname{Spec}_{X}$ denotes the relative spectrum functor over a
prestack $X$; see Definition 3.13.

  * •

$\mathbb{V}(\mathscr{E})$ or $\mathbb{V}_{X}(\mathscr{E})$ denotes the affine
cone of a connective quasi-coherent sheaf $\mathscr{E}$ over a prestack $X$
(§4.1); $\mathbb{P}(\mathscr{E})$ or $\mathbb{P}_{X}(\mathscr{E})$ denotes the
derived projectivization of a connective quasi-coherent sheaf $\mathscr{E}$
over a prestack $X$ (§4.2).

  * •

The Eagon–Northcott complexes $\mathbf{EN}_{d}(X,\sigma)$ are defined in
Remark 5.9.

  * •

For an perfect stack $X$ (Definition 6.3), a $X$-linear stable
$\infty$-category $\mathcal{C}$,
$\underline{\mathrm{Map}}_{X}(C,D)=\underline{\mathrm{Map}}_{\mathcal{C}/X}(C,D)\in\operatorname{QCoh}(X)$
denotes the mapping object defined in Construction 6.6. For a relative
exceptional object $E\in\mathcal{C}$ over $X$ (Definition 6.8),
$\mathbb{L}_{E}$ and $\mathbb{R}_{E}$ denote the left and right mutation
functors, respectively, defined in Definition 6.12.

  * •

The universal incidence locus $\widehat{Z}$ and the incidence diagram is
defined in Notation 7.4.

  * •

The notations $\mathfrak{M}$, $\mathfrak{Z}$, $\mathfrak{Z}_{\lambda}$, and
their variants, denote the moduli functors that appear in Hecke
correspondences, which we will define and only use in §8.

### Acknowledgment

The author thanks Arend Bayer for numerous helpful discussions throughout this
project, Bertrand Toën for asking him the question of viewing the
projectivization formula from the DAG perspective during his visit to Toulouse
two years ago, Fei Xie for helpful discussions on derived categories of nodal
curves and nodal threefolds, and Dougal Davis for discussions on DAG,
carefully reading the paper, and many helpful suggestions. The author is
grateful to Richard Thomas, Yukinobu Toda, Sasha Kuznetsov, Andrei Neguţ,
Adeel Khan, Yunfeng Jiang, Jeroen Hekking, and Yu Zhao for helpful
communications and comments on an earlier draft of this paper. This work is
supported by the Engineering and Physical Sciences Research Council
[EP/R034826/1].

##  2\. Derived symmetric algebras

This section reviews Lurie’s construction of derived symmetric algebras and
studies their basic properties. The constructions and results of this section
play an essential role in our later study of affine cones and derived
projectivizations in this paper.

In §2.1, we briefly review the classical constructions of the symmetric,
exterior, and divided powers. §2.2 reviews Lurie’s general theory of non-
abelian derived functors, and §2.3 reviews the constructions and properties of
the derived symmetric, exterior, and divided powers. In §2.4, we define
derived symmetric algebras and study their basic properties; see Propositions
2.16 and 2.17. Finally, in §2.5, we apply Lurie’s machinery §2.2 to construct
two canonical series of functorial fiber sequences associated to symmetric
powers.

######  Remark 2.1.

The material of this section has a direct generalization to a theory of
derived Schur functors, which we will explore in a forthcoming paper.

###  2.1. Classical symmetric, exterior and divided power algebras

For the readers’ convenience, we will briefly review the classical definitions
of symmetric, exterior and divided power algebras. In this subsection, we make
the following assumption:

  * •

We let $R$ be a(n ordinary) commutative ring, and $M$ a discrete $R$-module.

######  Definition 2.2 (Classical tensor algebras; see [Bou, Chapter III,
§5]).

For a pair of discrete $R$-modules $M$ and $N$, we let
$M\otimes_{R}^{\mathrm{cl}}N$ denote the classical tensor product
$\operatorname{Tor}_{0}^{R}(M,N)$ over $R$. For an integer $n\geq 0$, we let
$\operatorname{T}^{n}(M)$ denote the $n$-fold classical tensor product
$M\otimes_{R}^{\mathrm{cl}}M\otimes_{R}^{\mathrm{cl}}\cdots\otimes_{R}^{\mathrm{cl}}M$.
By convention, we set $\operatorname{T}^{0}(M)=R$,
$\operatorname{T}^{1}(M)=M$. Then the classical tensor algebra of $M$ over
$R$, denoted by $\operatorname{T}^{*}(M)$, or $\operatorname{T}_{R}^{*}(M)$,
is the (ordinary) graded associative $R$-algebra
$\operatorname{T}^{*}(M)=\bigoplus_{n\geq 0}\operatorname{T}^{n}(M)$, with the
multiplication map

$m_{p,q}\colon\operatorname{T}^{p}(M)\times\operatorname{T}^{q}(M)\to\operatorname{T}^{p+q}(M),$

where $p,q\geq 0$, given by the formula:

$m_{p,q}(x_{1}\otimes\cdots\otimes x_{p},x_{p+1}\otimes\cdots\otimes
x_{p+q})=x_{1}\otimes\cdots\otimes x_{p}\otimes x_{p+1}\otimes\cdots\otimes
x_{p+q}.$

######  Definition 2.3 (Classical symmetric algebras; see [Bou, Chapter III,
§6]).

The clasical symmetric algebra of $M$ over $R$, denoted by $S^{*}(M)$,
$S_{R}^{*}(M)$, or $\operatorname{Sym}_{\mathrm{cl}}^{*}(M)$, is the
(ordinary) commutative $R$-algebra obtained as the quotient of the tensor
algebra $\operatorname{T}_{R}(M)$ by the two-sided ideal
$\mathfrak{C}=\mathfrak{C}_{M}$, where the ideal $\mathfrak{C}$ is generated
by the elements of the form $x\otimes y-y\otimes x$ for $x,y\in M$. Since
$\mathfrak{C}$ is a graded ideal generated by homogeneous elements of degree
$2$, the algebra $S_{R}^{*}(M)$ is graded and equipped with a direct sum
decomposition $S_{R}^{*}(M)=\bigoplus_{n\geq 0}S_{R}^{n}(M)$, where
$S_{R}^{n}(M)=\operatorname{T}^{n}(M)/(\mathfrak{C}\cap\operatorname{T}^{n}(M))$.
We will refer to the $R$-module $S_{R}^{n}(M)$ as the classical $n$th
symmetric power of $M$ over $R$. From $\mathfrak{C}^{0}=\mathfrak{C}^{1}=0$,
we obtain canonical identifications $S_{R}^{0}(M)=R$ and $S_{R}^{1}(M)=M$. The
canonical map $M=S_{R}^{1}(M)\hookrightarrow S_{R}^{*}(M)$ is referred to as
the canonical injection.

######  Definition 2.4 (Classical exterior algebras, see [Bou, Chapter III,
§7]).

The exterior algebra of $M$ over $R$, denoted by
$\bigwedge_{\mathrm{cl}}^{*}(M)$, or $\bigwedge_{\mathrm{cl},R}^{*}(M)$, is
the ordinary $R$-algebra defined as the quotient of the tensor algebra
$\operatorname{T}_{R}(M)$ by the two-sided ideal
$\mathfrak{A}=\mathfrak{A}_{M}$, where $\mathfrak{A}$ is generated by elements
of the form $x\otimes x$ for $x\in M$. Since $\mathfrak{A}$ is a graded ideal
generated by homogeneous elements of degree $2$, the algebra
$\bigwedge_{\mathrm{cl}}^{*}(M)$ is graded, and we have a direct sum
decomposition $\bigwedge_{\mathrm{cl}}^{*}(M)=\bigoplus_{n\geq
0}\bigwedge_{\mathrm{cl}}^{n}(M)$, where
$\bigwedge_{\mathrm{cl}}^{n}(M)=\operatorname{T}^{n}(M)/(\mathfrak{A}\cap\operatorname{T}^{n}(M))$.
We will refer to $\bigwedge_{\mathrm{cl}}^{n}(M)$ as the classical $n$th
exterior power of $M$ over $R$. From $\mathfrak{A}^{0}=\mathfrak{A}^{1}=0$, we
obtain canonical identifications $\bigwedge_{\mathrm{cl}}^{0}(M)=R$ and
$\bigwedge_{\mathrm{cl}}^{1}(M)=M$. We will refer to the canonical map
$M=\bigwedge_{\mathrm{cl}}^{1}(M)\hookrightarrow\bigwedge_{\mathrm{cl}}^{*}(M)$
as the canonical injection.

######  Definition 2.5 (Classical divided power algebras, [Ro, Chapter III,
§1, p. 248] ).

The classical divided power (or P.D., puissances divisées) algebra of $M$ over
$R$, denoted by $\Gamma_{\mathrm{cl}}^{*}(M)$,
$\Gamma_{\mathrm{cl},R}^{*}(M)$, or $\mathbb{D}_{\mathrm{cl}}^{*}(M)$, is the
(ordinary) commutative $R$-algebra obtained as the quotient algebra of the
free polynomial algebra $R[\\{\gamma^{n}(x)\\}_{n\geq 0,x\in M}]$ on symbols
$\\{\gamma^{n}(x)\\}_{n\geq 0,x\in M}$ by the ideal generated by following
relations:

$\displaystyle\gamma^{0}(x)=1,$ $\displaystyle\forall x\in M;$
$\displaystyle\gamma^{n}(rx)=r^{n}\gamma^{n}(x)$ $\displaystyle\forall r\in
R,x\in M^{\prime},n\geq 0;$
$\displaystyle\gamma^{m}(x)\cdot\gamma^{n}(x)=\binom{m+n}{m}\gamma^{m+n}(x)$
$\displaystyle\forall x\in M,n\geq 0;$
$\displaystyle\gamma^{n}(x+y)=\sum_{i=0}^{n}\gamma^{i}(x)\gamma^{n-i}(y),$
$\displaystyle\forall x,y\in M,n\geq 0.$

Since the ideal generated by above relations are homogeneous, the algebra
$\Gamma_{\mathrm{cl}}^{*}(M)$ is graded, and we have a direct sum
decomposition $\Gamma_{\mathrm{cl}}^{*}(M)=\bigoplus_{n\geq
0}\Gamma_{\mathrm{cl}}^{n}(M)$, where $\Gamma_{\mathrm{cl}}^{0}(M)=R$ and
$\Gamma_{\mathrm{cl}}^{1}(M)=M$. We will refer to the $R$-module
$\Gamma_{\mathrm{cl}}^{n}(M)=\Gamma_{\mathrm{cl},R}^{n}(M)$ as the classical
$n$th divided power of $M$ over $R$. The assignment $x\mapsto\gamma^{n}(x)$
defines a map of $R$-modules $\gamma^{n}\colon
M\to\Gamma_{\mathrm{cl}}^{n}(M)$. Notice that $\gamma^{0}\equiv 1\colon M\to
R$ is the constant map, and $\gamma^{1}=\operatorname{id}\colon M\to M$ is the
identity map on $M$. For an element $x\in M$, we will also use the notation
$x^{(n)}:=\gamma^{n}(x)\in\Gamma_{\mathrm{cl}}^{n}(M)$; we will refer to
$x^{(n)}=\gamma^{n}(x)$ as the $n$th divided power of $x$.

We refer the readers to [Bou, Chapter III, §6, §7] and [Ro] for the basic
properties of the above classical constructions. Here, we will list some of
them so that the reader can easily compare them with the corresponding
properties of their derived counterparts:

  1. (1)

(Base change) Let $R\to R^{\prime}$ be a homomorphism of (ordinary)
commutative rings, and let $M$ be a discrete $R$-module. Then there are
canonical isomorphisms

$\displaystyle R^{\prime}\otimes_{R}^{\mathrm{cl}}S_{R}^{*}(M)$
$\displaystyle\simeq
S_{R^{\prime}}^{*}(R^{\prime}\otimes_{R}^{\mathrm{cl}}M),$ $\displaystyle
R^{\prime}\otimes_{R}^{\mathrm{cl}}\bigwedge\nolimits_{\mathrm{cl},R}^{*}(M)$
$\displaystyle\simeq\bigwedge\nolimits_{\mathrm{cl},R^{\prime}}^{*}(R^{\prime}\otimes_{R}^{\mathrm{cl}}M),$
$\displaystyle
R^{\prime}\otimes_{R}^{\mathrm{cl}}\Gamma_{\mathrm{cl},R}^{*}(M)$
$\displaystyle\simeq\Gamma_{\mathrm{cl},R^{\prime}}^{*}(R^{\prime}\otimes_{R}^{\mathrm{cl}}M)$

of graded $R^{\prime}$-algebras; see [Bou, Chapter III, §6.4, Proposition 7,
§7.5, Proposition 8] and [Ro, Theorem III.3, p. 262].

  2. (2)

(Freeness) If $M$ is a (locally) free $R$-module of rank $m$, then for any
integer $n\geq 0$, $S^{n}(M)$, $\bigwedge_{\mathrm{cl}}^{n}M$, and
$\Gamma_{\mathrm{cl}}^{n}(M)$ are (locally) free $R$-modules of ranks
$\binom{n+m-1}{n}$, ${m\choose n}$, and $\binom{n+m-1}{n}$, respectively. In
this case, we will drop the subscript $\mathrm{cl}$ and denote them by
$S^{n}(M)$, $\bigwedge^{n}M$, and $\Gamma^{n}(M)$; these notations will be
justified in the next section.

  3. (3)

(Duality) If $M$ is a locally free $R$-module, then for any integer $n\geq 0$,
there are canonical isomorphisms of $R$-modules
$(\bigwedge^{n}M)^{\vee}\simeq\bigwedge^{n}(M^{\vee})$,
$(S^{n}(M))^{\vee}\simeq\Gamma(M^{\vee})$. (Here, $N^{\vee}$ denote the
$R$-linear dual of a locally free $R$-module $N$.)

  4. (4)

(Right exactness) The formations of the graded algebras $S^{*}(M)$,
$\bigwedge_{\mathrm{cl}}^{*}M$ and $\Gamma_{\mathrm{cl}}^{*}(M)$ are right
exact in the variable $M$ in the following sense: suppose we are given a short
exact sequence of discrete $R$-modules:

$0\to P\to M\xrightarrow{u}N\to 0.$

Then there are induced short exact sequences

$\displaystyle 0\to I_{1}\to S^{*}(M)\xrightarrow{S^{*}(u)}S^{*}(N)\to 0,$
$\displaystyle 0\to
I_{2}\to\bigwedge\nolimits_{\mathrm{cl}}^{*}M\xrightarrow{\bigwedge\nolimits_{\mathrm{cl}}^{*}(u)}\bigwedge\nolimits_{\mathrm{cl}}^{*}N\to
0,$ $\displaystyle 0\to
I_{3}\to\Gamma_{\mathrm{cl}}^{*}(M)\xrightarrow{\Gamma_{\mathrm{cl}}^{*}(u)}\Gamma_{\mathrm{cl}}^{*}(N)\to
0,$

where $I_{1}$ is the ideal of the commutative algebra $S^{*}(M)$ generated by
$P$ ([Bou, Chapter III, §6.2, Proposition 4]), $I_{2}$ is the two-sided ideal
of the algebra $\bigwedge_{\mathrm{cl}}^{*}(M)$ generated by $P$ ([Bou,
Chapter III, §7.2, Proposition 3]), and $I_{3}$ is the ideal of the
commutative algebra $\Gamma_{\mathrm{cl}}^{*}(M)$ generated by elements of the
form $x^{(i)}$ for $x\in P$, $i\geq 1$ ([Ro, Proposition IV.8, p. 284]),
respectively.

######  Remark 2.6.

A word of caution: if $n\geq 2$, the functors $M\mapsto
S^{n}(M),\bigwedge_{\mathrm{cl}}^{n}(M),\Gamma_{\mathrm{cl}}^{n}(M)$ are not
right exact in the usual sense as they are not additive. However, they are
“right exact” in a “non-abelian sense” that they commute with reflexive
coequalizers, which is one of the underlying reasons they have a well-behaved
non-abelian derived theory. We will explore their derived functors in the
following subsections.

###  2.2. Non-abelian derived functors

To define and study the derived functors of symmetric, exterior, and divided
powers, we will use Lurie’s general theory of non-abelian derived functors as
developed in [HTT, §5.5.8], which we review in this subsection.

To keep our exposition brief, we introduce the following notations:

  * •

We let $\mathcal{C}\mathrm{at}_{\infty}^{\amalg}$ (resp.
$\mathcal{C}\mathrm{at}_{\infty}^{\pi}$) denote the subcategory of
$\mathcal{C}\mathrm{at}_{\infty}$ spanned by those $\infty$-categories which
admit finite coproducts (resp. finite products) and those functors which
preserve finite coproducts (resp. finite products). For
$\mathcal{C},\mathcal{D}\in\mathcal{C}\mathrm{at}_{\infty}^{\amalg}$ (resp.
$\mathcal{C}\mathrm{at}_{\infty}^{\pi}$), we let
$\mathrm{Fun}^{\amalg}(\mathcal{C},\mathcal{D})$ (resp.
$\mathrm{Fun}^{\pi}(\mathcal{C},\mathcal{D})$) denote the full subcategory of
$\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by those functors which
preserve finite coproducts (resp. finite products).

  * •

We let $\mathcal{C}\mathrm{at}_{\infty}^{\Sigma}$ denote the subcategory of
$\mathcal{C}\mathrm{at}_{\infty}$ spanned by those $\infty$-categories which
admit small sifted colimits and those functors which preserve small sifted
colimits. For
$\mathcal{C},\mathcal{D}\in\mathcal{C}\mathrm{at}_{\infty}^{\Sigma}$, we let
$\mathrm{Fun}_{\Sigma}(\mathcal{C},\mathcal{D})$ denote the full subcategory
of $\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by those functors which
preserve small sifted colimits.

  * •

We let $\widehat{\mathcal{C}\mathrm{at}}_{\infty}$ denote the “very large
version” of $\mathcal{C}\mathrm{at}_{\infty}$, that is, it is the
$\infty$-category of not necessarily small $\infty$-categories. We let
$\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\rm L}$ denote the full
subcategory of $\widehat{\mathcal{C}\mathrm{at}}_{\infty}$ spanned by those
$\infty$-categories which admit all small colimits and those functors which
preserve all small colimits. For
$\mathcal{C},\mathcal{D}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\rm L}$,
we let $\mathrm{Fun}^{\rm L}(\mathcal{C},\mathcal{D})$ denote the full
subcategory of $\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by those
functors which preserve all small colimits.

  * •

Similarly, we let $\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{?}$ denote the
“very large version” of the $\infty$-category
$\mathcal{C}\mathrm{at}_{\infty}^{?}$, where $?=\amalg,\pi,\Sigma$ (that is,
$\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{?}\subseteq\widehat{\mathcal{C}\mathrm{at}}_{\infty}$
is defined in a same way as the above-defined
$\mathcal{C}\mathrm{at}_{\infty}^{?}$ with the only exception that its objects
are not assumed to be small).

Let $\mathcal{C}\in\mathcal{C}\mathrm{at}_{\infty}^{\amalg}$ be an
$\infty$-category that admits finite coproducts, and let
$\mathcal{P}_{\Sigma}(\mathcal{C})=\mathrm{Fun}^{\pi}(\mathcal{C}^{\mathrm{op}},\mathcal{S})$
denote the $\infty$-category spanned by those functors which preserve finite
products. We can think of $\mathcal{P}_{\Sigma}(\mathcal{C})$ as obtained from
$\mathcal{C}$ by formally adjoining sifted colimit. More precisely:

######  Proposition 2.7 (Lurie [HTT, §5.5.8]).

For any $\mathcal{C}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\amalg}$,
there exists a Yoneda functor
$j\colon\mathcal{C}\to\mathcal{P}_{\Sigma}(\mathcal{C})$, such that the
natural transform
$\\{j\colon\mathcal{C}\to\mathcal{P}_{\Sigma}(\mathcal{C})\\}_{\mathcal{C}}$
exhibits the functor

$(\mathcal{C}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\amalg})\mapsto(\mathcal{P}_{\Sigma}(\mathcal{C})\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\rm
L})$

as a left adjoint to the inclusion
$\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\rm
L}\subseteq\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\amalg}$. Moreover,
given any $\mathcal{C}\in\mathcal{C}\mathrm{at}_{\infty}^{\amalg}$:

  1. (1)

The Yoneda functor $j\colon\mathcal{C}\to\mathcal{P}_{\Sigma}(\mathcal{C})$ is
fully faithful and preserves finite coproducts.

  2. (2)

For any $\mathcal{D}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\Sigma}$,
composition with $j$ induces an equivalence of $\infty$-categories

$\mathrm{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})\to\mathrm{Fun}(\mathcal{C},\mathcal{D}).$

Moreover, a functor $F\colon\mathcal{P}_{\Sigma}(\mathcal{C})\to\mathcal{D}$
belongs to
$\mathrm{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})$ if and
only if $F$ is a left Kan extension of $f=F\circ j$ along $j$.

  3. (3)

For any $\mathcal{D}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty}^{\rm L}$,
composition with $j$ restricts to an equivalence of $\infty$-categories

$\mathrm{Fun^{L}}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})\to\mathrm{Fun}^{\amalg}(\mathcal{C},\mathcal{D}).$

Combining with (2), we obtain that for any given functor
$F\in\mathrm{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})$,
$F$ preserves all small colimits if and only if $f=F\circ j$ preserves finite
coproducts.

###### Proof.

This is a reformulation of [HTT, Propositions 5.5.8.10, 5.5.8.15] in view of
the adjunction provided by [HTT, Corollary 5.3.6.10]. ∎

######  Definition 2.8.

In the situation of Proposition 2.7 (2) or (3), we will refer to the functor
$F$ as the derived functor of $f=F|\mathcal{C}$. By virtue of Proposition 2.7,
the functor $F$ is essentially uniquely determined by $f$ and preserves sifted
colimits.

###  2.3. Derived symmetric, exterior, and divided powers

In this subsection, we apply Lurie’s machinery of §2.2 to the classical
functors of §2.1.

######  Notation 2.9 ([SAG, §25.2.1]).

Let $A$ be a simplicial commutative ring and let $A^{\circ}$ be its underlying
$\mathbb{E}_{\infty}$-ring. We let $\operatorname{SCRMod}^{\rm
cn}:=\operatorname{CAlg}^{\Delta}\times_{\operatorname{CAlg}}\operatorname{Mod}^{\mathrm{cn}}$
denote the $\infty$-category of pairs $(A,M)$, where $A$ is a simplicial
commutative ring and $M$ is a connective $A^{\circ}$-module. Then the
forgetful functor $q\colon\operatorname{SCRMod}^{\rm
cn}\to\operatorname{CAlg}^{\Delta}$ is a coCartesian ﬁbration which classifies
the functor
$(A\in\operatorname{CAlg}^{\Delta})\mapsto(\operatorname{Mod}^{\mathrm{cn}}_{A}\in\widehat{\mathcal{C}\mathrm{at}}_{\infty})$.
Let ${\rm PolyMod}^{\rm ff}\subseteq\operatorname{SCRMod}^{\rm cn}$ denote the
full subcategory spanned by elements of the form
$(A=\mathbb{Z}[x_{1},\ldots,x_{m}],M=A^{n})$ for $m,n\geq 0$. Then the
inclusion ${\rm PolyMod}^{\rm ff}\hookrightarrow\operatorname{SCRMod}^{\rm
cn}$ extends to an equivalence of $\infty$-categories
$\mathcal{P}_{\Sigma}({\rm PolyMod}^{\rm ff})\simeq\operatorname{SCRMod}^{\rm
cn}$ ([SAG, Proposition 25.2.1.2]).

######  Construction 2.10 (Derived symmetric, exterior and divided powers;
Lurie [SAG, §25.2.2]).

In the situation of Proposition 2.7, we let $\mathcal{C}={\rm PolyMod}^{\rm
ff}$, $\mathcal{D}=\operatorname{SCRMod}^{\rm cn}$ (Notation 2.9) and let
$f\colon{\rm PolyMod}^{\rm ff}\to{\rm PolyMod}^{\rm
ff}\subseteq\operatorname{SCRMod}^{\rm cn}$ be one of classical functors
considered in §2.1. Then we obtain the derived functor of $f$. More
concretely, let $n\geq 0$ be an integer:

  1. (1)

Let $f$ be the functor $(R,M)\mapsto(R,S_{R}^{n}(M))$, where $S_{R}^{n}(M)$ is
the classical $n$th symmetric power of $M$ over $R$ (Definition 2.3). Then the
derived functor of $f$ has the form $F(A,M)=(A,\operatorname{Sym}_{A}^{n}M)$,
where $A\in\operatorname{CAlg}^{\Delta}$ and
$\operatorname{Sym}_{A}^{n}M\in\operatorname{Mod}_{A}^{\mathrm{cn}}$. The
connective $A$-module $\operatorname{Sym}_{A}^{n}M$ is called the derived
$n$th symmetric power of $M$ over $A$.

  2. (2)

Let $f$ be the functor $(R,M)\mapsto(R,\bigwedge_{\mathrm{cl},R}^{n}(M))$,
where $\bigwedge_{\mathrm{cl},R}^{n}(M))$ is the classical $n$th exterior
power of $M$ over $R$ (Definition 2.4). Then the derived functor of $f$ has
the form $F(A,M)=(A,\bigwedge_{A}^{n}M)$, where
$A\in\operatorname{CAlg}^{\Delta}$ and
$\bigwedge_{A}^{n}M\in\operatorname{Mod}_{A}^{\mathrm{cn}}$. We will refer to
the connective $A$-module $\bigwedge_{A}^{n}M$ as the derived $n$th exterior
power of $M$ over $A$.

  3. (3)

Let $f$ be the functor $(R,M)\mapsto(R,\Gamma_{\mathrm{cl},R}^{n}(M))$, where
$\Gamma_{\mathrm{cl},R}^{n}(M))$ is the classical $n$th divided power of $M$
over $R$ (Definition 2.5). Then the derived functor of $f$ has the form
$F(A,M)=(A,\Gamma_{A}^{n}M)$, where $A\in\operatorname{CAlg}^{\Delta}$ and
$\Gamma_{A}^{n}M\in\operatorname{Mod}_{A}^{\mathrm{cn}}$. We will refer to the
connective $A$-module $\Gamma_{A}^{n}M$ as the derived $n$th divdied power of
$M$ over $A$.

For readers’ convenience, we list some of the basic properties of these
functors that are frequently used in this paper; all these results in our
context are due to Lurie:

######  Proposition 2.11 (Lurie [SAG, §25.2]).

The derived symmetric, exterior, and divided power functors defined in
Construction 2.10 satisfy the following properties:

  1. (1)

(Base change) Let $A\to B$ be a morphism of simplicial commutative rings. Then
for any connective $A$-module $M$ and an integer $n\geq 0$, there are
canonical equivalences:

$\displaystyle B\otimes_{A}\operatorname{Sym}_{A}^{n}(M)$
$\displaystyle\xrightarrow{\sim}\operatorname{Sym}_{B}^{n}(B\otimes_{A}M);$
$\displaystyle B\otimes_{A}\bigwedge\nolimits_{A}^{n}(M)$
$\displaystyle\xrightarrow{\sim}\bigwedge\nolimits_{B}^{n}(B\otimes_{A}M);$
$\displaystyle B\otimes_{A}\Gamma_{A}^{n}(M)$
$\displaystyle\xrightarrow{\sim}\Gamma_{B}^{n}(B\otimes_{A}M).$

  2. (2)

(Freeness) For any simplicial commutative ring
$A\in\operatorname{CAlg}^{\Delta}$ and an integer $n\geq 0$, if the $A$-module
$M$ is (locally) free of ﬁnite rank $r$, then $\operatorname{Sym}_{A}^{n}M$,
$\bigwedge_{A}^{n}M$ and $\Gamma_{A}^{n}M$ are (locally) free of ﬁnite ranks
${n+r-1\choose n}$, ${r\choose n}$, and respectively ${n+r-1\choose n}$. If
$M$ is flat, then $\operatorname{Sym}_{A}^{n}M$, $\bigwedge_{A}^{n}M$ and
$\Gamma_{A}^{n}M$ are flat.

  3. (3)

(Flat modules over discrete rings) If $A=R$ is an ordinary commutative ring,
and $M$ is a flat $A$-module, then there are canonical isomorphisms

$\operatorname{Sym}_{R}^{n}M\simeq\operatorname{Sym}_{\mathrm{cl},R}^{n}M,\qquad\bigwedge\nolimits_{R}^{n}M\simeq\bigwedge\nolimits_{\mathrm{cl},R}^{n}M,\qquad\Gamma_{R}^{n}(M)\simeq\Gamma_{\mathrm{cl},R}^{n}(M).$

(In this situation, we will often abuse notation and use
$\operatorname{Sym}_{R}^{n}M$, $\bigwedge_{A}^{n}M$ and $\Gamma_{A}^{n}M$ to
denote the corresponding classical constructions.)

  4. (4)

(Illusie’s proposition) Let $A$ be an simplicial commutative ring, then for
any connective $A$-module $M$ and an integer $n\geq 0$, there are canonical
equivalences of connective $A$-modules $\operatorname{Sym}_{A}^{n}(\Sigma
M)\simeq\Sigma^{n}\bigwedge_{A}^{n}(M)$ and $\bigwedge_{A}^{n}(\Sigma
M)\simeq\Sigma^{n}\Gamma_{A}^{n}(M)$.

###### Proof.

The assertion (1) is [SAG, Proposition 25.2.3.1], the assertion (2) is [SAG,
Corollary 25.2.3.2, Corollary 25.2.3.3], the assertion (3) is [SAG,
Proposition 25.2.3.4], and the assertion (4) is [SAG, Proposition 25.2.4.2]. ∎

###  2.4. Derived symmetric algebras

In this subsection, we review the construction of derived symmetric algebras
and study their basic properties.

######  Definition 2.12 (Derived symmetric algebras; [SAG, Construction
25.2.2.6]).

Let $\operatorname{SCRMod}^{\rm cn}$ be defined as in Notation 2.9, and let
$\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$ denote the functor
$\infty$-category whose objects are morphisms of simplicial commutative rings
$A\to B$. Then the forgetful functor

${\rm
forg}\colon\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})\to\operatorname{SCRMod}^{\rm
cn},\quad(A\to B)\mapsto(A,B^{\circ})$

admits a left adjoint functor of the following form:

$\operatorname{SCRMod}^{\rm
cn}\to\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta}),\quad(A,M)\mapsto(A,\operatorname{Sym}_{A}^{*}(M)).$

The $A$-algebra
$\operatorname{Sym}_{A}^{*}(M)\in\operatorname{CAlg}^{\Delta}_{A}$ is called
derived symmetric algebra of $M$ over $A$. By virtue of [SAG, Construction
25.2.2.6], there is a canonical equivalence of $A$-modules

$\operatorname{Sym}_{A}^{*}(M)\simeq\bigoplus_{n\geq
0}\operatorname{Sym}_{A}^{n}(M).$

######  Remark 2.13 (The underlying commutative algebra objects of derived
symmetric algebras).

Let $A$ be a simplicial commutative ring, and let
$\Theta\colon\operatorname{CAlg}^{\Delta}\to\operatorname{CAlg}_{\mathbb{Z}}^{\mathrm{cn}}$
denote the functor of taking underlying $\mathbb{E}_{\infty}$-rings ([SAG,
Construction 25.1.2.1]). Then $\Theta$ induces a forgetful functor

$\Theta_{A}\colon\operatorname{CAlg}^{\Delta}_{A}\to\operatorname{CAlg}^{\mathrm{cn}}_{\Theta(A)/}\simeq\operatorname{CAlg}(\operatorname{Mod}_{A}^{\mathrm{cn}}),$

where the last equivalence is given by [HA, Corollary 3.4.1.7]. For an object
$B\in\operatorname{CAlg}^{\Delta}_{A}$, we will refer to its image
$\Theta_{A}(B)\in\operatorname{CAlg}(\operatorname{Mod}_{A}^{\mathrm{cn}})$ as
the underlying commutative algebra object in
$\operatorname{Mod}_{A}^{\mathrm{cn}}$ of $B$. For any connective $A$-module
$M$, we will generally abuse notations by using the notation
$\operatorname{Sym}_{A}^{*}(M)$ to denote the derived symmetric algebra of $M$
over $A$, its underlying commutative algebra object in
$\operatorname{Mod}_{A}^{\mathrm{cn}}$, and its underlying connective
$A$-module. In particular, the underlying connective $A$-module
$\operatorname{Sym}_{A}^{*}(M)\simeq\bigoplus_{n\geq
0}\operatorname{Sym}_{A}^{n}(M)$ is equipped with a multiplication map

$m\colon\operatorname{Sym}_{A}^{*}(M)\otimes_{A}\operatorname{Sym}_{A}^{*}(M)\to\operatorname{Sym}_{A}^{*}(M)$

which is unital, commutative, and associative up to homotopy. Furthermore, the
multiplication is graded, that is, $m$ restricts to a map of connective
$A$-modules

$\operatorname{Sym}_{A}^{n}(M)\otimes_{A}\operatorname{Sym}_{A}^{n^{\prime}}(M)\to\operatorname{Sym}_{A}^{n+n^{\prime}}(M).$

(To prove this assertion, observe that since the relative tensor product
$\otimes_{A}\colon\operatorname{Mod}_{A}^{\mathrm{cn}}\times\operatorname{Mod}_{A}^{\mathrm{cn}}\to\operatorname{Mod}_{A}^{\mathrm{cn}}$
preserves sifted colimits separately in each variable, we might regard the
multiplication $m$ as a functor from $\operatorname{SCRMod}^{\rm cn}$ to
$\mathcal{E}$, where $\mathcal{E}$ is defined as in Notation 2.18. Then since
$m$ preserves sifted colimits, it is determined by its restrictions to ${\rm
PolyMod}^{\rm ff}$, for which the desired statements are straightforward.)

######  Remark 2.14 (Coproducts in $\operatorname{SCRMod}^{\rm cn}$ and
$\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$).

Let $\mathcal{C}_{0}={\rm PolyMod}^{\rm ff}$ denote the full subcategory of
$\operatorname{SCRMod}^{\rm cn}$ spanned by objects of the form
$(A=\mathbb{Z}[x_{1},\ldots,x_{m}],M=A^{n})$, as in Notation 2.9. Then
$\mathcal{C}_{0}$ admits finite coproducts given by the formula
$(A,M=A^{n})\coprod(B,N=B^{t})=(A\otimes B,M\boxplus N)$, where
$A=\mathbb{Z}[x_{1},\ldots,x_{m}]$, $B=\mathbb{Z}[y_{1},\ldots,y_{s}]$,
$A\otimes B\simeq\mathbb{Z}[x_{1},\ldots,x_{m},y_{1},\ldots,y_{s}]$, and
$M\boxplus N\simeq(A\otimes B)^{n+t}$. By virtue of Proposition 2.7, the
coproduct structure of $\mathcal{C}_{0}$ extends to a coproduct structure on
$\operatorname{SCRMod}^{\rm cn}\simeq\mathcal{P}_{\Sigma}(\mathcal{C}_{0})$
for which the inclusion
$\mathcal{C}_{0}\hookrightarrow\operatorname{SCRMod}^{\rm cn}$ preserves
finite coproducts. The forgetful functor $\operatorname{SCRMod}^{\rm
cn}\to\operatorname{CAlg}^{\Delta}$ preserves finite coproducts, therefore
coproducts in $\operatorname{SCRMod}^{\rm cn}$ takes the form
$(A,M)\coprod(B,N)=(A\otimes B,M\boxplus N)$, where $A\otimes B$ is the
pushout of the diagram of simplicial commutative rings
$A\leftarrow\mathbb{Z}\rightarrow B$, and $M\boxplus N$ is an $A\otimes
B$-module. We will refer $M\boxplus N\in\operatorname{Mod}_{A\otimes B}$ as
the external direct sum of $M$ and $N$.

We denote the coproduct of two objects $(A\to A^{\prime})$ and $(B\to
B^{\prime})$ in $\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$ by
$(A\to B)\coprod(A^{\prime}\to B^{\prime})=(A\otimes A^{\prime},B\boxtimes
B^{\prime})$, where $A\otimes B$ is the pushout of the diagram of simplicial
commutative rings $A\leftarrow\mathbb{Z}\rightarrow A^{\prime}$ as before, and
$B\boxtimes B^{\prime}\in\operatorname{CAlg}^{\Delta}_{A\otimes A^{\prime}}$
denotes the simplicial commutative ring $B\otimes B^{\prime}$ with the induced
$A\otimes A^{\prime}$-algebra structure. We will refer to $B\boxtimes
B^{\prime}\in\operatorname{CAlg}^{\Delta}_{A\otimes A^{\prime}}$ as the
external tensor product of $B$ and $B^{\prime}$.

######  Definition 2.15.

Let $A$ be a simplicial commutative ring and $n\geq 0$ an integer. We say an
$A$-module $M$ is perfect (resp., perfect to order $n$, in the sense of [DAG,
Proposition 2.5.7]) if the functor
$\mathrm{Map}_{\operatorname{Mod}_{A}}(M,\underline{\hphantom{A}})$ preserve
filtered colimits (resp., if the restriction of functor
$\mathrm{Map}_{\operatorname{Mod}_{A}}(M,\underline{\hphantom{A}})$ to
$\tau_{\leq n}\operatorname{Mod}_{A}$ preserves filtered colimits). We $M$ is
almost perfect if it is perfect to order $n$ for all $n\geq 0$. Let
$\phi\colon A\to B$ be a map of simplicial commutative rings. We say $\phi$ is
locally of finite presentation (resp., of finite presentation to order $n$, in
the sense of [DAG, Proposition 3.1.5]) if the functor
$\mathrm{Map}_{\operatorname{CAlg}^{\Delta}_{A}}(B,\underline{\hphantom{A}})$
preserve filtered colimits (resp., if the restriction of functor
$\mathrm{Map}_{\operatorname{CAlg}^{\Delta}_{A}}(B,\underline{\hphantom{A}})$
to $\tau_{\leq n}\operatorname{CAlg}^{\Delta}_{A}$ preserves filtered
colimits). We say $\phi$ is almost of ﬁnite presentation if it is of finite
presentation to order $n$ for all $n\geq 0$.

The following proposition is a list of basic properties of derived symmetric
algebras:

######  Proposition 2.16.

Let $A$ be a simplicial commutative ring and $M$ a connective $A$-module.

  1. (1)

(Classical truncations) There is a canonical equivalence of discrete
$\pi_{0}A$-algebras

$\pi_{0}(\operatorname{Sym}_{A}^{*}(M))\simeq\pi_{0}(\operatorname{Sym}_{\pi_{0}A}^{*}(\pi_{0}M))\simeq
S^{*}_{\pi_{0}A}(\pi_{0}M),$

where $S_{\pi_{0}A}^{*}(\pi_{0}M)$ denotes the classical symmetric algebra of
the discrete module $\pi_{0}(M)$ over the discrete commutative ring
$\pi_{0}A$.

  2. (2)

(Finiteness conditions) The map of simplicial commutative rings
$A\to\operatorname{Sym}_{A}^{*}(M)$ is of finite presentation to order $n$
(resp. almost of ﬁnite presentation, locally of finite presentation) if and
only if the $A$-module $M$ is perfect to order $n$ (resp. almost perfect,
perfect).

  3. (3)

(Relative cotangent complexes) Let $L_{\operatorname{Sym}_{A}^{*}(M)/A}$
(resp. $L_{A/\operatorname{Sym}_{A}^{*}(M)}$) denote the relative cotangent
complex of the map of simplicial commutative rings
$A\to\operatorname{Sym}_{A}^{*}(M)$ (resp. the augmentation map
$\operatorname{Sym}_{A}^{*}(M)\to A$). Then there are canonically equivalences
$L_{\operatorname{Sym}_{A}^{*}(M)/A}\simeq
M\otimes_{A}\operatorname{Sym}_{A}^{*}(M)\in\operatorname{Mod}_{\operatorname{Sym}_{A}^{*}(M)}^{\rm
cn}$ and
$L_{A/\operatorname{Sym}_{A}^{*}(M)}\simeq\Sigma\,M\in\operatorname{Mod}_{A}^{\rm
cn}$.

  4. (4)

(Base-change) The functor $(A,M)\in\operatorname{SCRMod}^{\rm
cn}\mapsto(A,\operatorname{Sym}_{A}^{*}M)\in\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$
carries $q$-coCartesian edges of $\operatorname{SCRMod}^{\rm cn}$ to
$\mathrm{ev}_{\\{0\\}}$-coCartesian edges of
$\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$. In other words, for
any map $A\to A^{\prime}$ in $\operatorname{CAlg}^{\Delta}$ and any connective
$A$-module $M$, there is a canonical equivalence
$A^{\prime}\otimes_{A}\operatorname{Sym}_{A}^{*}(M)\simeq\operatorname{Sym}_{A^{\prime}}^{*}(A^{\prime}\otimes_{A}M)\in\operatorname{CAlg}^{\Delta}_{A^{\prime}}$.

  5. (5)

(Direct sums) For $A,B\in\operatorname{CAlg}^{\Delta}$,
$M\in\operatorname{Mod}^{\rm cn}_{A}$, $N\in\operatorname{Mod}^{\rm cn}_{B}$,
there is a canonical equivalence $\operatorname{Sym}_{A\otimes
B}^{*}(M\boxplus
N)\simeq\operatorname{Sym}_{A}^{*}(M)\boxtimes\operatorname{Sym}_{B}^{*}(N)$
in $\operatorname{CAlg}^{\Delta}_{A\otimes B}$. In particular, for any
connective $A$-modules $M$ and $M^{\prime}$, there is a canonical equivalence
of algebras $\operatorname{Sym}_{A}^{*}(M\oplus
M^{\prime})\simeq\operatorname{Sym}_{A}^{*}(M)\otimes_{A}\operatorname{Sym}_{A}^{*}(M^{\prime})\in\operatorname{CAlg}^{\Delta}_{A}$.

###### Proof.

Assertion (5) follows from the combination of Remark 2.14 and Proposition 2.7:
let $\mathcal{C}_{0}={\rm PolyMod^{ff}}$ be as defined in Remark 2.14, then it
is direct to see that the restriction
$(\operatorname{Sym}^{*}){|\mathcal{C}_{0}}\colon\mathcal{C}_{0}\to\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$
preserves finite coproducts. Therefore, by virtue of Proposition 2.7 (3),
$\operatorname{Sym}^{*}\colon\operatorname{SCRMod}^{\rm
cn}\to\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})$ preserves finite
coproducts. The last statement of (5) follows from base change along the
multiplication map $A\otimes A\to A$.

We next prove assertion (4). To show the canonical map
$\alpha_{(M,A^{\prime})}\colon
A^{\prime}\otimes_{A}\operatorname{Sym}_{A}^{*}(M)\to\operatorname{Sym}_{A^{\prime}}^{*}(A^{\prime}\otimes_{A}M)$
is an equivalence, arguing as the proof of [SAG, Proposition 25.2.3.1], (since
the functor $M\mapsto\alpha_{(M,A^{\prime})}$ commutes with sifted colimits)
we may first reduce to the case where $M=A\times_{\mathbb{Z}}M_{0}$,
$M_{0}\simeq\mathbb{Z}^{m}$. Then we have a commutative diagram

${A^{\prime}\otimes_{A}A\otimes_{\mathbb{Z}}\operatorname{Sym}_{\mathbb{Z}}^{*}(M_{0})}$${A^{\prime}\otimes_{A}\operatorname{Sym}_{A}^{*}(A\otimes_{\mathbb{Z}}M_{0})}$${\operatorname{Sym}_{A^{\prime}}^{*}(A^{\prime}\otimes_{A}A\otimes_{\mathbb{Z}}M_{0}).}$$\scriptstyle{\alpha_{M}}$

To show $\alpha_{M}$ is an equivalence, it suffices to prove in the case where
$A=\mathbb{Z}$. Since the functor $A^{\prime}\mapsto\alpha_{M,A^{\prime}}$
commutes with sifted colimits, we may further reduce to the case where
$A^{\prime}=\mathbb{Z}[x_{1},\ldots,x_{n}]$. In this case, $\alpha_{M}$ is
evidently an isomorphism of polynomial algebras.

For assertion (3), the formula $L_{\operatorname{Sym}_{A}^{*}(M)/A}\simeq
M\otimes_{A}\operatorname{Sym}_{A}^{*}(M)$ is [SAG, Example 25.3.2.2]. By
virtue of [SAG, Remark 25.3.2.5], the composition
$A\to\operatorname{Sym}_{A}^{*}(M)\to A$ of simplicial commutative rings
induces a fiber sequence

$A\otimes_{\operatorname{Sym}_{A}^{*}(M)}L_{\operatorname{Sym}_{A}^{*}(M)/A}\to
L_{A/A}\to L_{A/\operatorname{Sym}_{A}^{*}(M)}.$

The formula $L_{A/\operatorname{Sym}_{A}^{*}(M)}\simeq\Sigma\,M$ follows from
the above fiber sequence and $L_{A/A}\simeq 0$.

For assertion (2), if the map $A\to\operatorname{Sym}_{A}^{*}(M)$ is of finite
presentation to order $n$ (resp. almost of ﬁnite presentation, locally of
finite presentation), then it follows from [DAG, Proposition 3.2.14] that
$L_{\operatorname{Sym}_{A}^{*}(M)/A}$ is perfect to order $n$ (resp., almost
perfect, perfect) as an $\operatorname{Sym}_{A}^{*}(M)$-module. Consequently,
$M=L_{\operatorname{Sym}_{A}^{*}(M)/A}\otimes_{\operatorname{Sym}_{A}^{*}(M)}A$
is perfect to order $n$ (resp., almost perfect, perfect) as an $A$-module. For
the converse direction, from the canonical equivalence of spaces

$\mathrm{Map}_{\operatorname{CAlg}^{\Delta}_{A}}(\operatorname{Sym}_{A}^{*}(M),B)\simeq\mathrm{Map}_{\operatorname{Mod}_{A}}(M,B^{\circ})$

for all $B\in\operatorname{CAlg}^{\Delta}_{A}$ and the fact that $(\tau_{\leq
n}B)^{\circ}\simeq\tau_{\leq n}(B^{\circ})$, we deduce that if $M$ is perfect
to order $n$ (resp. perfect), then $A\to\operatorname{Sym}_{A}^{*}(M)$ is of
finite presentation to order $n$ (resp. locally of finite presentation).

Finally, we prove assertion (1). Apply assertion (4) to the map
$A\to\pi_{0}(A)$, we obtain an equivalence
$\pi_{0}(A)\otimes_{A}\operatorname{Sym}_{A}^{*}(M)\simeq\operatorname{Sym}_{\pi_{0}A}^{*}(\pi_{0}A\otimes_{A}M)$.
Consequently, we have

$\pi_{0}(\operatorname{Sym}_{A}^{*}M)\simeq\pi_{0}(\operatorname{Sym}_{\pi_{0}A}^{*}(\pi_{0}A\otimes_{A}M)).$

Let $R=\pi_{0}A$. It remains to show that for
$N=\pi_{0}(A)\otimes_{A}M\in\operatorname{Mod}^{\rm cn}_{R}$, there is a
canonical equivalence $\pi_{0}(\operatorname{Sym}_{R}^{*}N)\simeq
S_{R}^{*}(\pi_{0}N)$, where $S_{R}^{*}(\pi_{0}N)$ is the classical symmetric
algebra. Let $P_{\bullet}$ be a simplicial resolution of $N$, where each
$P_{n}$ is a projective $R$-module. Since the derived symmetric algebra
functor $\operatorname{Sym}_{R}^{*}(\underline{\hphantom{A}})$ preserve sifted
colimits, we obtain equivalences

$\operatorname{Sym}_{R}^{*}(M)\simeq|\operatorname{Sym}_{R}^{*}(P_{\bullet})|\simeq|S_{R}^{*}(P_{\bullet})|.$

Consequently, $\pi_{0}(\operatorname{Sym}_{R}^{*}(N))$ is isomorphic to the
colimit of the diagram

${S_{R}^{*}(P_{1})}$${S_{R}^{*}(P_{0})}$$\scriptstyle{S_{R}^{*}(d_{1})}$$\scriptstyle{S_{R}^{*}(d_{0})}$

(where $d_{0},d_{1}\colon P_{1}\rightrightarrows P_{0}$ are the face maps). On
the other hand, the colimit of the above diagram is canonically isomorphic to
$S_{R}^{*}(\pi_{0}N)$, by the right-exactness of the classical symmetric
algebra functors (see, for example, [Bou, Chapter III, §6.2, Proposition 4,
p.499]). ∎

######  Proposition 2.17.

Let $R$ be a simplicial commutative ring, and let $\rho\colon M^{\prime}\to M$
be a map of connective $R$-modules. Then there is a pushout diagram in
$\operatorname{CAlg}^{\Delta}_{R}$:

${\operatorname{Sym}_{R}^{*}(M^{\prime})}$${\operatorname{Sym}_{R}^{*}(M)}$${R}$${\operatorname{Sym}_{R}^{*}({\rm
cofib}(\rho)).}$$\scriptstyle{\operatorname{Sym}_{R}^{*}(\rho)}$

###### Proof.

Being a left adjoint (Definition 2.12), the functor
$\operatorname{Sym}_{R}^{*}$ preserves small colimits. Consequently, the
desired result follows from applying $\operatorname{Sym}_{R}^{*}$ to the
pushout square

${M^{\prime}}$${M}$${0}$${\operatorname{cofib}(\rho).}$$\scriptstyle{\rho}$

∎

###  2.5. Universal fiber sequences associated to symmetric powers

In this subsection, we review and generalizes Lurie’s construction in his
proof of Illusie’s result [SAG, Proposition 25.2.4.1] to obtain universal
fiber sequences associated to symmetric powers.

######  Notation 2.18.

Let $\mathcal{E}$ denote the $\infty$-category
$\mathrm{Fun}(\Delta^{1},\operatorname{SCRMod}^{\rm
cn})\times_{\mathrm{Fun}(\Delta^{1},\operatorname{CAlg}^{\Delta})}\operatorname{CAlg}^{\Delta}$
whose objects are pairs $(A,\rho\colon M^{\prime}\to M)$, where $A$ is a
simplicial commutative ring and $\rho$ is a morphism of connective
$A$-modules. Let $\mathcal{D}$ denote the full subcategory of the
$\infty$-category
$\mathrm{Fun}(\Delta^{1}\times\Delta^{1},\operatorname{SCRMod}^{\rm
cn})\times_{\mathrm{Fun}(\Delta^{1}\times\Delta^{1},\operatorname{CAlg}^{\Delta})}\operatorname{CAlg}^{\Delta}$
spanned by objects $(A,\alpha)$, where $A$ is a simplicial commutative ring,
and $\alpha$ is a fiber sequence of connective $A$-modules:

${M^{\prime}}$${M}$${0}$${M^{\prime\prime}.}$$\scriptstyle{\rho}$$\scriptstyle{\varphi}$

As usual, we will abbreviate the above fiber sequence as $\alpha\colon
M^{\prime}\to M\to M^{\prime\prime}$. Let
$\theta\colon\mathcal{D}\to\mathcal{E}$ denote the forgetful functor which
carries $(A,\alpha\colon
M^{\prime}\xrightarrow{\rho}M\xrightarrow{\varphi}M^{\prime\prime})$ to
$(A,\rho\colon M^{\prime}\to M)$. Then by virtue of [HA, Remark 1.1.1.7], that
is, by applying [HTT, Proposition 4.3.2.15] twice, we deduce that $\theta$ is
a trivial Kan fibration.

Let $\mathcal{E}_{0}\subseteq\mathcal{E}$ be the full subcategory spanned by
those pairs $(R,\rho\colon M^{\prime}\to M)$, where $R$ is a polynomial ring
$\mathbb{Z}[x_{1},\ldots,x_{k}]$ and $\rho$ fits into a short exact sequence
$0\to M^{\prime}\xrightarrow{\rho}M\to M^{\prime\prime}\to 0$ of finitely
generated free $R$-modules. Similarly, let
$\mathcal{D}_{0}\subseteq\mathcal{D}$ be the full subcategory spanned by those
pairs $(R,\alpha)$, where $R$ is a polynomial ring
$\mathbb{Z}[x_{1},\ldots,x_{k}]$, and $\alpha$ is a short exact sequence $0\to
M^{\prime}\xrightarrow{\rho}M\xrightarrow{\rho^{\prime}}M^{\prime\prime}\to 0$
of finitely generated free $R$-modules. Then the argument of the proof of
[SAG, Proposition 25.2.4.1] shows that the inclusions
$\mathcal{E}_{0}\hookrightarrow\mathcal{E}$ and
$\mathcal{D}_{0}\hookrightarrow\mathcal{D}$ induce equivalences of
$\infty$-categories $\mathcal{P}_{\Sigma}(\mathcal{E}_{0})\simeq\mathcal{E}$
and $\mathcal{P}_{\Sigma}(\mathcal{D}_{0})\simeq\mathcal{D}$, respectively.

######  Construction 2.19 (Koszul complexes).

Let $R$ be a discrete commutative ring, and suppose we are given a morphism
$\rho\colon M^{\prime}\to M$ of finite generated locally free $R$-modules. Let
$S_{R}^{*}M$ denote the ordinary symmetric algebra generated by $M$ over $R$.
Then there is canonically a Koszul complex of locally free
$S_{R}^{*}(M)$-modules of finite rank:

(2.1)
$\displaystyle\cdots\to(\bigwedge\nolimits_{R}^{2}M^{\prime})\otimes_{R}(S_{R}^{*}M)\to(\bigwedge\nolimits_{R}^{1}M^{\prime})\otimes_{R}(S_{R}^{*}M)\to(\bigwedge\nolimits_{R}^{0}M^{\prime})\otimes_{R}(S_{R}^{*}M)$

associated to the cosection $M^{\prime}\otimes_{R}S_{R}^{*}(M)\to
S_{R}^{*}(M)$ induced by $\rho$. Restricting to homogeneous elements of degree
$n$, we obtain a complex of locally free $R$-modules of finite rank:

(2.2) $\displaystyle\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to
M)\colon\quad\bigwedge\nolimits^{n}_{R}M^{\prime}\xrightarrow{d_{n}}\cdots\xrightarrow{d_{2}}\bigwedge\nolimits_{R}^{1}M^{\prime}\otimes_{R}(S_{R}^{n-1}M)\xrightarrow{d_{1}}S_{R}^{n}M.$

The construction $(R,\rho)\mapsto(R,\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to
M))$ determines a functor from the ordinary category of pairs $(R,\rho\colon
M^{\prime}\to M)$, where $R$ is a (discrete) commutative ring and $\rho$ is a
morphism of finite generated locally free $R$-modules, to the ordinary
category category of pairs $(R,M_{*})$, where $R$ is a (discrete) commutative
ring and $M_{*}$ is a non-negatively graded chain complexes of finitely
generated locally free $R$-modules. By construction, the formation of
$\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ commutes with base change $R\to
R^{\prime}$ of discrete commutative rings. For any integer $n^{\prime}\geq 0$,
from the $S_{R}^{*}(M)$-module structure of (2.1), multiplication with
elements of $S_{R}^{n^{\prime}}M$ induces a morphism of non-negatively graded
chain complexes of finitely generated locally free $R$-modules:

$\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to
M)\otimes_{R}(S_{R}^{n^{\prime}}M)\to\mathbf{S}^{n+n^{\prime}}(R,\rho\colon
M^{\prime}\to M).$

######  Remark 2.20.

By choosing local basis for the locally free modules $M^{\prime}$ and $M$,
respectively, the differentials $d_{i}$ of the complex
$\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ of (2.2) can be described
explicit as follows: for all local elements $x_{1},\ldots,x_{i}\in M^{\prime}$
and $y\in S_{R}^{d-i}M$, we have

$\displaystyle d_{i}(x_{1}\wedge\cdots\wedge x_{i}\otimes
y)=\sum_{j=1}^{i}(-1)^{j-1}(x_{1}\wedge\cdots\widehat{x_{j}}\cdots\wedge
x_{i})\otimes(\rho(x_{j})\cdot
y)\in\bigwedge\nolimits_{R}^{i-1}M^{\prime}\otimes_{R}S_{R}^{d-i+1}M.$

######  Variant 2.21.

There is an analogous construction for exterior algebra. Let $R$ be a discrete
commutative ring, and suppose we are given a morphism $\rho\colon
M^{\prime}\to M$ of finite generated locally free $R$-modules. For any $n\geq
0$, there is a Koszul complex of exterior type

(2.3) $\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to
M)\colon\quad\Gamma_{R}^{n}(M^{\prime})\xrightarrow{d_{n}^{\prime}}\cdots\xrightarrow{d_{2}^{\prime}}\Gamma_{R}^{1}(M^{\prime})\otimes_{R}\bigwedge\nolimits_{R}^{n-1}(M)\xrightarrow{d_{1}^{\prime}}\bigwedge\nolimits_{R}^{n}M$

which could be defined by the following formula:

$\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to
M)=\Sigma^{n}\,(\mathbf{S}^{n}(R,\rho^{\vee}\colon M^{\vee}\to
M^{\prime\vee}))^{\vee}.$

(Here
$(\underline{\hphantom{A}})^{\vee}=\operatorname{Hom}_{R}(\underline{\hphantom{A}},R)=\operatorname{Ext}^{0}_{R}(\underline{\hphantom{A}},R)$
denotes the operation of taking duals over $R$ in the classical sense.)
Concretely, the differentials $d_{j}^{\prime}$ of the complex (2.3) are
described by the following formula: for all local elements
$g\in\bigwedge^{d-j}M$ and
$\varepsilon_{1}^{(\nu_{1})}\cdots\varepsilon_{m^{\prime}}^{(\nu_{m^{\prime}})}\in\Gamma^{j}(M^{\prime})$
(where $\varepsilon_{1},\ldots,\varepsilon_{m^{\prime}}$ is a local basis of
$M^{\prime}$, $\nu_{i}\geq 0$ and $\sum\nu_{i}=j$), we have

$d_{j}^{\prime}(g\otimes\varepsilon_{1}^{(\nu_{1})}\cdots\varepsilon_{m^{\prime}}^{(\nu_{m^{\prime}})})=\sum_{i=1}^{m^{\prime}}(\rho(\varepsilon_{i})\wedge
g)\otimes\varepsilon_{1}^{(\nu_{1})}\cdots\varepsilon_{i}^{(\nu_{i}-1)}\cdots\varepsilon_{m^{\prime}}^{(\nu_{m^{\prime}})}\in\bigwedge\nolimits_{R}^{d-j+1}M\otimes_{R}\Gamma_{R}^{j-1}M^{\prime}.$

The formation of $\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to M)$ enjoys
similar functorial properties as the formation of $\mathbf{S}^{n}(R,\rho\colon
M^{\prime}\to M)$, and all the constructions and results of this subsection
have direct analogue for the exterior powers. The complex
$\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to M)$ will not play an
essential role in this paper; However, it is useful for computations in
various situations.

######  Remark 2.22.

The totality $\mathbf{S}^{*}(R,\rho\colon M^{\prime}\to M)=\bigoplus_{n\geq
0}\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ has a canonical graded
$R$-Hopf algebra structure and a bi-graded dga structure; The totality
$\mathbf{\Lambda}^{*}(R,\rho\colon M^{\prime}\to M)=\bigoplus_{n\geq
0}\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to M)$ has a canonical bigraded
$R$-Hopf algebra structure and a bi-graded dga structure. We will not use
these algebra structures in this paper; but they will play a role in the
theory of derived Schur functors in our forthcoming paper.

######  Construction 2.23.

Following Lurie [SAG, proof of Proposition 25.2.4.1], we let $R$ be a discrete
commutative ring, and consider a short exact sequence of finite generated free
$R$-modules $0\to M^{\prime}\xrightarrow{\rho}M\to M^{\prime\prime}\to 0.$
Then the Koszul complex of (2.1) is an exact sequence of finite generated free
$S_{R}^{*}(M)$-modules that resolves $S_{R}^{*}M^{\prime\prime}$. Restricting
to homogeneous elements of degree $n$, we obtain that the complex
$\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ of (2.2) is a resolution of the
module $S_{R}^{n}M^{\prime\prime}$. Let $f_{i,n}(R,\rho)$ denote the image of
the differential $d_{i}$ of $\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$,
and let $\alpha_{i,n}(R,\rho)$ denote the short exact sequence:

$0\to
f_{i+1,n}(R,\rho)\xrightarrow{a_{i}}(\bigwedge\nolimits^{i}_{R}M^{\prime})\otimes_{R}(S_{R}^{n-i}M)\xrightarrow{b_{i}}f_{i,n}(R,\rho)\to
0.$

Then $a_{i-1}\circ b_{i}=d_{i}$, and we have canonical equivalences:

$f_{0,n}(R,\rho\colon M^{\prime}\to M)\simeq S_{R}^{n}M^{\prime\prime},\qquad
f_{n,n}(R,\rho\colon M^{\prime}\to
M)\simeq\bigwedge\nolimits_{R}^{n}M^{\prime}.$

For any integer $n^{\prime}\geq 0$, multiplication with elements of
$S_{R}^{n^{\prime}}M$ induces functors

$\alpha_{i,n}(R,\rho)\otimes_{R}S_{R}^{n^{\prime}}M\to\alpha_{i,n+n^{\prime}}(R,\rho)$

for which the induced map on middle terms coincides with the map induced by
the multiplication map $S_{R}^{n-i}M\otimes_{R}S_{R}^{n^{\prime}}M\to
S_{R}^{n+n^{\prime}-i}M$. Since
$\mathcal{P}_{\sigma}(\mathcal{E}_{0})\simeq\mathcal{E}$, and $\mathcal{D}$
admits sifted colimits, by virtue of Proposition 2.7, for each integer $0\leq
i\leq n-1$, the functors $\mathcal{E}_{0}\to\mathcal{E}$ and
$\mathcal{E}_{0}\to\mathcal{D}$,

$(R,\rho\colon M^{\prime}\to
M)\mapsto(R,d_{i+1}\text{~{}of~{}}\mathbf{S}^{n}(R,\rho))\qquad(R,\rho\colon
M^{\prime}\to M)\mapsto(R,\alpha_{i,n}(R,\rho))$

have essentially unique extensions $D_{i+1,n}\colon\mathcal{E}\to\mathcal{E}$
and $U_{i,n}\colon\mathcal{E}\to\mathcal{D}$,

$(A,\rho\colon M^{\prime}\to M)\mapsto(A,d_{i+1,n}(A,\rho))\qquad(A,\rho\colon
M^{\prime}\to M)\mapsto(A,\alpha_{i,n}(A,\rho)),$

respectively; here $d_{i+1,n}(A,\rho)$ is a morphism of connective
$A$-modules:

$(\bigwedge\nolimits^{i+1}_{A}M^{\prime})\otimes_{R}(\operatorname{Sym}_{A}^{n-i-1}M)\to(\bigwedge\nolimits^{i}_{A}M^{\prime})\otimes_{R}(\operatorname{Sym}_{A}^{n-i}M),$

and $\alpha_{i,n}(A,\rho)$ is a fibre sequence of connective $A$-modules:

$F_{i+1,n}(A,\rho)\xrightarrow{a_{i,n}(A,\rho)}(\bigwedge\nolimits^{i}_{A}M^{\prime})\otimes_{A}(\operatorname{Sym}_{A}^{n-i}M)\xrightarrow{b_{i,n}(A,\rho)}F_{i,n}(A,\rho).$

Moreover, there are canonical equivalences

$d_{i,n}(A,\rho)\circ d_{i+1,n}(A,\rho)\simeq 0,\qquad a_{i,n}(A,\rho)\circ
b_{i+1,n}(A,\rho)\simeq d_{i+1,n}(A,\rho),$ $F_{0,n}(A,\rho\colon
M^{\prime}\to
M)\simeq\operatorname{Sym}_{A}^{n}(\operatorname{cofib}(\rho)),\qquad
F_{0,n}(A,\rho\colon M^{\prime}\to
M)\simeq\bigwedge\nolimits_{A}^{n}M^{\prime}.$

Furthermore, for any integer $n^{\prime}\geq 0$, there are canonical morphisms
of fiber sequences

$\alpha_{i,n}(A,\rho)\otimes_{A}\operatorname{Sym}_{A}^{n^{\prime}}M\to\alpha_{i,n+n^{\prime}}(A,\rho)$

for which the induced map on middle terms coincides with the map induced by
the multiplication map
$\operatorname{Sym}_{A}^{n-i}M\otimes_{A}\operatorname{Sym}_{A}^{n^{\prime}}M\to\operatorname{Sym}_{A}^{n+n^{\prime}-i}M$
of Remark 2.13.

######  Remark 2.24.

The functors $D_{i,n}\colon\mathcal{E}\to\mathcal{E}$,
$(A,\rho)\mapsto(A,d_{i,n}(A,\rho))$, where $1\leq i\leq n$, by construction
commute with sifted colimits, and the diagrams

${\mathcal{E}}$${\mathcal{E}}$${\operatorname{CAlg}^{\Delta}}$${\operatorname{CAlg}^{\Delta}}$$\scriptstyle{D_{i,n}}$$\scriptstyle{q}$$\scriptstyle{q}$$\scriptstyle{\operatorname{id}}$

commute up to canonical equivalences (here the vertical arrow $q$ is the
natural forgetful map the carries $(A,\rho)$ to $A$). Furthermore, [SAG,
Proposition 25.2.3.1] implies that the functor
$D_{i,n}\colon\mathcal{E}\to\mathcal{E}$ carries $q$-coCartesian edges to
$q$-coCartesian edges.

######  Remark 2.25 (Base change).

The functors $U_{i,n}\colon\mathcal{E}\to\mathcal{D}$,
$(A,\rho)\mapsto(A,\alpha_{i,n}(A,\rho))$, where $0\leq i\leq n-1$, by
construction commute with sifted colimits, and the diagrams

${\mathcal{E}}$${\mathcal{D}}$${\operatorname{CAlg}^{\Delta}}$${\operatorname{CAlg}^{\Delta}}$$\scriptstyle{U_{i,n}}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\operatorname{id}}$

commute up to canonical equivalences (here the vertical arrows $q$ and
$q^{\prime}$ are the natural forgetful maps). We claim that each functor
$U_{i,n}\colon\mathcal{E}\to\mathcal{D}$ carries $q$-coCartesian edges to
$q^{\prime}$-coCartesian edges. That is, for every morphism $\phi\colon A\to
B$ of simplicial commutative rings and every morphism $\rho\colon
M^{\prime}\to M$ of connective $A$-modules, the canonical map

$B\otimes_{A}\alpha_{i,n}(A,\rho)\to\alpha_{i,n}(B,B\otimes_{A}\rho)$

is an equivalence in
$\mathcal{D}\times_{\operatorname{CAlg}^{\Delta}}\\{B\\}$. In fact, for a
given $n\geq 0$, we prove by induction on $0\leq i\leq n-1$. In the case
$i=0$, the equivalence
$B\otimes_{A}\alpha_{0}(A,\rho)\simeq\alpha_{0}(B,B\otimes_{A}\rho)$ follows
from the fact that the functor
$\operatorname{Mod}_{A}\to\operatorname{Mod}_{B}$, $M\mapsto B\otimes_{A}M$ is
exact, and the formations of $\operatorname{Sym}_{A}^{n}M$ and
$\operatorname{Sym}_{A}^{n}(\operatorname{cofib}(\rho))$ commute with base
change (see [SAG, Proposition 25.2.3.1]). For the induction step, the case of
$(i+1)$ follows from the case $i$ by virtue of loc. cit. and the fact that the
final vertex of $\alpha_{i+1}(A,\rho)$ coincides with the initial vertex of
$\alpha_{i,n}(A,\rho)$.

######  Construction 2.26.

Let $R$ be a discrete commutative ring, and suppose we are given a short exact
sequence $0\to M^{\prime}\xrightarrow{\rho}M\to M^{\prime\prime}\to 0$ of
finite generated free $R$-modules. For integers $n\geq 0$ and $0\leq i\leq n$,
we let $g_{i,n}(R,\rho)\in\operatorname{Mod}_{R}^{\mathrm{cn}}$ denote the
class of the “brutal” truncation

$(\bigwedge\nolimits^{i}_{R}M^{\prime})\otimes_{R}(S_{R}^{n-i}M)\xrightarrow{d_{i}}\cdots\to
M^{\prime}\otimes_{R}S_{R}^{n-1}M\xrightarrow{d_{1}}S_{R}^{n}M$

of the complex (2.2). Then there are canonical equivalences

$g_{0,n}(R,\rho)=S_{R}^{n}M,\qquad g_{0,n}(R,\rho)\simeq
S_{R}^{n}M^{\prime\prime}.$

Since the $R$-modules

$\operatorname{Ext}^{s}_{R}\left((\bigwedge\nolimits_{R}^{i}M^{\prime})\otimes_{R}(S_{R}^{n-i}M),(\bigwedge\nolimits_{R}^{j}M^{\prime})\otimes_{R}(S_{R}^{n-j}M)\right)$

vanish for all $s\neq 0$ and $i,j\in\mathbb{Z}$ (here $M,M^{\prime}$ are
finite generated free $R$-modules), the exact sequence (2.2) determines
essentially unique fiber sequences

$\beta_{i,n}(R,\rho)\colon\quad\Sigma^{i}(\bigwedge\nolimits_{R}^{i+1}M^{\prime})\otimes_{R}(S_{R}^{n-i-1}M)\to
g_{i,n}(R,\rho)\to g_{i+1}(R,\rho).$

For any integer $n^{\prime}\geq 0$, multiplication with elements of
$S_{R}^{n^{\prime}}M$ induces functors

$\beta_{i,n}(R,\rho)\otimes_{R}S_{R}^{n^{\prime}}M\to\beta_{i,n+n^{\prime}}(R,\rho)$

for which the induced map on middle terms coincides with the map induced by
the multiplication map $S_{R}^{n-i-1}M\otimes_{R}S_{R}^{n^{\prime}}M\to
S_{R}^{n+n^{\prime}-i-1}M$. As in Construction 2.23, by virtue of Proposition
2.7, the functors $\mathcal{E}_{0}\to\mathcal{D}$,

$(R,\rho\colon M^{\prime}\to M)\mapsto(R,\beta_{i,n}(R,\rho))$

extend essentially uniquely to functors
$V_{i,n}\colon\mathcal{E}\to\mathcal{D}$,

$(A,\rho\colon M^{\prime}\to M)\mapsto(A,\beta_{i,n}(A,\rho)),$

where $\beta_{i,n}(A,\rho)$ is a fibre sequence of connective $A$-modules:

$\Sigma^{i}(\bigwedge\nolimits_{A}^{i+1}M^{\prime})\otimes_{A}(\operatorname{Sym}_{A}^{n-i-1}M)\xrightarrow{c_{i,n}(A,\rho)}G_{i,n}(A,\rho)\to
G_{i+1,n}(A,\rho).$

Let $e_{i,n}(A,\rho)$ denote the morphism
$G_{i+1,n}(A,\rho)\to\Sigma^{i+1}(\bigwedge\nolimits_{A}^{i+1}M^{\prime})\otimes_{R}(\operatorname{Sym}^{n-i-1}M)$
obtained from $\beta_{i,n}(A,\rho)$. Then the composition of the natural maps

$\Sigma^{i}(\bigwedge\nolimits_{A}^{i+1}M^{\prime})\otimes_{A}(\operatorname{Sym}_{A}^{n-i-1}M)\xrightarrow{c_{i,n}(A,\rho)}G_{i,n}(A,\rho)\xrightarrow{e_{i-1,n}(A,\rho)}\Sigma^{i}(\bigwedge\nolimits_{A}^{i}M^{\prime})\otimes_{A}(\operatorname{Sym}_{A}^{n-i}M)$

is canonically equivalent to $\Sigma^{i}\circ d_{i+1,n}(A,\rho)$, and there
are canonical equivalences:

$G_{0,n}(A,\rho\colon M^{\prime}\to M)\simeq\operatorname{Sym}_{A}^{n}M,\qquad
G_{n,n}(A,\rho\colon M^{\prime}\to
M)\simeq\operatorname{Sym}_{A}^{n}(\operatorname{cofib}(\rho)).$

For any integer $n^{\prime}\geq 0$, there are canonical morphisms of functors

$\beta_{i,n}(A,\rho)\otimes_{A}\operatorname{Sym}_{A}^{n^{\prime}}M\to\beta_{i,n+n^{\prime}}(A,\rho)$

for which the induced map on the initial vertices coincides with the map
induced by the multiplication
$\operatorname{Sym}_{A}^{n-i-1}M\otimes_{A}\operatorname{Sym}_{A}^{n^{\prime}}M\to\operatorname{Sym}_{A}^{n+n^{\prime}-i-1}M$
of Remark 2.13.

######  Remark 2.27 (Base change).

For all $0\leq i\leq n-1$, by construction the functors
$V_{i,n}\colon\mathcal{E}\to\mathcal{D}$, $(A,\rho)\mapsto(A,\beta(A,\rho))$
commute with sifted colimits, and the following diagrams

${\mathcal{E}}$${\mathcal{D}}$${\operatorname{CAlg}^{\Delta}}$${\operatorname{CAlg}^{\Delta}}$$\scriptstyle{V_{i,n}}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\operatorname{id}}$

commute up to canonical equivalences (where the vertical arrows $q$ and
$q^{\prime}$ are the natural forgetful maps). By a similar argument as in
Remark 2.25, we deduce that each functor
$V_{i,n}\colon\mathcal{E}\to\mathcal{D}$ carries $q$-coCartesian edges to
$q^{\prime}$-coCartesian edges. In other words, for every morphism $\phi\colon
A\to B$ of simplicial commutative rings and every morphism $\rho\colon
M^{\prime}\to M$ of connective $A$-modules, the canonical map

$B\otimes_{A}\beta_{i,n}(A,\rho)\to\beta_{i,n}(B,B\otimes_{A}\rho)$

is an equivalence in
$\mathcal{D}\times_{\operatorname{CAlg}^{\Delta}}\\{B\\}$.

######  Lemma 2.28.

Let $\mathcal{C}$ be a stable $\infty$-category (or a prestable
$\infty$-category in the sense of [SAG, Deﬁnition C.1.2.1]), let $n\geq 1$ be
an integer, and suppose we are given a composable sequence of morphisms in
$\mathcal{C}$:

$C_{n}\xrightarrow{d_{n}}C_{n-1}\xrightarrow{d_{n-1}}\cdots\to
C_{1}\xrightarrow{d_{1}}C_{0}$

for which the successive compositions satisfy $d_{i}\circ d_{i+1}\simeq 0$,
$1\leq i\leq n-1$. Assume that

$\operatorname{Ext}^{-k}_{\mathcal{C}}(C_{i+j},C_{i}):=\pi_{k}(\mathrm{Map}_{\mathcal{C}}(C_{i+j},C_{i}))\simeq
0\quad\text{for}\quad i\geq 0,j\geq 1,k\geq 1.$

Then there are essentially unique objects $\\{D_{i}\in\mathcal{C}\\}_{0\leq
i\leq n}$ that fit into essentially unique cofiber sequences
$\beta_{i}\colon\Sigma^{i}C_{i+1}\xrightarrow{f_{i}}D_{i}\xrightarrow{g_{i}}D_{i+1}$
such that $D_{0}=C_{0}$, $f_{0}=d_{1}$, and $h_{i-1}\circ
f_{i}\simeq\Sigma^{i}\,d_{i+1}$, where $h_{i}\colon
D_{i+1}\to\Sigma^{i+1}C_{i+1}$ denotes the connecting morphism determined by
$\beta_{i}$.

###### Proof.

This result is well-known in the context of triangulated categories (see, for
example, [O97, Lemma 1.5]), and the same proof strategy works here. We will
illustrate with the case $n=2$; the general situation follows from a similar
argument and induction. Let $D_{1}$ be the cofiber of $d_{1}\colon C_{1}\to
C_{0}$, let $\beta_{0}\colon
C_{1}\xrightarrow{f_{0}=d_{1}}C_{0}\xrightarrow{g_{0}}D_{1}$ denote the
corresponding cofiber sequence, and let $h_{0}\colon D_{1}\to\Sigma C_{1}$
denote the essentially unique connecting morphism determined by $\beta_{0}$.
From the fiber sequences of spaces

$\mathrm{Map}_{\mathcal{C}}(\Sigma
C_{2},D_{1})\xrightarrow{h_{0}\circ}\mathrm{Map}_{\mathcal{C}}(\Sigma
C_{2},\Sigma C_{1})\xrightarrow{\Sigma
d_{1}\circ}\mathrm{Map}_{\mathcal{C}}(\Sigma C_{2},\Sigma C_{0}),$

there exists an element $f_{1}\in\mathrm{Map}_{\mathcal{C}}(\Sigma
C_{2},D_{1})$ lifting the element $\Sigma
d_{2}\in\mathrm{Map}_{\mathcal{C}}(\Sigma C_{2},\Sigma C_{1})$. The space
$\Omega\mathrm{Map}(\Sigma C_{2},C_{0})$ of such liftings is contractible by
our assumption and Whitehead’s theorem ([HTT, p.16]). Therefore, $f_{1}$ is
determined up to contractible choices. Let $D_{2}$ be the cofiber of $f_{1}$,
and $\beta_{1}$ the corresponding cofiber sequence. Then $D_{2}$ (resp.
$\beta_{1}$) exists and is determined up to contractible choices. This proves
the desired assertion. ∎

######  Remark 2.29.

We see from the above proof Lemma 2.28 that the conditions of the lemma are
much stronger than necessary (for example, only the conditions
$\operatorname{Ext}_{\mathcal{C}}^{-k}(C_{2},C_{0})\simeq 0$, $k\geq 1$, were
used in the proof of the case $n=2$). However, the above version is enough for
the purpose of applications in this paper; in a forthcoming paper, we plan to
include a more systematical study of these systems and stronger versions of
Lemma 2.28.

######  Corollary 2.30 (Illusie).

Let $R$ be a discrete commutative ring, and let $\rho\colon M^{\prime}\to M$
be a morphism of finitely generated locally free $R$-modules. Let $n\geq 0$ be
an integer, and let $\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ denote the
Koszul complex (2.2). Then there is a canonical equivalence

$[\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to
M)]\simeq\operatorname{Sym}_{R}^{n}(\operatorname{cofib}(M^{\prime}\xrightarrow{\rho}M))\in\operatorname{Mod}_{R}^{\mathrm{cn}}$

where $[\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)]$ denotes the image of
the complex $\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)$ under the canonical
equivalence $\mathcal{D}_{\geq
0}(\operatorname{Ch}(\operatorname{Mod}_{R}^{\heartsuit}))\simeq\operatorname{Mod}_{R}^{\mathrm{cn}}$
of [HA, Proposition 7.1.1.15].

###### Proof.

By virtue of Lemma 2.28, we have a canonical equivalence
$[\mathbf{S}^{n}(R,\rho\colon M^{\prime}\to M)]\simeq G_{n,n}(R,\rho\colon
M^{\prime}\to M)$ (where $G_{i,n}(R,\rho)$ is defined as in Construction
2.26). Then the desired assertion follows from the properties of the fiber
sequences $\beta_{i}(A,\rho)$ of Construction 2.26. ∎

######  Variant 2.31 (Illusie).

In the same situation of as Corollary 2.30, let
$\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to M)$ denote the Koszul complex
of exterior type (2.3). Then there is a canonical equivalence

$[\mathbf{\Lambda}^{n}(R,\rho\colon M^{\prime}\to
M)]\simeq\bigwedge\nolimits_{R}^{n}(\operatorname{cofib}(M^{\prime}\xrightarrow{\rho}M))\in\operatorname{Mod}_{R}^{\mathrm{cn}}.$

###### Proof.

The proof is completely analogous to the proof of Corollary 2.30. ∎

###  2.6. Filtrations of symmetric powers

In this subsection, we review and slightly generalizes Lurie’s [SAG,
Construction 25.2.5.4]. Let $R$ be a discrete commutative ring, and suppose we
are given a short exact sequence of finite generated free $R$-modules:

$0\to M^{\prime}\xrightarrow{\varphi}M\to M^{\prime\prime}\to 0.$

For each non-negative integer $n$ and an integer $0\leq i\leq n$, there are
natural morphisms $(S_{R}^{n-i}M^{\prime})\otimes_{R}(S_{R}^{i}M)\to
S_{R}^{n}M$,
$(\bigwedge_{R}^{n-i}M^{\prime})\otimes_{R}(\bigwedge_{R}^{i}M)\to\bigwedge_{R}^{n}M$,
and
$(\Gamma_{R}^{i}M^{\prime})\otimes_{R}(\Gamma_{R}^{n-i}M)\to\Gamma_{R}^{n}M$
given by the respective multiplication maps; We let
$\overline{F}^{i}(R,\varphi)\subseteq S_{R}^{n}M$,
$\overline{G}^{i}(R,\varphi)\subseteq\bigwedge_{R}^{n}M$, and
$\overline{H}^{i}(R,\varphi)\subseteq\Gamma_{R}^{n}M$ denote the corresponding
images. Then we obtain finite filtrations

$\displaystyle
0=\overline{F}^{-1}(R,\varphi)\subseteq\overline{F}^{0}(R,\varphi)\subseteq\cdots\subseteq\overline{F}^{n}(R,\varphi)=S_{R}^{n}(M),$
$\displaystyle
0=\overline{G}^{-1}(R,\varphi)\subseteq\overline{G}^{0}(R,\varphi)\subseteq\cdots\subseteq\overline{G}^{n}(R,\varphi)=\bigwedge\nolimits_{R}^{n}(M),$
$\displaystyle
0=\overline{H}^{-1}(R,\varphi)\subseteq\overline{H}^{0}(R,\varphi)\subseteq\cdots\subseteq\overline{H}^{n}(R,\varphi)=\Gamma_{R}^{n}(M),$

respectively, such that the induced maps to the respective successive
quotients

$\displaystyle S_{R}^{n-i}M^{\prime}\otimes S_{R}^{i}M^{\prime\prime}$
$\displaystyle\to\overline{F}^{i}(R,\varphi)/\overline{F}^{i-1}(R,\varphi),$
$\displaystyle\bigwedge\nolimits_{R}^{n-i}M^{\prime}\otimes\bigwedge\nolimits_{R}^{i}M^{\prime\prime}$
$\displaystyle\to\overline{G}^{i}(R,\varphi)/\overline{G}^{i-1}(R,\varphi),$
$\displaystyle\Gamma_{R}^{n-i}M^{\prime}\otimes\Gamma_{R}^{i}M^{\prime\prime}$
$\displaystyle\to\overline{H}^{i}(R,\varphi)/\overline{H}^{i-1}(R,\varphi)$

are isomorphisms. In particular, for each $0\leq i\leq n$, we obtain canonical
fiber sequences

$\displaystyle\alpha^{i}(R,\varphi)\colon\qquad\overline{F}^{i-1}(R,\varphi)\to\overline{F}^{i}(R,\varphi)\to(S_{R}^{n-i}M^{\prime})\otimes_{R}(S_{R}^{i}M^{\prime\prime}),$
$\displaystyle\beta^{i}(R,\varphi)\colon\qquad\overline{G}^{i-1}(R,\varphi)\to\overline{G}^{i}(R,\varphi)\to(\bigwedge\nolimits_{R}^{n-i}M^{\prime})\otimes_{R}(\bigwedge\nolimits_{R}^{i}M^{\prime\prime}),$
$\displaystyle\gamma^{i}(R,\varphi)\colon\qquad\overline{H}^{i-1}(R,\varphi)\to\overline{H}^{i}(R,\varphi)\to(\Gamma_{R}^{n-i}M^{\prime})\otimes_{R}(\Gamma_{R}^{i}M^{\prime\prime}).$

Let $\mathcal{E}_{0},\mathcal{E},\mathcal{D}$ be the $\infty$-categories
defined in §2.5, Notation 2.18, then by virtue of Proposition 2.7, the above
functors extend essentially uniquely to functors

$\mathcal{E}\to\mathrm{Fun}(\Delta^{n},\operatorname{SCRMod}^{\rm
cn})\times_{\mathrm{Fun}(\Delta^{n},\operatorname{CAlg}^{\Delta})}\operatorname{CAlg}^{\Delta},$
$\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,F^{0}(A,\varphi)\xrightarrow{d_{1}}F^{1}(A,\varphi)\to\cdots\to
F^{n}(A,\varphi))$ $\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,G^{0}(A,\varphi)\xrightarrow{e_{1}}G^{1}(A,\varphi)\to\cdots\to
G^{n}(A,\varphi))$ $\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,H^{0}(A,\varphi)\xrightarrow{f_{1}}H^{1}(A,\varphi)\to\cdots\to
H^{n}(A,\varphi))$

and functors $\mathcal{E}\to\mathcal{D}$,

$\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,\alpha^{i}(A,\varphi)\colon
F^{i-1}(A,\varphi)\xrightarrow{d_{i}}F^{i}(A,\varphi)\to(\operatorname{Sym}_{A}^{n-i}M^{\prime})\otimes_{A}(\operatorname{Sym}_{A}^{i}M^{\prime\prime})),$
$\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,\beta^{i}(A,\varphi)\colon
G^{i-1}(A,\varphi)\xrightarrow{e_{i}}G^{i}(A,\varphi)\to(\bigwedge\nolimits_{A}^{n-i}M^{\prime})\otimes_{A}(\bigwedge\nolimits_{A}^{i}M^{\prime\prime})),$
$\displaystyle(A,\varphi\colon M^{\prime}\to
M)\mapsto(A,\gamma^{i}(A,\varphi)\colon
H^{i-1}(A,\varphi)\xrightarrow{f_{i}}H^{i}(A,\varphi)\to(\Gamma_{A}^{n-i}M^{\prime})\otimes_{A}(\Gamma_{A}^{i}M^{\prime\prime})),$

where $\alpha_{i,n}(A,\varphi)$, $\beta_{i,n}(A,\varphi)$, and
$\gamma_{i}(A,\varphi)$ are cofiber sequences, and $M^{\prime\prime}$ is the
cofiber of $\rho$. Moreover, we have canonical equivalences:

$\displaystyle F^{0}(A,\varphi)=\operatorname{Sym}^{n}_{A}M^{\prime}$
$\displaystyle F^{n}(A,\varphi)=\operatorname{Sym}^{n}_{A}M,$ $\displaystyle
G^{0}(A,\varphi)=\bigwedge\nolimits^{n}_{A}M^{\prime}$ $\displaystyle
G^{n}(A,\varphi)=\bigwedge\nolimits^{n}_{A}M,$ $\displaystyle
H^{0}(A,\varphi)=\Gamma^{n}_{A}M^{\prime}$ $\displaystyle
H^{n}(A,\varphi)=\Gamma^{n}_{A}M.$

######  Example 2.32.

Let $A$ be a simplicial commutative ring, and $\rho\colon M\to M^{\prime}$ a
morphism of connective $A$-modules such that $\operatorname{cofib}(A,\varphi)$
is a line bundle (that is, a locally free module of rank $1$). Then for each
$n\geq 1$, we obtain a fiber sequence:

$\beta^{1}(A,\rho)\colon\qquad\bigwedge\nolimits_{A}^{n}M^{\prime}\xrightarrow{\bigwedge^{n}\varphi}\bigwedge\nolimits_{A}^{n}M\to\left(\bigwedge\nolimits_{A}^{n-1}M^{\prime}\right)\otimes_{A}\operatorname{cofib}(A,\varphi).$

##  3\. Quasi-coherent sheaves and algebras on prestacks

In this section, we first review Lurie’s machinery in [SAG, §6.2.1], which
allows one to “globalize” constructions from affine cases to the cases of
general prestacks (§3.1). In §3.2, we review the theory of quasi-coherent
sheaves on prestacks. In §3.3, we apply the machinery of §3.1 to construct the
$\infty$-category of quasi-coherent algebras
$\operatorname{QCAlg}^{\Delta}(X)$ on a prestack $X$, study its basic
properties and define relative spectra. Finally, in §3.4 and §3.5, we apply
the machinery of §3.1 to globalize the constructions and results of §2 on
symmetric, exterior, and divided powers, and symmetric algebras, respectively,
to the case of prestacks.

###  3.1. $\infty$-categories of sheaves of objects on prestacks

Let $q\colon\mathcal{C}\to\operatorname{CAlg}^{\Delta}$ be a coCartesian
fibration. In this subsection, we will apply the machinery developed by Lurie
in [SAG, §6.2.1] to construct an $\infty$-category
$\underline{\mathcal{C}}(X)$ of “sheaf of objects of $\mathcal{C}$ on a
prestack $X$”. Recall that a prestack $X$ is functor from the
$\infty$-category of simplicial commutative rings
$\operatorname{CAlg}^{\Delta}$ to the $\infty$-category of spaces
$\mathcal{S}$. By virtue of [HTT, Proposition 3.3.2.5], a prestack $X$ is
classified by a left fibration which we denote by
$\int_{\operatorname{CAlg}^{\Delta}}X\to\operatorname{CAlg}^{\Delta}$. By
virtue of [SAG, Construction 6.2.1.7], the coCartesian fibration
$q\colon\mathcal{C}\to\operatorname{CAlg}^{\Delta}$ determines a functor

$\Phi(q)\colon\mathrm{Fun}(\operatorname{CAlg}^{\Delta},\mathcal{S})^{\mathrm{op}}\to\widehat{\mathcal{C}\mathrm{at}}_{\infty}$

which carries a prestack
$X\in\mathrm{Fun}(\operatorname{CAlg}^{\Delta},\mathcal{S})$ to the
$\infty$-category

$\underline{\mathcal{C}}(X)=\mathrm{Fun}_{/\operatorname{CAlg}^{\Delta}}^{\rm
CCart}(\int_{\operatorname{CAlg}^{\Delta}}X,\mathcal{C}),$

where $\mathrm{Fun}_{/\mathcal{E}}^{\rm
CCart}(\int_{\operatorname{CAlg}^{\Delta}}X,\mathcal{C})$ denotes the
$\infty$-category of functors from $\int_{\operatorname{CAlg}^{\Delta}}X$ to
$\mathcal{C}$ which commute with their projections to
$\operatorname{CAlg}^{\Delta}$ and carry each edge of
$\int_{\operatorname{CAlg}^{\Delta}}X$ to a coCartesian edge of
$q\colon\mathcal{C}\to\operatorname{CAlg}^{\Delta}$ (see [SAG, Definition
6.2.1.1]).

######  Remark 3.1.

By virtue of [SAG, Remark 6.2.1.8], more informally,
$\underline{\mathcal{C}}(X)$ can be described as the limit
$\varprojlim_{A,\eta\in X(A)}\mathcal{C}_{A}$, where $\mathcal{C}_{A}$ is the
fiber of $q$ over $A\in\operatorname{CAlg}^{\Delta}$. In other words, we can
think of an object of $C\in\underline{\mathcal{C}}(X)$ as a functor which