2.1 Smooth deformations

Recall that a deformation of an algebra R is a pair of a commutative augmented algebra Z and an algebra map $Q\to R$ from a flat Z-algebra Q which reduces to an isomorphism $k\otimes _ZQ\cong R$ . Note that, as part of this definition, we require that the structure map $Z\to Q$ has central image.

By a (formally) smooth deformation of an algebra R, we mean a deformation $Q\to R$ parameterized by a (formally) smooth algebra Z over our base field k. In the formal setting, we require that Z is complete with finite-dimensional tangent space at its distinguished point. This implies an isomorphism of local algebras $Z\cong k[\!\hspace {.1mm}[{y_1,\dots ,y_n}]\!\hspace {.1mm}]$ (see [Reference Negron and Pevtsova28, §3] for a more detailed account). Throughout we let $m_Z\subset Z$ denote the maximal ideal corresponding to the augmentation $1:Z\to k$ .

Note that a deformation parameterized by a smooth, that is, finite type, algebra Z can be completed to produce a formally smooth analog. This is technically convenient, because working with local rings is convenient.

2.2 The specific setup

Throughout this work, we consider specifically a deformation $Q\to R$ of a finite-dimensional algebra R parameterized by a formally smooth algebra Z, which is isomorphic to a power series algebra in finitely many variables. We assume additionally that Q is Noetherian and that the quotient $\Lambda =R/\operatorname {Jac}(R)$ is separable over k, so that all base changes $\Lambda _K$ along arbitrary field extensions $k\to K$ are also semisimple. One should always consider the deformation Q to be Gorenstein—although this assumption is not strictly necessary for many results.

We make a further assumption that the deformation $Q\to R$ admits an integral form, by which we mean that Q is obtained from a finite-type deformation of R via completion. The relevance of this assumption does not appear until §5, at which point we explain the notion in more detail.

2.3 New deformations from old

Given a deformation $Q\to R$ as in §2.2, and $f\in m_Z$ with nonzero reduction $\bar {f}\in m_Z/m_Z^2$ , we can produce another deformation $Q/(f)\to R$ parameterized by the algebra $Z/(f)$ . The algebra $Z/(f)$ will still be a power series ring, so that $Q/(f)\to R$ is still of the type demanded in §2.2.

In the early sections of this text, when we consider a deformation $Q\to R$ , we are considering either a deformation $Q\to R$ with Q Noetherian and of finite global dimension, or some associated hypersurface $Q/(f)\to R$ thereof. So we have these two flavors of deformation to consider: one of finite global dimension and one which is (generally) singular.

At the point at which we begin to discuss hypersurface support specifically, we fix Q to be of finite global dimension, and speak explicitly of the associated hypersurface deformations $Q/(f)\to R$ .

2.4 Smooth deformations and actions on categories

Given a (formally) smooth deformation $Q\to R$ , parameterized by a given (formally) smooth commutative algebra Z, we consider $A_Z$ the dg algebra

$$ \begin{align*} A_Z:=\operatorname{\mathrm{Sym}}\big(\Sigma^{-2}(m_Z/m_Z^2)^\ast\big), \end{align*} $$

with vanishing differential. Bezrukavnikov and Ginzburg [Reference Bezrukavnikov and Ginzburg12] show that the deformation Q specifies an action of the algebra $A_Z$ on the derived category of R-modules

(1) $$ \begin{align} \iota_Q:A_Z\to Z\left(D(R)\right). \end{align} $$

This action lifts to an action on a certain dg category $D_{coh}(R)$ [Reference Negron and Pevtsova28, §3.4], and manifests concretely as a collection of compatible algebra maps

$$ \begin{align*} \iota_Q^M:A_Z\to \operatorname{\mathrm{REnd}}_R(M), \end{align*} $$

at arbitrary M, where the image of $A_Z$ lands in the (graded) center of $\operatorname {\mathrm {REnd}}_R(M)$ .

The algebra $A_Z$ , along with its action on $D(R)$ , can be seen as a generalization of the algebra of cohomological operators associated with a local complete intersection [Reference Avramov and Buchweitz1], [Reference Eisenbud17], [Reference Gulliksen21].

2.5 Hypersurface support: a preliminary report

Consider $Q\to R$ a deformation as in §2.2, with Q additionally of finite global dimension. For a closed point $c\in \mathbb {P}(m_Z/m_Z^2)$ , we consider an associated hypersurface algebra $Q/(f)$ , with $f\in m_Z$ such that $\bar {f}\in m_Z/m_Z^2$ is a (nonzero) representative for c. We speak of $f\in m_Z$ , specifically, as our representative for c. When the choice of representative is irrelevant, we write $Q_c$ for an arbitrary expression $Q_c=Q/(f)$ . We similarly define hypersurface algebras $Q_c$ at nonclosed points in $\mathbb {P}(m_Z/m_Z^2)$ by employing base change (see §5).

We would like to define the hypersurface support of R, relative to this deformation Q, as follows.

Proto-definition 2.2. For M an arbitrary R-module, we define the hypersurface support of M as

$$ \begin{align*} \operatorname{\mathrm{supp}}^{hyp}_{\mathbb{P}}(M)=\{c\in \mathbb{P}(m_Z/m_Z^2): M_{k(c)}\ \text{is of infinite projective dimension over }Q_c\}. \end{align*} $$

Here, $M_{k(c)}$ is simply the base change $M\otimes k(c)$ at the residue field for c. We would then like to prove that this hypersurface support satisfies many desirable properties. In pursuing this line of thinking, there are two primary issues which one has to deal with:

1. (a) One would like to show that this definition carries no ambiguity, in the sense that the projective dimension of $M_{k(c)}$ over $Q_c$ is independent of the choice of representative f for c, and so independent of the specific choice of representing hypersurface algebra $Q_c=Q/(f)$ .

2. (b) One would like to understand that the projective dimension for M is encoded globally on $\mathbb {P}(m_Z/m_Z^2)$ .

For (b), we might mean, for example, that there is a sheaf $\mathcal {E}_M$ on $\mathbb {P}(m_Z/m_Z^2)$ whose fibers contain information about the projective dimension of M at various $Q_c$ . In Proposition 3.1 and Theorem 5.4 below, we deal with these fundamental issues (see also Proposition 6.7). Having addressed these points, we formally define the hypersurface support in §5, and address some of its basic properties in §6.

3.1 Application to Gorenstein deformations

As remarked previously, we generally consider Q to be a Gorenstein deformation of R. Recall that Q is called Gorenstein (resp. d-Gorenstein) if $\operatorname {injdim}_Q(Q)<\infty $ (resp. $\operatorname {injdim}_Q(Q)\leq d$ ). Specifically, Q should be of finite injective dimension over itself both on the left and the right. In this case, Proposition 3.1 proliferates as

3.2 Primes and associated primes

We recall that a prime ideal in Q is a (two-sided) proper ideal $\mathfrak {p}$ in Q such that the product of two ideals lies in $\mathfrak p$ , $IJ\subset \mathfrak p$ , if and only if I is contained in $\mathfrak p$ or J is contained in  $\mathfrak p$ . By considering principal ideals, we see that primeness is equivalent to the implication

$$ \begin{align*} aQb\subset \mathfrak p \ \Rightarrow \ a\in \mathfrak p\ \text{or}\ b\in \mathfrak p,\ \ \text{for any }a,b\in Q. \end{align*} $$

From this latter description, it is clear that prime ideals in commutative rings, in the above sense, are prime in the usual sense. This expression of primeness also implies

One also has the notion of a prime module over Q. This is a nonzero module N over Q such that any nonzero submodule $N'\subset N$ has $\operatorname {ann}(N')=\operatorname {ann}(N)$ (see, e.g., [Reference Goodearl and Warfield20]). One can see that the annihilator of a prime module is prime, and by considering the quotient $N=Q/\mathfrak p$ of Q by a prime ideal, we see also that all prime ideals appear as annihilators of prime modules.

By considering cyclic submodules of prime modules, and Noetherianity of Q, we have

3.3 Proof of Proposition 3.1

The following proof is adapted from [Reference Avramov and Foxby2, Prop. 5.5].

Proof of Proposition 3.1

We prove the statement for $\operatorname {\mathrm {Tor}}$ first. We have

$$ \begin{align*} \operatorname{flatdim}(M)=\operatorname{min}\{d:\operatorname{\mathrm{Tor}}_{>d}^Q(M',M)=0\ \forall\ \text{fin gen'd }M'\}, \end{align*} $$

or $\operatorname {flatdim}(M)=\infty $ if no such d exists [Reference Weibel35, Prop. 3.2.4]. By Corollary 3.6, we find further that

$$ \begin{align*} \operatorname{flatdim}(M)=\operatorname{min}\{d:\operatorname{\mathrm{Tor}}_{>d}^Q(N,M)=0\ \forall\ \text{cyclic prime modules }N\}, \end{align*} $$

or $\operatorname {flatdim}(M)=\infty $ if no such d exists.

Now, let us assume that $\operatorname {\mathrm {Tor}}_{>d}^Q(\Lambda ,M)=0$ as in the statement of Proposition 3.1, and that d is the minimal nonnegative integer with this property. We want to show that $\operatorname {flatdim}(M) =d$ . Consider now the possibly empty collection of prime ideals in Q,

$$ \begin{align*} \mathscr{P}=\{\operatorname{ann}(N):N\text{ cyclic prime with }\operatorname{\mathrm{Tor}}_{>d}^Q(N,M)\neq 0\}. \end{align*} $$

We want to show that this collection is empty. Suppose this is not the case, and choose $\mathfrak p$ a maximal element in $\mathscr {P}$ , which must exist by Noetherianity of Q. Let $N_{\mathfrak p}$ be its associated cyclic prime module. Note that we have a surjection $Q/\mathfrak p\to N_{\mathfrak p}$ , by cyclicity of $N_{\mathfrak p}$ . We claim that the maximal ideal $m_Z$ of Z must be contained in $\mathfrak p$ , in which case $Q/\mathfrak p$ and $N_{\mathfrak p}$ must be finite-dimensional.

If $m_Z\nsubseteq \mathfrak p$ , then chose $x\in m_Z-\mathfrak p$ . Applying Lemma 3.4, we observe an exact sequence of modules

(2) $$ \begin{align} 0\to N_{\mathfrak p}\overset{x}\to N_{\mathfrak p}\to N_{\mathfrak p}/xN_{\mathfrak p}\to 0. \end{align} $$

Note that $\mathfrak p\subsetneq xQ+\mathfrak p\subset \operatorname {ann}(N_{\mathfrak p}/xN_{\mathfrak p})$ . So now, by considering a filtration of $N_{\mathfrak p}/xN_{\mathfrak p}$ as in Corollary 3.6 and maximality of $\mathfrak p$ in $\mathscr {P}$ , we conclude that $\operatorname {\mathrm {Tor}}_{>d}^Q(N_{\mathfrak p}/xN_{\mathfrak p},M)=0$ . The long exact sequence on cohomology obtained from (2) therefore implies that the action map

$$ \begin{align*} x\cdot-:\operatorname{\mathrm{Tor}}_{>d}^Q(N_{\mathfrak p},M)\to \operatorname{\mathrm{Tor}}_{>d}^Q(N_{\mathfrak p},M) \end{align*} $$

is an isomorphism. But this cannot happen, as our torsion hypothesis implies that $\operatorname {\mathrm {Tor}}_\ast ^Q(N_{\mathfrak p},M)$ is $m_Z$ -torsion, and hence any $x\in m_Z$ annihilates elements in each nonzero Tor group. So we conclude that $m_Z\subset \mathfrak p$ , and that $N_{\mathfrak p}$ is finite-dimensional. In that case, $N_{\mathfrak p}$ has a finite composition series where the simple factors are direct summands of $\Lambda $ . Hence,

$$ \begin{align*} \operatorname{\mathrm{Tor}}_{>d}^Q(\Lambda,M)=0\ \Rightarrow\ \operatorname{\mathrm{Tor}}_{>d}^Q(N_{\mathfrak p},M)=0, \end{align*} $$

which contradicts our assumption that $\mathfrak p \in \mathscr {P}$ , and hence contradicts our assumption that $\mathscr {P}$ is nonempty. It follows that $\operatorname {flatdim}(M)=d$ .

As for the injective dimension, by Baer’s criterion, we have

$$ \begin{align*} \operatorname{injdim}(M)=\operatorname{min}\{d:\operatorname{\mathrm{Ext}}^{>d}_Q(M',M)=0\ \forall\ \text{fin gen'd }M'\}, \end{align*} $$

or $\operatorname {injdim}(M)=\infty $ if no such d exists. So we may proceed exactly as above, with $\operatorname {\mathrm {Tor}}$ replaced by $\operatorname {\mathrm {Ext}}$ , to obtain the desired result.

4.1 Singularity categories

For R a Gorenstein ring, we define the big singularity category as

$$ \begin{align*} \operatorname{Sing}(R):=D^b(R\text{-Mod})/\langle \operatorname{\mathrm{Proj}}(R)\rangle, \end{align*} $$

where R-Mod is the category of arbitrary R-modules. This is the Verdier quotient of the bounded derived category by the thick subcategory of objects of finite projective = finite injective dimension. We also have the small singularity category, which is the quotient of the derived category of finitely generated modules by the thick subcategory of perfect complexes

$$ \begin{align*} \operatorname{sing}(R):=D^b(R\text{-mod})/\langle \operatorname{proj}(R)\rangle. \end{align*} $$

Although we do not explicitly use the following result, it clarifies the manner in which the theory of the present paper is an extension of that of [Reference Negron and Pevtsova28].

Proof. The map $R\text {-Mod}\to \operatorname {Sing}(R)$ restricted to the subcategory $\operatorname {GProj}(R)$ of Gorenstein projectives [Reference Enochs and Jenda18] induces an equivalence from the stable category

$$ \begin{align*} \underline{\operatorname{GProj}}(R)\overset{\cong}\longrightarrow \operatorname{Sing}(R) \end{align*} $$

(see [Reference Iyengar and Krause23, Th. 5.6 and Cor. 6.6], [Reference Bergh, Oppermann and Jorgensen11, Th. 3.1]). Similarly, we have an equivalence between the stable category of finitely generated Gorenstein projectives and the small singularity category $\underline {\operatorname {Gproj}}(R)\cong \operatorname {sing}(R)$ [Reference Buchweitz13]. It is known now that $\underline {\operatorname {GProj}}(R)$ is compactly generated with compact objects identified with $\underline {\operatorname {Gproj}}(R)$ , via the functor induced by the inclusion $\operatorname {Gproj}(R)\to \operatorname {GProj}(R)$ [Reference Benson, Iyengar, Krause and Pevtsova9, Prop. 2.10]. So one considers the square connecting the two equivalences above to obtain the claimed result.

As a more terrestrial comment, one can see by partially resolving bounded complexes that any object in $\operatorname {Sing}(R)$ is isomorphic to a shift $\Sigma ^n M$ of some R-module M. Furthermore, by considering syzygies and cosyzygies, we see that the objects $R\text {-Mod}\subset \operatorname {Sing}(R)$ are stable under shifting. So we see that the additive map

$$ \begin{align*} R\text{-Mod}\to \operatorname{Sing}(R) \end{align*} $$

is essentially surjective. The following is well known to experts and can be deduced, for example, by identifying the singularity category with the stable category of Gorenstein projectives and applying [Reference Enochs and Jenda18, Th. 10.2.14].

4.2 All of our algebras are Gorenstein

Our proof is adapted from [Reference Cassidy, Conner, Kirkman and Moore14, Lem. 5.3].

Proof. Take I the ideal generated by the $f_i$ in Q, and take $d=\operatorname {gldim}(Q)$ . Via the Koszul resolution $Kos=Kos(f_1,\dots ,f_m)$ over Z, and the corresponding Koszul resolution $Kos_Q=Q\otimes _Z Kos$ of $Q/I$ over Q, we see that $\operatorname {\mathrm {RHom}}_Q(Q/I,Q)\cong \Sigma ^{-m}Q/I$ . Hence,

$$ \begin{align*} \begin{array}{rl} \operatorname{\mathrm{RHom}}_{Q/I}(-,Q/I)& \cong \Sigma^m\operatorname{\mathrm{RHom}}_{Q/I}(-,\operatorname{\mathrm{RHom}}^\ast_Q(Q/I,Q))\\[3pt] & \cong \Sigma^m\operatorname{\mathrm{RHom}}_Q(-,Q). \end{array} \end{align*} $$

This functor has cohomology vanishing in degrees $>d$ , so that $\operatorname {injdim}(Q/I)\leq d$ .

Let us close the subsection with a remark about singularity categories. By considering the Koszul resolution for R over Q, we see that the restriction map $D^b(R)\to D^b(Q)$ sends the thick ideal $\langle \operatorname {\mathrm {Proj}}(R)\rangle $ into $\langle \operatorname {\mathrm {Proj}}(Q)\rangle $ , and hence descends to a triangulated functor

(3) $$ \begin{align} \operatorname{res}:\operatorname{Sing}(R)\to \operatorname{Sing}(Q). \end{align} $$

5.1 Integral forms

As mentioned in §2.2, we assume that our deformation $Q\to R$ admits an integral form $\mathcal {Q}\to R$ . By this, we mean that R admits a deformation $\mathcal {Q}\to R$ with $\mathcal {Q}$ of finite global dimension for which the parameterizing algebra $\mathcal {Z}$ is smooth, and in particular of finite type over k, and such that the completion at the distinguished point $1:\mathcal {Z}\to k$ recovers our original deformation. That is to say, $Z=\widehat {\mathcal {Z}}_1$ , the completion of $\mathcal {Z}$ at the kernel of the augmentation map $1: \mathcal {Z} \to k$ , and $Q\cong Z\otimes _{\mathcal {Z}}\mathcal {Q}$ .

The point of this integral form is to pacify certain conflicts which arise when attempting to deal simultaneously with base change and completion.

5.2 Topological base change

Consider a deformation $Q\to R$ as in §2.2, with some integral form $\mathcal {Q}\to R$ . For a field extension $k\to K$ , we let $\mathcal {Z}_K$ , $\mathcal {Q}_K$ , and $R_K$ denote the usual base change $-_K=K\otimes -$ , so that we have the base changed deformation $\mathcal {Q}_K\to R_K$ parameterized by $\mathcal {Z}_K$ . We have the standard base change operation

$$ \begin{align*} (-)_K:R\text{-Mod}\to R_K\text{-Mod},\ \ M\mapsto M_K=K\otimes M. \end{align*} $$

We complete at the distinguished point $1:\mathcal {Z}_K\to K$ to obtain the corresponding formal deformation $Q_K\to R_K$ parameterized by $Z_K=\widehat {(\mathcal {Z}_K)}_1$ . So, from another perspective, we obtain the formal deformation $Q_K\to R_K$ via topological base change $Z_K=\varprojlim _n (Z/m_Z^n)\otimes K$ and $Q_K=\varprojlim _n (Q/m_Z^nQ)\otimes K$ . For $Z=k[\!\hspace {.1mm}[{y_1,\dots ,y_n}]\!\hspace {.1mm}]$ , we have $Z_K=K[\!\hspace {.1mm}[{y_1,\dots ,y_n}]\!\hspace {.1mm}]$ .

Lemma 5.3. Consider a field extension $k\to K$ . For any finite-dimensional R-module V, and arbitrary R-module M, the natural maps

$$ \begin{align*} K\otimes\operatorname{\mathrm{Tor}}^Q_\ast(V,M)\to \operatorname{\mathrm{Tor}}^{Q_K}_\ast(V_K,M_K)\ \ \text{and}\ \ K\otimes\operatorname{\mathrm{Ext}}^\ast_Q(V,M)\to \operatorname{\mathrm{Ext}}^\ast_{Q_K}(V_K,M_K) \end{align*} $$

are isomorphisms.

5.3 Representative independence over hypersurfaces

We have the following infinite analog of [Reference Negron and Pevtsova28, Cor. 5.3]. The proof is taken almost directly from [Reference Avramov, Iyengar, Carlson, Iyengar and Pevtsova3], and so is delayed until the appendix.

Theorem 5.4 (cf. [Reference Avramov, Iyengar, Carlson, Iyengar and Pevtsova3, Th. 2.1])

Let $Q\to R$ be a deformation as in §2.2, and assume that Q is of finite global dimension. Consider M any R-module, and $f,g\in m_Z$ with equivalent, nonzero, classes $\bar {f}=\bar {g}$ in $m_Z/m_Z^2$ . Then, there is an identification of graded vector space

$$ \begin{align*} \operatorname{\mathrm{Tor}}_\ast^{Q/(f)}(\Lambda,M)=\operatorname{\mathrm{Tor}}_\ast^{Q/(g)}(\Lambda,M). \end{align*} $$

Similarly, there is an identification

$$ \begin{align*} \operatorname{\mathrm{Ext}}^\ast_{Q/(f)}(\Lambda,M)=\operatorname{\mathrm{Ext}}^\ast_{Q/(g)}(\Lambda,M). \end{align*} $$

In particular, these (co)homology groups vanish in high degree over $Q/(f)$ if and only if they vanish in high degree over $Q/(g)$ .

Of course, the hypersurface algebra $Q/(f)$ depends on f only up to nonzero scaling, $Q/(f)=Q/(\zeta f)$ for any $\zeta \in k^\times $ . So, to phrase things slightly differently, the claimed identification of (co)homology groups holds whenever the reductions $\bar {f}$ and $\bar {g}$ represent the same point in projective space $\mathbb {P}(m_Z/m_Z^2)$ .

5.4 Hypersurface support

Fix now R a finite-dimensional algebra with prescribed deformation $Q\to R$ as in §2.2, and suppose that Q is of finite global dimension.

Consider a point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ , which we may view as a map into the base change $\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)_K=\mathbb {P}(m_{Z_K}/m_{Z_K}^2)$ . We say a hypersurface algebra $Q_K/(f)$ is a hypersurface representative for c if the class of f in $m_{Z_K}/m_{Z_K}^2$ lies in the punctured line defined by c. As a corollary to Theorem 5.4, we have

When the distinction is unimportant, we take, for a given point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ ,

$$ \begin{align*} Q_c=Q_K/(f)\ \text{for any representative}\ f\ \text{of the point }c. \end{align*} $$

By the above lemma, the kernel of the functor

(4) $$ \begin{align} F_c:\operatorname{Sing}(R)\to \operatorname{Sing}(Q_c), \end{align} $$

defined as base change $(-)_K:\operatorname {Sing}(R)\to \operatorname {Sing}(R_K)$ composed with restriction (3), is independent of the choice of representative for the point c.

Lemma 5.7 says that if a given point $c':\operatorname {\mathrm {Spec}}(K')\to \mathbb {P}$ factors through some other point

$$ \begin{align*} \operatorname{\mathrm{Spec}}(K')\to \operatorname{\mathrm{Spec}}(K)\overset{c}\to \mathbb{P}(m_Z/m_Z^2), \end{align*} $$

then the kernels of the two functors $F_c:\operatorname {Sing}(R)\to \operatorname {Sing}(Q_c)$ and $F_{c'}:\operatorname {Sing}(R)\to \operatorname {Sing}(Q_{c'})$ agree.

Definition 5.8. We say an R-module M is supported at a given point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ if M is not in $\ker \{F_c: \operatorname {Sing}(R)\to \operatorname {Sing}(Q_c)\}$ . Equivalently, M is supported at c if $M_K$ , considered as a $Q_c$ -module via the deformation $Q_c\to R_K$ , is of infinite projective dimension over $Q_c$ . We take

$$ \begin{align*} \operatorname{\mathrm{supp}}^{hyp}_{\mathbb{P}}(M):=\left\{ \begin{array}{c} \text{the image of all points }c:\operatorname{\mathrm{Spec}}(K)\to \mathbb{P}(m_Z/m_Z^2)\\ \text{at which }\operatorname{projdim}_{Q_c}(M_K)=\infty \end{array}\right\}. \end{align*} $$

Corollary 5.5 and Lemma 5.7 tell us that there are no ambiguities in the above definition. We note that the support $\operatorname {\mathrm {supp}}^{hyp}_{\mathbb {P}}(M)$ is a subset in $\mathbb {P}(m_Z/m_Z^2)$ which is not necessarily closed. We see below that $\operatorname {\mathrm {supp}}^{hyp}_{\mathbb {P}}(M)$ is closed whenever M is finite-dimensional.

5.5 Support for finite-dimensional modules

Recall our deformation algebra $A_Z=\operatorname {\mathrm {Sym}}(\Sigma ^{-2}(m_Z/m_Z^2)^\ast )$ , from §2.4, and its corresponding action on $D^b(R)$ .

We have $\operatorname {\mathrm {Proj}}(A_Z)=\mathbb {P}(m_Z/m_Z^2)$ and the natural $A_Z$ -action on extensions provides, for any R-modules N and M, an associated quasicoherent sheaf $\operatorname {\mathrm {Ext}}^\ast _R(N,M)^\sim $ on $\mathbb {P}(m_Z/m_Z^2)$ . By [Reference Negron and Pevtsova28, Cor. 4.7], these extension sheaves are coherent when N and M are finite-dimensional.

Proposition 5.9. For a finite-dimensional R-module V over R, we have

$$ \begin{align*} \operatorname{\mathrm{supp}}^{hyp}_{\mathbb{P}}(V)=\operatorname{\mathrm{Supp}}_{\mathbb{P}}\operatorname{\mathrm{Ext}}^\ast_R(V,\Lambda)^\sim. \end{align*} $$

In particular, the support of a finite-dimensional R-module is closed in $\mathbb {P}(m_Z/m_Z^2)$ .

In the proof, we employ the following construction: to any point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ , we have an associated graded algebra map $\phi _c:A_Z\to K[t]$ which is nonzero in degree $2$ and annihilates the kernel of $c:A_Z^2=(m_Z/m_Z^2)^\ast \to K$ . We also consider the base change $\phi _{c,K}:(A_Z)_K\to K[t]$ , which is now a graded algebra surjection. Note that although the map c is really only defined up to scaling, its kernel is well defined and uniquely determined by the point c. Similarly, the map $\phi _c$ is uniquely determined up to the $K^\times $ -action on $K[t]$ by graded automorphisms.

Proof. Consider an arbitrary point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ . We want to show that V is supported at c if and only if the image $c(pt)\in \mathbb {P}$ of c is in the support of the graded $A_Z$ -module $\operatorname {\mathrm {Ext}}^\ast _R(V,\Lambda )$ . By Lemma 5.7, we are free to assume that c is a geometric point, that is, that $K=\overline {K}$ . (We make this assumption to conform to the setting of [Reference Negron and Pevtsova28].)

By [Reference Negron and Pevtsova28, Lem. 6.8], we have $\operatorname {\mathrm {Ext}}^{\gg 0}_{Q_K/(f)}(V_K,\Lambda _K)=0$ if and only if the fiber of

$$ \begin{align*} \operatorname{\mathrm{Ext}}^\ast_{R_K}(V_K,\Lambda_K)=K\otimes\operatorname{\mathrm{Ext}}^\ast_R(V,\Lambda) \end{align*} $$

along the localized map $\phi _{c,K}:K\otimes A_Z\to K[t,t^{-1}]$ vanishes. So, V is supported at the K-point c, in the sense of Definition 5.8, if and only if the pullback $\pi ^\ast \operatorname {\mathrm {Ext}}^\ast _R(V,\Lambda )^\sim $ along the projection $\pi :\mathbb {P}_K\to \mathbb {P}$ is supported at c. Since, for any coherent sheaf $\mathcal {F}$ on $\mathbb {P}$ ,

$$ \begin{align*} \operatorname{\mathrm{Supp}}(\pi^\ast\mathcal{F})=\pi^{-1}\operatorname{\mathrm{Supp}}(\mathcal{F}) \end{align*} $$

[Reference Tate34, Tag 056H], we see that V is supported at c if and only if $\operatorname {\mathrm {Ext}}^\ast _R(V,\Lambda )^\sim $ is supported at $c(pt)$ , as desired.

6.1 Dg modules and sheaves on $\mathbb {P}$

Fix $S=k[\xi _1,\dots , \xi _n]$ a polynomial ring in n variables, and $\mathbb {P}=\operatorname {\mathrm {Proj}}(S)$ . We consider S as a dg algebra, with $\deg (\xi _i)=2$ , for all i, and vanishing differential. Following standard notation, we take $S_{(f)}$ the degree $0$ portion of the localization $S_f$ , for homogenous f, and $N_{(f)}=(N_f)^{0}$ for any graded S-module N. For a degree $2$ function $f\in S$ , let $U_f\subset \mathbb {P}$ denote the basic open $U_f=\operatorname {\mathrm {Spec}}(S_{(f)})$ .

For any dg S-module N, the differential $d:N\to N$ is a degree $1 S$ -linear homomorphism, and each localization $N_f$ is a dg module over $S_f$ . We can consider each $N_f$ as a dg module over the (non-dg) ring $S_{(f)}$ , in which case it becomes an unbounded complex of $S_{(f)}$ -modules and we are in a classical (non-dg) situation. The dg modules $N_f$ glue over the localizations $S_{(f)}$ , $\deg (f)=2$ , to produce a sheaf $\mathcal {F}_N$ of quasi-coherent dg $\mathscr {O}_{\mathbb {P}}$ -modules on projective space $\mathbb {P}=\operatorname {\mathrm {Proj}}(S)$ . Indeed, as an $\mathscr {O}_{\mathbb {P}}$ -module, $\mathcal {F}_N$ is assembled from the usual sheaves associated with $N^{ev}$ , $N^{odd}$ , and their shifts.

The above construction is natural in N, so that in total we obtain an exact functor from dg S-modules to dg sheaves on projective space.

Consider a geometric point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}$ , which determines (up to $K^\times $ -action) a graded surjection $\phi _c:S_K\to K[t]$ , $\deg (t)=2$ . If this point lies in $U_f\subset \mathbb {P}_K$ , then $f\in S_K^2$ maps to a nonzero scaling of t, and we may generally assume that this scalar is $1$ .

Lemma 6.4. Consider a geometric point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}$ , and a standard open $U_f\subset \mathbb {P}_K$ containing c. Then, there is a natural isomorphism

$$ \begin{align*} \mathrm{L}c^\ast \mathcal{F}_N\cong K[t,t^{-1}]\otimes_{S_K}^{\textrm{L}}N_K \end{align*} $$

in $D(K)$ , where we change base along the (localized) map $\phi _c:S_K\to K[t,t^{-1}]$ .

Proof. By changing base initially, we may assume $K=k$ , so that c is a closed point, and by replacing N with a semiprojective resolution, we may assume that N is K-flat (see, e.g., [Reference Negron and Pevtsova28, §3.2] for a discussion of a semiprojective resolutions and [Reference Spaltenstein32, Def. 5.1] for the definition of K-flat). In this case, the derived fiber and derived product are identified with their underived counterparts. It suffices to provide an isomorphism $k\otimes _{S_{(f)}}N_f\cong k[t,t^{-1}]\otimes _{S}N$ . The latter space is isomorphic to $k[t,t^{-1}]\otimes _{S_f}N_f$ , since f maps to a nonzero scaling of t, $\phi (f)=\zeta t$ .

We have the two maps

$$ \begin{align*} k\otimes_{S_{(f)}}N_f\to k[t,t^{-1}]\otimes_{S_f}N_f,\ \ \xi\otimes m\mapsto \xi\otimes m, \end{align*} $$

and

$$ \begin{align*} k[t,t^{-1}]\otimes_{S_f}N_f\to k\otimes_{S_{(f)}}N_f,\ \ \xi t^n\otimes m\mapsto \xi\otimes \zeta^{-n}f^nm. \end{align*} $$

One observes directly that these maps are mutually inverse, and so provide the desired isomorphism.

Proof. We have $H^\ast (\mathcal {F}_N)|_{U_f}=H^\ast (\mathcal {F}_N|_{U_f})=H^\ast (N_f)$ . By exactness of localization, and the fact that the differential $d:N\to N$ is S-linear, we localize the exact sequences

$$ \begin{align*} 0\to Z(N)\to N\overset{d}\to N,\ \ N\overset{d}\to N\to N/B(N)\to 0, \end{align*} $$

to get exact sequences

$$ \begin{align*} 0\to Z(N)_f\to N_f\overset{d}\to N_f,\ \ N_f\overset{d}\to N_f\to N_f/B(N)_f\to 0. \end{align*} $$

This gives $Z(N)_f\cong Z(N_f)$ , $B(N)_f\cong B(N_f)$ , and therefore $H^\ast (N_f)\cong H^\ast (N)_f$ .

6.2 A vanishing theorem

To be clear, by a geometric point, we mean a graded algebra map $S\to K[t]$ , with $\deg (t)=2$ and $K=\bar {K}$ , such that the base change $S_K\to K[t]$ is surjective.

Proof. The proof is by induction on the number of generators of S. Before proceeding with the induction argument, we first claim that for N as in the statement of the theorem, all of the localizations of cohomology $H^\ast (N)_f$ , $f\in S^2$ , vanish.

Indeed, boundedness of the fibers $K[t]\otimes ^{\textrm {L}}_S N$ implies that all of the localizations $K[t,t^{-1}]\otimes ^{\textrm {L}}_S N$ vanish, and by Lemma 6.4, we conclude that all of the derived fibers $\mathrm {L}c^\ast \mathcal {F}_N$ of the associated sheaf vanish, at all geometric points $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}$ . By Neeman [Reference Neeman and Bökstedt26, Lem. 2.12], this implies that $H^\ast (\mathcal {F}_N)=0$ , and by Lemma 6.5, we conclude $H^\ast (N)_f=0$ at all $f\in S^2$ .

Now, let us proceed with our induction argument. Take $n=\dim (S^2)$ , and consider for the base case the algebra $k[t]$ , $\deg (t)=2$ . If the base change $\bar {k}[t]\otimes _{k[t]}N=N_{\overline {k}}$ of any dg $k[t]$ -module along the map $k[t]\to \overline {k}[t]$ , $t\mapsto t$ , has bounded cohomology, then obviously N has bounded cohomology, as desired. Now, suppose that the claimed result holds for all polynomial rings $S'$ in $<n$ variables.

Consider $S'=S/(f)$ for any nonzero $f\in S^2$ , and N as prescribed. Then, the derived fibers of $S'\otimes ^{\textrm {L}}_S N$ have uniformly bounded cohomology, and thus $S'\otimes ^{\textrm {L}}_S N$ has bounded cohomology by our induction hypothesis. By replacing N with a semiprojective resolution, we may assume that f acts as a nonzero divisor on N, and that $N/fN=S'\otimes ^{\textrm {L}}_SN$ . We now have the exact sequence

$$ \begin{align*} 0\to N\overset{f}\to \Sigma^2 N\to \Sigma^2 N/\Sigma^2fN\to 0, \end{align*} $$

with $H^{>r-2}(N/fN)=H^{>r-2}(S\otimes ^{\textrm {L}}_SN)=0$ at some large r. By considering the long exact sequence on cohomology, the map

$$ \begin{align*} f\cdot-:H^i(N)\to H^{i+2}(N) \end{align*} $$

is therefore seen to be an isomorphism for all $i>r$ . It follows that $H^{>r}(N)$ either vanishes or has no f-torsion. As the localization $H^\ast (N)_f$ vanishes, the latter case is impossible. We conclude that $H^{\gg 0}(N)=0$ .

6.3 Proof of Theorem 6.1

As a final ingredient, we have the following result, which was essentially proved in [Reference Negron and Pevtsova28, §6.3]. In the statement, we employ the maps $\phi _c:A_Z\to K[t]$ discussed in §5.5.

Proposition 6.7. At any geometric point $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}(m_Z/m_Z^2)$ , with corresponding map $\phi _c:A_Z\to K[t]$ from $A_Z$ , we have

$$ \begin{align*} K[t]\otimes^{\textrm{L}}_{A_Z}\operatorname{\mathrm{RHom}}_R(\Lambda,M)\cong \operatorname{\mathrm{RHom}}_{Q_c}(\Lambda_K,M_K). \end{align*} $$

Proof. Take $\phi _{c,K}:(A_Z)_K\to K[t]$ the base change of $\phi _c$ , $A=A_Z$ , and $A_c$ the corresponding algebra for the hypersurface deformation $Q_c\to R_K$ . We have

$$ \begin{align*} \operatorname{\mathrm{RHom}}_{R_K}(\Lambda_K,M_K)\cong A_c\otimes^t\operatorname{\mathrm{RHom}}_{Q_c}(\Lambda_K,M_K), \end{align*} $$

in the derived category of dg $A_c$ -modules, with $A_c\otimes ^t\operatorname {\mathrm {RHom}}_{Q_c}(\Lambda _K,M_K)$ a K-flat dg module [Reference Negron and Pevtsova28, Lems. 4.1 and 4.2]. Recall that $A_c^2=\ker (\phi _{c,K}^2)$ . Thus,

$$ \begin{align*} K[t]\otimes^{\textrm{L}}_{A}\operatorname{\mathrm{RHom}}_Q(\Lambda,M)&\cong K[t]\otimes^{\textrm{L}}_{A_K}K\otimes\operatorname{\mathrm{RHom}}_R(\Lambda,M)\\[6pt]&\cong K[t]\otimes^{\textrm{L}}_{A_K}\operatorname{\mathrm{RHom}}_{R_K}(\Lambda_K,M_K)\\[6pt]&\cong K\otimes^{\textrm{L}}_{A_c}\operatorname{\mathrm{RHom}}_{R_K}(\Lambda_K,M_K)\\[6pt]&\cong K\otimes_{A_c} (A_c\otimes^t\operatorname{\mathrm{RHom}}_{Q_c}(\Lambda_K,M_K))\cong \operatorname{\mathrm{RHom}}_{Q_c}(\Lambda_K,M_K).\end{align*} $$

We now prove our theorem.

Proof of Theorem 6.1

Fix $d=\operatorname {gldim}(Q)$ . Recall that $\operatorname {\mathrm {Ext}}^{\gg 0}_{Q_c}(\Lambda _K,M_K)=0$ if and only if $\operatorname {\mathrm {Ext}}^{>d}_{Q_c}(\Lambda _K,M_K)=0$ , by Corollary 3.2. We have the formula

$$ \begin{align*} K[t]\otimes^{\textrm{L}}_{A_Z}\operatorname{\mathrm{RHom}}_R(\Lambda,M)\cong\operatorname{\mathrm{RHom}}_{Q_c}(\Lambda_K,M_K), \end{align*} $$

at arbitrary points $c:\operatorname {\mathrm {Spec}}(K)\to \mathbb {P}$ . So, we apply Theorem 6.6, with $S=A_Z$ and N the dg $A_Z$ -module $\operatorname {\mathrm {RHom}}_R(\Lambda ,M)$ , and Corollary 3.2, to find that the hypersurface support for M vanishes if and only if $H^\ast (\operatorname {\mathrm {RHom}}_R(\Lambda ,M))=\operatorname {\mathrm {Ext}}^\ast _R(\Lambda ,M)$ vanishes in high degree. Such vanishing of cohomology occurs if and only if M is of finite projective dimension over R, by Corollary 3.2. So, we have the desired relation, $\operatorname {\mathrm {supp}}^{hyp}_{\mathbb {P}}(M)=\emptyset $ , if and only if M vanishes in $\operatorname {Sing}(R)$ .