Multigraded linear series and recollement

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by  Alastair Craw

MULTIGRADED LINEAR SERIES AND RECOLLEMENT
ALASTAIRCRAW,YUKARIITO, ANDJOSEPH KARMAZYN
Abstract. GivenaschemeY equippedwithacollection ofglobally generatedvectorbundles
E1,...,En, we study the universal morphism from Y to a fine moduli space M(E) of cyclic
modules over the endomorphism algebra of E := OY ⊕E1⊕···⊕En. This generalises the
classicalmorphismtothelinearseriesofabasepoint-freelinebundleonascheme. Wedescribe
7 the image of the morphism and present necessary and sufficient conditions for surjectivity in
1 terms of a recollement of a module category. When the morphism is surjective, this gives a
0 finemoduli space interpretation of the image, and as an application we show that for a small,
2
finitesubgroup G⊂GL(2,k), everysub-minimal partial resolution of A2k/G is isomorphic toa
n finemodulispaceM(EC)whereEC isasummandofthebundleE definingthereconstruction
a
algebra. WealsoconsiderapplicationstoGorensteinaffinethreefolds,whereReid’srecipesheds
J
somelight ontheclasses ofalgebrafrom which onecanreconstruct agivencrepantresolution.
8
1

]
G
A Contents
.
h
1. Introduction 1
t
a 2. Multigraded linear series 5
m
3. The cornering category and recollement 8
[
4. Cornering the reconstruction algebra 12

2
5. Cornering noncommutative crepant resolutions 16
v
9 6. The toric case in dimension three 19
7 7. On Reid’s recipe and surface essentials 23
6
Appendix A. The numerical Grothendieck group for compact support 26
1
0 References 28
.
1
0
7
1
v: 1. Introduction
i
X The study of an algebraic variety in terms of the morphisms to the linear series of basepoint-
r free line bundles has always been a central tool in algebraic geometry. Here we extend this
a
notion to the multigraded linear series of a collection of globally generated vector bundles on
a scheme, thereby unifying several constructions from the literature (see [CS08, Cra11b] and
[Cra11a, Section 5]). Ourprimarygoal is to providenew,geometrically significant modulispace
descriptions of any given scheme, and we illustrate this in several families of examples.
Date: January 19, 2017.
2010 Mathematics Subject Classification. 14A22 (Primary); 14E16, 16G20, 18F30 (Secondary).
Key words and phrases. Linearseries,modulispaceofquiverrepresentations,specialMcKaycorrespondence,
noncommutativecrepant resolutions.
1
Multigraded linear series. To be more explicit, let Y be a scheme that is projective over an
affine scheme of finite type over k, an algebraically closed field of characteristic zero. Given a
collection E ,...,E of effective vector bundles on Y, define E := E where E is the
1 n 0≤i≤n i 0
trivial bundle on Y. Let A := End (E) denote the endomorphism algebra and consider the
Y
L
dimension vector v := (v ) given by v := rk(E ) for 0 ≤ i ≤ n. We definethe multigraded linear
i i i
series of E to be the fine moduli space M(E) of 0-generated A-modules of dimension vector v
(see Definition 2.5). The universal family on M(E) is a vector bundle T = T together
0≤i≤n i
with a k-algebra homomorphism A → End(T), where T is a tautological vector bundle of rank
i
L
v for 1≤ i ≤ n and T is the trivial bundle.
i 0
Our first main result (see Theorem 2.6) generalises the classical morphism ϕ : Y → |L| to
|L|
the linear series of a single basepoint-free line bundle L on Y, or the morphism to a Grassman-
nian defined by a globally generated vector bundle on a projective variety:
Theorem 1.1. If the vector bundles E ,...,E are globally generated, then there is a morphism
1 n
f: Y → M(E) satisfying E = f∗(T ) for 0 ≤i ≤ n whose image is isomorphic to the image of
i i
the morphism ϕ : Y → |L| to the linear series of L := det(E )⊗j for any j >> 0.
|L| 1≤i≤n i
When the line bundle det(E ) is ample, the unNiversal morphism f: Y → M(E) from
1≤i≤n i
Theorem 2.6 is a closed immersion and it is natural to ask whether f is surjective, in which
N
case f presents Y as the fine moduli space M(E). Even when Y is isomorphic to M(E), one
can sometimes gain more insight by deleting summands of E. Indeed, if C ⊆ {0,1,...,n} is
a subset containing 0 then the subbundle E := E of E has the trivial bundle E as a
C i∈C i 0
summand, and Theorem 1.1 gives a universal morphism
L
g : M(E) −→ M(E ) (1.1)
C C
between multigraded linear series which can lead to a more geometrically significant moduli
space description of Y. A moduli construction determined by a tilting bundle E on Y is of
C
clear geometric significance, but one need not demand this much; after all, one does not require
every indecomposable summand of Beilinson’s tilting bundle in order to reconstruct Pn.
Astatement similartoTheorem1.1holdsifwereplaceM(E)byaproductofGrassmannians
over Γ(O ). However, the dimension of this product is higher than that of M(E) in general,
Y
and f is almost never an isomorphism, i.e., f cannot provide a moduli space description of Y.
The Special McKay correspondence. Our second main result illustrates this phenomenon.
Let G ⊂ GL(2,k) be a finite subgroup without pseudo-reflections, write Irr(G) for the set of
isomorphismclasses ofirreduciblerepresentations ofG, andletY denotetheminimalresolution
of A2/G. Generalising the work of Ito–Nakamura [IN99] for a finite subgroup of SL(2,k),
k
Kidoh [Kid01] and Ishii [Ish02] proved that Y is isomorphic to the G-Hilbert scheme, that is,
the fine moduli space of G-equivariant coherent sheaves of the form O for subschemes Z ⊂ A2
Z k
such that Γ(O ) is isomorphic to the regular representation of G. If we write
Z
T := T⊕dim(ρ)
ρ
ρ∈Irr(G)
M
for the tautological bundle on the G-Hilbert scheme, then since End (T) is isomorphic to the
Y
skew group algebra (see Lemma 4.1), it follows that the minimal resolution Y ∼= G-Hilb is
isomorphic to the multigraded linear series M(T). When G is a finite subgroup of SL(2,k), T
2
is a tilting bundle on Y by work of Kapranov–Vasserot [KV00], so Y is derived equivalent to
the category of modules over the endomorphism algebra of T. However, this is false in general;
put simply, the G-Hilbert scheme is the wrong moduli description of the minimal resolution of
A2/G unless G is a finite subgroup of SL(2,k).
k
Amorenaturalmodulispacedescription ofY comes fromtheSpecialMcKay correspondence
of the finite subgroup G ⊂ GL(2,k). For the set Sp(G) := {ρ ∈ Irr(G) | H1(T∨)= 0} of special
ρ
representations, it follows from Van den Bergh [VdB04b] that the reconstruction bundle
E := T
ρ
ρ∈Sp(G)
M
is a tilting bundleon Y, so Y is derived equivalent to themodulecategory of theendomorphism
algebra of E, that is, to the category of modules over the reconstruction algebra studied by
Wemyss [Wem11b]. Our second main result (see Proposition 4.2 and Theorem 4.4) shows that
E contains enoughinformation to reconstructY, and henceprovides a modulispace description
that trumps the G-Hilbert scheme in general:
Theorem 1.2. Let G ⊂ GL(2,k) be a finite subgroup without pseudo-reflections. Then:
(i) the minimal resolution Y of A2/G is isomorphic to the multigraded linear series M(E)
k
of the reconstruction bundle; and
(ii) for any partial resolution Y′ such that the minimal resolution Y → A2/G factors via
k
Y′, there is a summand E ⊆ E such that Y′ is isomorphic to M(E ).
C C
In other words, for a finite subgroup G ⊂ GL(2,k), the minimal resolution of A2/G can be
k
obtained directly from the special representations. The statement of part (i) is due originally
to Karmazyn [Kar14, Corollary 5.4.5], while an analogue of part (ii) in the complete local
setting can be deduced by combining Iyama–Kalck–Wemyss–Yang [KIWY15, Theorem 4.6]
with [Kar14, Corollary 5.2.5]. Note however that our approach is completely different in each
case, and is closer in spirit to the geometric construction of the Special McKay correspondence
for cyclic subgroups of GL(2,k) given by Craw [Cra11b].
Main tools. Thekey to the proofof Theorem 1.2 is a homological criterion to decide whenthe
morphism g from (1.1) is surjective. In this situation, any subset C ⊆ {0,1,...,n} containing
C
0 determines a subbundle E of E, and the module categories of the algebras A := End (E)
C Y
andA := End (E )arelinkedbyarecollement(seeSection3). Inparticular,thereisanexact
C Y C
functor j∗: A-mod → A -mod with left adjoint j : A -mod → A-mod. These functors capture
C ! C
informationaboutthemorphismg : M(E) → M(E )from(1.1): closedpointsy ∈ M(E)and
C C
x ∈ M(E ) correspond to 0-generated modules M ∈ A-mod and N ∈ A -mod of dimension
C y x C
vectors v := (v ) and v := (v ) respectively, and
i 0≤i≤n C i i∈C
g (y) = x ⇐⇒ j∗M = N .
C y x
Since the functor j lifts 0-generated A -modules to 0-generated A-modules, the question of
! C
whether x lies in the image of g reduces to the following (see Proposition 3.6):
C
Proposition 1.3. The morphism g : M(E) → M(E ) is surjective iff for each x ∈ M(E ),
C C C
the A-module j(N ) admits a surjective map onto an A-module of dimension vector v.
! x
3
A second key ingredient is that a derived equivalence
Ψ(−):= E∨⊗ −: Db(A) −→ Db(Y)
A
induces an isomorphism between the lattice of dimension vectors for A and the numerical
Grothendieck group for compact supportKnum(Y), introduced by Bayer–Craw–Zhang [BCZ16]
c
(see Appendix A). In particular, understanding the class of the object Ψ(j(N )) in Knum(Y)
! x c
reveals the dimension vector of the A-module j(N ) for each closed point x ∈ M(E ), and this
! x C
provides a tool to help determine whether the above the homological criterion applies. More
explicitly, we prove the following result (see Theorem 5.4):
Theorem 1.4. Suppose that for each x ∈ M(E ), the class [Ψ(j(N ))] ∈ Knum(Y) can be
C ! x c
written as a positive combination of the classes of sheaves on Y. Then g is surjective.
C
Examples from NCCRs in dimension three. These tools are well adapted to studying
crepant resolutions of Gorenstein affinethreefolds. For any such singularity X, Van denBergh’s
construction [VdB04a] of an NCCR (satisfying Assumption 5.1) produces a crepant resolution
of X as a fine moduli space of stable representations for an algebra A (see Proposition A.3).
The choice of 0-generated stability condition chooses a particular crepant resolution Y and a
globally generated bundleT on Y. In this case, Y is isomorphic to the multigraded linear series
M(T); sinceT isatilting bundle,thismoduliconstructioniscertainly geometrically significant.
Nevertheless, motivated by work of Takahashi [Tak11], we ask whether one can reconstruct
Y using only a proper summand of T (in general, none of the indecomposable summands of T
is ample). To state the result, we say that a vertex i ∈ Q = {0,1,...,n} is essential if there is
0
a 0-generated A-module of dimension vector v that contains the vertex simple A-module S in
i
its socle. The following result combines Propositions 6.1 and 6.5:
Proposition 1.5. Let M(T) be the crepant resolution of the Gorenstein, affine toric threefold
X picked out by the choice of 0-generated stability condition as above. Then:
(i) for any subset C ⊆ Q containing 0, the image of g : M(T) → M(T ) is an irreducible
0 C C
component of M(T ); and
C
(ii) if C is the union of {0} with the set of essential vertices, then g is an isomorphism
C
onto its image.
Here, part(ii)generalises theresultof Takahashi[Tak11]beyondthecasewhereX is anabelian
quotient. Example 6.2 shows that Proposition 1.5 is optimal: in general M(T ) is reducible.
C
We conclude with several examples that are orthogonal in spiritto Proposition 1.5(ii), with a
view to strengthening the statement to an isomorphism M(T) ∼= M(T ). Rather than keep the
C
summands of T corresponding to essential vertices, instead we use Reid’s recipe as a guide to
help us choose which essential vertices to remove. For an essential vertex i ∈ Q , derived Reid’s
0
recipe [CL09, Log10, CCL12, BCQ15] proves that the image under the derived equivalence Ψ
of the vertex simple A-module S is a sheaf. This gives enough information to compute the
i
class of Ψ(j(N )) in Knum(Y), and under a dimension condition (see Corollary 5.6), we can
! x c
apply Theorem 1.4 to deduce that g : M(T) → M(T ) is an isomorphism. We showcase this
C C
constructioninExamples7.5and7.7,andinthelatter examplewealsoshowthatthedimension
condition can fail if we remove two indecomposable summands of T corresponding to essential
vertices that mark the same surface by Reid’s recipe. Put more geometrically, surjectivity can
fail if the summands of T do not generate the Picard group of M(T).
C
4
Notation. Let k be an algebraically closed field of characteristic zero. For any quasiprojective
k-scheme Y and k-algebra A, we write Db(Y) and Db(A) for the bounded derived categories of
coherent sheaves on Y and finitely generated left A-modules respectively. We use the phrase
vector bundle as shorthand for a locally-free sheaf of finite rank.
Acknowledgements. ThefirstauthorthanksStefanSchro¨erforastimulatingdiscussion. The
first and third authors were supported by EPSRC grants EP/J019410/1 and EP/M017516/1
respectively, and the second author was supported by JSPS Grant-in-Aid (C) No. 23540045.
2. Multigraded linear series
An associative k-algebra A that is presented in the form A ∼= kQ/I for some finite connected
quiver Q and two-sided ideal I ⊂ kQ determines a choice of idempotents e ∈ A, one for each
i
vertex i ∈ Q . Let ZQ0 denote the free abelian group generated by the vertex set of Q. A
0
dimension vector v = (v ) ∈ NQ0 determines the rational vector space
i
Θv := v⊥ = θ = (θi)∈ Hom(ZQ0,Q)| θivi =0
( )
iX∈Q0
of stability parameters for A-modules of dimension vector v. For θ ∈ Θv, an A-module M of
dimension vector v is θ-semistable if θ(N) ≥ 0 for every nonzero proper A-submodule N of
M. The notion of θ-stability is defined by replacing ≥ with >, and we say θ ∈ Θv is generic
if every θ-semistable A-module is θ-stable. There is a wall and chamber decomposition on Θv,
where two generic parameters θ,θ′ ∈ Θv lie in the same chamber if and only if the notions of
θ-stability and θ′-stability coincide.
When v is indivisible and θ ∈ Θv is generic, King [Kin94] constructs the fine moduli space
M(A,v,θ) of isomorphism classes of θ-stable A-modules of dimension vector v. The universal
family on M(A,v,θ) is a tautological vector bundle
T = T
i
iM∈Q0
satisfying rk(T ) = v for i ∈ Q , together with a k-algebra homomorphism A → End(T), such
i i 0
that the fibre of T at any closed point of M(A,v,θ) is the corresponding θ-stable A-module of
dimension vector v. In fact, T is defined only up to tensor product by an invertible sheaf, but
we remove this ambiguity by choosing once and for all a vertex of the quiver that we denote
0 ∈ Q and working only with dimension vectors v satisfying v = 1; we normalise T by fixing
0 0
T to be the trivial bundle.
0
These moduli spaces arise naturally in geometry as follows. Let R be a finitely generated
k-algebra, let Y be a projective R-scheme and let E ,...,E be nontrivial, effective vector
1 n
bundles on Y. In addition, for E := O , write A := End( E ) for the endomorphism
0 Y 0≤i≤n i
algebra. This decomposition of E := E gives a complete set of orthogonal idempotents
0≤i≤n i L
e = id ∈ End(E ) of A such that 1 = e . Then A admits a presentation
i i L 0≤i≤n i
P A ∼= kQ/I (2.1)
such that the vertex set is Q = {0,1,...,n}. Indeed, introduce a set of loops at each vertex
0
corresponding to a finite set of k-algebra generators of R, and for 0 ≤ i,j ≤ n we introduce
5
arrows from i to j corresponding to a finite generating set for Hom(E ,E ) as an R-module.
i j
This determines a surjective k-algebra homomorphism kQ → End(E) with kernel I.
Remark 2.1. The ideal I ⊂ kQ constructed in this way need not be admissible, and relations
may even involve idempotents. For example, if E ∼= E then the isomorphisms correspond to
i j
relations of the form aa′−e ,a′a−e ∈ I for some a,a′ ∈ kQ.
i j
Thedimension vector v = (v ) definedby setting v := rk(E ) for 0 ≤ i≤ n is indivisible, and
i i i
E is a flat family of A-modules of dimension vector v. If there exists generic θ ∈ Θv such that
for each closed point y ∈ Y, the fibre E of E over y is θ-stable, then the universal property of
y
M(A,v,θ) determines a morphism
f: Y −→ M(A,v,θ)
satisfyingE = f∗(T )foralli ∈ Q ; notethatf dependsonE andtheGIT chambercontaining
i i 0
θ. The following result is known to experts but we were unable to find a reference.
Lemma 2.2. Given a vector bundle E on Y, suppose there exists a generic θ = (θi)∈ Θv such
that E is a flat family of θ-stable A-modules of dimension vector v. The image of the universal
morphism f: Y → M(A,v,θ) is isomorphic to the image of the morphism from Y to the linear
series of the globally generated line bundle det(E )⊗θi.
0≤i≤n i
Proof. Since pullback commutes with tensNor operations on T , the universal property of the
i
morphism f gives det(E )⊗θi = det(f∗(T ))⊗θi ∼= f∗(det(T )⊗θi) for 0≤ i ≤ n, and hence
i i i
det(E )⊗θi ∼= f∗ det(T )⊗θi . (2.2)
i i
 
0≤i≤n 0≤i≤n
O O
 
Theline bundleL′ := det(T )⊗θi on M(A,v,θ) is the polarising ample bundleobtained
0≤i≤n i
from the GIT construction. After taking a multiple if necessary, if g: M(A,v,θ)→ |L′| denotes
N
the closed immersion to the linear series of L′, then equation (2.2) gives
L := (g◦f)∗(O(1)) = det(E )⊗θi.
i
0≤i≤n
O
It follows that g◦f coincides with the classical morphism ϕ : Y → |L| to the linear series of
|L|
L. In particular, det(E )⊗θi is globally generated. (cid:3)
0≤i≤n i
The problem wNith Lemma 2.2 is that it is a difficult problem in general to find a suitable
parameter θ ∈Θv for which a given vector bundleE defines a flat family of θ-stable A-modules.
Herewehighlightaspecialsituation wherethisproblemhasasimplesolution. It’seasytosee
that any stability parameter θ = (θi) ∈ Θv satisfying θi > 0 for all i 6= 0 is generic, so there is
a GIT chamber Θ+v ⊂ Θv containing all such stability parameters. Given an A-module M that
is θ-stable for θ ∈ Θ+, it follows directly from the definition that there exists a surjective A-
v
module homomorphism Ae → M. More generally, we say that an A-module M is 0-generated
0
if there exists a surjective A-module homomorphism Ae → M. It is sometimes advantageous
0
to use this latter notion because it is well-defined without having to make explicit reference to
a dimension vector v.
Proposition 2.3. Let E ,...,E be vector bundles on Y and set E = O . Then E
1 n 0 Y 0≤i≤n i
is a flat family of 0-generated A-modules if and only if E is globally generated for all 1 ≤ i≤ n.
i
L
6
Proof. The bundle E := E on Y is a flat family of A-modules of dimension vector
0≤i≤n i
v = (rk(E )) , and for any closed point y ∈Y, the A-module structure in the fibre E(y) of
i 0≤i≤n
L
E over y is obtained by evaluating all homomorphisms between the bundles E (for 0 ≤ i ≤ n)
i
at y. Choose θ ∈ Θ+ satisfying θ > 0 for i 6= 0. Since θ = − θ , an A-submodule W
v i 0 1≤i≤n i
of E(y) is destabilising if and only if dim(W ) = 1 and there exists i > 0 such that the sum
0
P
of all maps from W to W determined by paths in the quiver from 0 to i is not surjective. In
0 i
particular, E(y) is θ-unstable for all y ∈ Y if and only if there exists i > 0 such that E cannot
i
be written as the quotient of O⊕k for some k ∈ N; equivalently, E(y) is θ-stable for all y ∈Y if
Y
and only if E is globally generated for 1 ≤ i≤ n. (cid:3)
i
Corollary 2.4. For any θ ∈ Θ+v, the locally-free sheaf Ti on M(A,v,θ) is globally generated
for all 0 ≤ i ≤n.
Proof. The sheaf T is a flat family of θ-stable A-modules of dimension vector v on
0≤i≤n i
M(A,v,θ). The rLesult follows from Proposition 2.3 and the fact that T0 ∼= OM(A,v,θ). (cid:3)
We now introduce the main definition.
Definition 2.5. Let E ,...,E be globally generated vector bundles on Y. For E := O ,
1 n 0 Y
write E = E , and define A := End (E) and v = (rk(E )) . The multigraded
0≤i≤n i Y i 0≤i≤n
linear series of E is the fine moduli space
L
M(E) := M(A,v,θ)
of θ-stable A-modules of dimension vector v for any θ ∈ Θ+v.
Corollary 2.4 implies that each direct summand of the tautological bundleT = T on
0≤i≤n i
M(E) is globally generated. To justify the terminology ‘multigraded linear series’, we present
L
the following key result (compare also Example 2.7).
Theorem 2.6. Let E ,...,E be globally generated vector bundles on Y. There is a morphism
1 n
f: Y → M(E) satisfying E = f∗(T ) for 1 ≤i ≤ n whose image is isomorphic to the image of
i i
the morphism ϕ : Y → |L| to the linear series of L := det(E )⊗j for any j >> 0.
|L| 1≤i≤n i
Proof. Apply Proposition 2.3 and Lemma 2.2 to any parNameter θ ∈ Θ+v, noting that E0 = OY.
The isomorphism class of the image of f does not depend on the choice of θ ∈ Θ+, and the final
v
statement follows by choosing the parameter θ ∈ Θ+ satisfying θ = j for i 6= 0 and j >>0. (cid:3)
v i
Example 2.7 (Linear series of higher rank). When Y is projective and E has rank r, then
1
M (v,θ) is isomorphic to the Grassmannian Gr(H0(E ),r) of rank r quotients of H0(E ), and
A 1 1
f coincides with the morphism ϕ to the linear series of higher rank that recovers E as the
|E1| 1
pullback of the tautological quotient bundle of rank r; see Mukai [Muk10, Section 3]. When
r = 1, this is the classical linear series of a basepoint-free line bundle.
Example 2.8 (Quiver flag varieties). Let Q bea finite, acyclic, connected quiver with a unique
source denoted 0 ∈ Q , and let v = (v ) ∈ NQ0 be a dimension vector satisfying v = 1. For
0 i 0
θ ∈ Θ+v, the fine moduli space M(kQ,v,θ) is called a quiver flag variety [Cra11a], and for
i ∈Q , the tautological bundle T of rank v is globally generated by Corollary 2.4. Every such
0 i i
variety is an iterative Grassmann-bundle [Cra11a, Theorem 3.3], and T is simply the pullback
i
of the tautological quotient bundle on one of the Grassmann bundles in the tower.
7
WeclaimthatM(kQ,v,θ)isthemultigradedlinearseriesM(T)associatedtoT = T .
i∈Q0 i
Since v = rk(T ) for i∈ Q , it suffices to show that the tautological k-algebra homomorphism
i i 0
L
h: kQ→ End(T)
isanisomorphism. Theproofof[Cra11a,Lemma5.3]establishes thathissurjective, sosuppose
c p ∈ Ker(h), where paths p ,...,p in Q have common tail at i ∈ Q and common
1≤α≤k α α 1 k 0
head at j ∈ Q . For any path p in Q from vertex 0 to i, the sum c p p lies in the
P 0 1≤α≤k α α
kernel of the map e (kQ)e → Hom(T ,T ) ∼= Γ(T ) of k-vector spaces induced by h. However,
0 i 0 i i
P
this map is an isomorphism [Cra11a, Corollary 3.5(ii)], so h is injective as required.
We may now apply Theorem 2.6 to the determinants of the tautological bundles. Indeed,
det(T ) is globally generated for each i ∈ Q , and the line bundle L = det(T ) is ample
i 0 i∈Q0 i
by [Cra11a, Lemma 3.7], so for E = det(T ), the induced morphism
i∈Q0 i N
f:LM(kQ,v,θ)→ M(E)
isaclosedimmersion. Thisprovidesasimpleconstruction ofthemultigraded Plu¨cker embedding
of the quiver flag variety from [Cra11a, Proposition 6.1].
Remark 2.9. Multigraded linear series were introduced by Craw–Smith [CS08] when each E
i
has rank one and Y is a projective toric variety.
(1) The construction of [CS08] is phrased in terms of a quiver with relations that gives a
natural presentation A ∼= kQ/I, leading to an explicit GIT description of the image of
f embedded in a quiver flag variety. However, we now assume only that Y is projective
over an affine, and in this generality no such natural presentation exists, leading us to
place greater emphasis on A rather than on a quiver. While we sacrifice an explicit
description of the image of f, Theorem 2.6 nevertheless determines the image of f up
to isomorphism.
(2) Theobservation in Theorem 2.6 that the image of f is determined by det(E )⊗j
1≤i≤n i
for j >> 0 renders redundant the assumption from [CS08, Corollary 4.10] (and hence
N
also from [Cra11a, Proposition 5.5] and Prabhu-Naik [Pra17, Proposition 5.5]) that a
map obtained by multiplication of global sections is surjective.
3. The cornering category and recollement
In this section we use Theorem 2.6 repeatedly to produce a compatible family of morphisms
between different multigraded linear series. We then introduce a homological criterion that is
sufficient to guarantee that any of these morphisms is surjective.
We continue to assume that R is a finitely generated k-algebra, Y is a projective R-scheme
and that E ,...,E are nontrivial globally generated vector bundles on Y. For E = O ,write
1 n 0 Y
E := E and define A:= End(E). To produce new endomorphism algebras from A, let
0≤i≤n i
C ⊆ {0,1,...,n} be any subset containing {0}. Define both the idempotent e := e of
L C i∈C i
A and the k-algebra A := e Ae . Since A ∼= End(E), the locally-free sheaf E := E
C C C C P i∈C i
on Y satisfies
L
A ∼= End(E ).
C C
8
The process of passing from A to A is called cornering the algebra A. For v := (rk(E ))
C C i i∈C
and for any 0-generated stability parameter θC ∈ Θ+v , Theorem 2.6 gives a morphism
C
f : Y −→ M(E )= M(A ,v ,θ ). (3.1)
C C C C C
Before studying these morphisms in detail, we record the fact that if we’re interested only
in the scheme underlying a multigraded linear series, we may assume without loss of generality
that the globally generated vector bundles E ,...,E are pairwise non-isomorphic:
1 n
Lemma 3.1. For a ,...,a ≥ 1, we have M( E ) ∼= M( E⊕ai) as R-schemes.
1 n 0≤i≤n i 0≤i≤n i
Proof. The endomorphism algebra of E′ := L0≤i≤nEi is MoritLa equivalent to the endomor-
phism algebra of E := E⊕ai. By increasing the multiplicities of the summands in the
0≤i≤n i L
tautological bundle on M(E′), we obtain a flat family T⊕ai of 0-generated End(E)-
L 0≤i≤n i
modules of dimension vector v = (a rk(E )), so there is a morphism f: M(E′) → M(E). For
i i
L
the other direction, we have E′ = E for some cornering subset C, so (3.1) defines a morphism
C
f : M(E) → M(E′). Universality ensures that these morphisms are mutually inverse. (cid:3)
C
Our next result encodes the fact that the morphisms (3.1) are compatible as we vary the
choice of the subset C. These moduli spaces and universal morphisms between them form a
poset with maximal element M(E) and minimal element M(E ) ∼= SpecΓ(O ). To state the
0 Y
result, regard the poset of subsets of {0,1,...,n} that contain {0} as a category C in which
the morphisms are the set-theoretic inclusion maps between subsets.
Proposition 3.2. LetE ,...,E begloballygenerated vectorbundlesonY andsetR := Γ(O ).
1 n Y
There is a contravariant functor from C to the category of projective R-schemes that sends a
set C to the multigraded linear series M(E ).
C
Proof. Given subsets C′ ⊆ C ⊆ {0,1,...,n} that both contain {0}, we first construct a mor-
phism fC,C′: M(EC) → M(EC′) that fits into a commutative diagram
Y (3.2)
M(EC{{✇)f✇C✇✇✇✇✇✇✇fC,C❍′❍❍❍❍❍f//❍CM❍′❍$$ (EC′).
Toavoid aproliferation of indices,wewriteT = T andT′ = T′ forthetautological
i∈C i i∈C′ i
bundles on M(EC) and M(EC′) respectively, wLhere rk(Ti′) = rk(LTi) = rk(Ei) for all i ∈ C′.
Since T is a flat family of 0-generated A -modules on M(E ), it follows that T is a
C C i∈C′ i
flat family of 0-generated AC′-modules on M(EC). The universal property of M(EC′) as in
L
Lemma 2.2 defines a morphism fC,C′: M(EC) → M(EC′) satisfying fC∗,C′(Ti′) = Ti for i ∈ C′.
The morphism f is also universal, so
C
(fC,C′ ◦fC)∗(Ti′) = fC∗(Ti) = Ei
for alli ∈ C′. Thispropertycharacterises fC′, sodiagram (3.2)commutes as required. Thatthe
assignment sending the inclusion C′ ֒→ C to the morphism fC,C′ respects composition follows
similarly from the universal property of fC,C′. Our assignment is therefore a functor.
It remains to prove that fC,C′ is a morphism of projective R-schemes. For C′′ := {0}, we
have AC′′ = End(E0) = R and hence M(EC′′) = SpecR. The result follows by commutativity
of the morphisms fC′,C′′ ◦fC,C′ = fC,C′′ established above. (cid:3)
9
Definition 3.3. Let C ⊆ {0,1,...,n} be a subset containing 0. The cornering morphism for
C is the universal morphism
g := f : M(E) −→ M(E ) (3.3)
C {0,...,n},C C
obtained by applying the functor from Proposition 3.2 to the inclusion C ֒→ {0,1,...,n}.
Theorem 2.6 ensures that we understand the image of each cornering morphism. The next
example illustrates that while these morphisms may be surjective, they need not be.
Example 3.4. For P2 equipped with the tilting bundle E := O ⊕ O(1) ⊕O(2), the algebra
End (E) can be presented using the Beilinson quiver with relations:
P2
x y = y x
1 0 1 0
x0 x1
0 y0 1 y1 2 y1z0 = z1y0
z0 z1
z x = x z
1 0 1 0
Consider the category C from Proposition 3.2 viewed as a poset:
C = {0,1,2}
3
⊂
⊂
C1 := {0,1} ⊂ C2 := {0,2}
⊂
⊂
C := {0}
0
It is easy to calculate that M(E) ∼= P2. Example 2.7 implies that g : M(E) → M(E ) is
C1 C1
an isomorphism, whereas g : M(E) → M(E ) ∼= P5 is the Veronese embedding ϕ . For
C2 C2 |O(2)|
1 ≤ i≤ 3, we have that M(E ) → M(E ) ∼= Speck is the structure morphism.
Ci C0
In order to introduce the surjectivity criterion we continue to assume that C ⊆ {0,1,...,n}
contains 0 and where the idempotent e = e determines the algebra A = e Ae . In
C i∈C i C C C
this situation there are six functors forming a recollement of the abelian module category:
P
i∗ j∗
A/AeCA−mod i∗ A−mod j∗ AC −mod
i! j!
where the functors are defined by
i∗(−) := A/(AeCA)⊗A(−) j∗(−) := HomAC(eA,−)
i∗(−) := inc and j∗(−) := eCA⊗A(−)∼= HomA(AeC,−)
i!(−) := HomA(A/AeCA,−) j!(−) := AeC ⊗eCAeC (−)
such that (i∗,i ,i!) and (j,j∗,j ) are adjoint triples, i , j , and j are fully faithful, and Imi =
∗ ! ∗ ∗ ∗ ! ∗
kerj∗. In particular, j∗ and i are exact. The module j(N) is maximally extended by objects
∗ !
supported on A/AeA. Indeed, for any A/AeA-module M, we have that Exti(j(N),i (M)) =
! ∗
Exti(N,j∗i (M)) = 0.
∗
Recall that in order to define the multigraded linear series M(E ) from (3.1), we introduced
C
the dimension vector v = (rk(E )) .
C i i∈C
Lemma 3.5. Let N be an A -module of dimension vector v .
C C
(i) If N is a 0-generated A -module, then jN is a 0-generated A-module.
C !
10