Associative Schemes

TL;DR

This paper develops a noncommutative deformation theory framework for modules over associative algebras, defining associative varieties with a topology and sheaf structure, extending classical algebraic geometry to a noncommutative setting.

Contribution

It introduces the concept of associative varieties, extending classical varieties, and shows they can be studied via associative algebras, bridging noncommutative algebra and geometry.

Findings

Defined a topology on spectral modules with simple modules as closed points

Constructed a sheaf of rings on the topological space of modules

Proved associative varieties generalize classical algebraic varieties

Abstract

We state results from noncommutative deformation theory of modules over an associative $k$-algebra $A,$ $k$ a field, necessary for this work. We define a set of $A$-modules $\operatorname{aSpec}A$ containing the simple modules, whose elements we call spectral, for which there exists a topology where the simple modules are the closed points. Applying results from deformation theory we prove that there exists a sheaf of associative rings $\mathcal O_X$ on the topological space $X=\operatorname{aSpec}A$ giving it the structure of a pointed ringed space. In general, an associative variety $X$ is a ringed space with an open covering $\{U_i=\operatorname{aSpec}{A_i}\}_{i\in I}.$ When $A$ is a commutative $k$-algebra, $\operatorname{aSpec}A\simeq\spec A,$ and so the category $\cat{aVar}_k$ of associative varieties is an extension of the category of varieties $\cat{Var}_k,$ i.e. there exists a faithfully full functor $I:\cat{Var}_k\rightarrow\cat{aVar}_k.$ Our main result says that any associative variety $X$ is $\operatorname{aSpec}(\mathcal O_X(X))$ for the $k$-algebra $\mathcal O_X(X),$ and so any study of varieties can be reduced to the study of the associative algebra $\mathcal O_X(X).$

Equations

$$k^r\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\cdots \stackrel{{\pi }_n}{\to }{R}_n\stackrel{{\pi }_{n-1}}{\to }{R}_{n-1}\stackrel{{\pi }_{n-2}}{\to }\cdots \stackrel{{\pi }_2}{\to }{R}_2\stackrel{{\pi }_1}{\to }k$$

$$k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\cdots \stackrel{{\pi }_n}{\to }{R}_n\stackrel{{\pi }_{n-1}}{\to }{R}_{n-1}\stackrel{{\pi }_{n-2}}{\to }\cdots \stackrel{{\pi }_2}{\to }{R}_2\stackrel{{\pi }_1}{\to }{k}^r$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$O^{O^A(\tilde{M})}(\tilde{M})=O^A(\tilde{M}).$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$O_X(U)=\underset{\underset{\tilde{M}\subseteq U}{\leftarrow }}{\lim }{O}^A(\tilde{M})$$

$${\mathcal{O}}_X(U)=\underset{\underset{V⫋U}{\leftarrow }}{\lim }{O}_X(V).$$

$$F_{X,\tilde{P}}=\underset{\underset{\tilde{P}\subseteq U}{\to }}{\lim }{F}_X(U).$$

$$\eta :A\to O^A(\mathrm{Simp}A)$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{O}}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1 Acknowledgements

This article could not have been written without the cooperation with Severin Barmeier, who had the idea and insight in noncommutative algebraic geometry necessary for the generalization to noncommutative schemes. Also, I thank Gunnar Fløystad, David Ploog, Eivind Eriksen, Jon Eivind Vatne and Ragni Piene for their comments when lecturing different versions during the winter 2022/2023. As always, my supervisor O. Arnfinn Laudal has been the greatest inspiration and moderator of this work.

2 Notation and Preliminaries

All rings $A$ are assumed to be associative unital rings. For a right $A$-module $M$ we denote the structure morphism by $\eta^{A}_{M}:A\rightarrow\operatorname{End}_{\mathbb{Z}}(M),$ that is, for $m\in M,a\in A,\ m\cdot a=\eta^{A}_{M}(a)(m).$

Recall from [1] that an $A$-module $M$ is simple if it has no proper sub-modules. When $k$ is a division ring, $A$ is a $k$-algebra and $M$ is of finite dimension over $k,$ then $\eta^{A}_{M}$ surjective implies $M$ simple, and the converse holds if $\operatorname{End}_{A}(M)=k,$ in particular this holds when $k$ is an algebraically closed field.

3 Deformation Theory

Let $k$ be a field.

Definition 1**.**

Let $k$ be in the centre of $R.$ Then the $k^{r}$-algebra $R$ is called $r$-pointed if it is augmented over $k^{r}$ such that the diagram

[TABLE]

commutes. A morphism between $r$-pointed algebras is a $k^{r}$-algebra homomorphism commuting with the augmentation. The category of $r$-pointed $k$-algebras is denoted $\mathbf{Alg}^{r}_{k}$.

Recall from [2] that an associative ring is called local if it has a unique maximal right ideal $\mathfrak{m}$. Then this ideal is also a left ideal, and this is equivalent to the condition that $\mathfrak{m}$ is a two-sided ideal such that every element not in $\mathfrak{m}$ is a unit.

Lemma 1**.**

Let $R$ be a $1$-pointed $k$-algebra, i.e. fitting in the diagram

[TABLE]

If $(\ker\pi)^{n}=0$ for some $n\geq 1,$ then if $\pi(r)\neq 0,$ then $r$ is a unit in $R.$ In particular, $\mathfrak{m}=\ker\pi$ is a unique maximal ideal and $R$ is a local ring.

Proof.

$r\notin\ker\pi\Rightarrow\pi(r)=\alpha=\pi(\iota(\alpha))\Rightarrow r-\iota(\alpha)\in\ker\pi\Rightarrow r=u-x$ where $u=\iota(\alpha)$ is a unit and $x\in\ker\pi.$ Let $t=\sum_{i=1}^{n}u^{-i}x^{i-1}.$ Then $rt=tr=1.$ Any proper ideal in $R$ consists of non-units, hence is contained in $\ker\pi.$ Thus this is the unique ideal in $R,$ which is local. ∎

Proposition 1**.**

Consider a sequence of $1$-pointed $k$-algebras

[TABLE]

such that for each $n,$ $(\ker\pi_{n-1})\mathfrak{m}_{n}=0,\ \mathfrak{m}_{n}=\ker\pi_{n-1}.$ Then $\underset{\underset{n>0}{\leftarrow}}{\lim}\ R_{n}=\hat{R}$ is a local ring.

Proof.

By definition, $\hat{R}$ fits in the diagram

[TABLE]

With the notation from the proof of Lemma 1, it follows that when $r\notin\ker\pi,$ we can put $t=\sum_{n=0}^{\infty}u^{-(n+1)}x^{n}\in\hat{R}$ and then $rt=tr=1.$ When every $r\notin\ker\pi$ is a unit, $\hat{R}$ is a local ring with (unique) maximal ideal $\ker\pi.$ ∎

Definition 2**.**

An $r$-pointed ring $\hat{R}$ is called formal if it is the projective limit of a sequence of $r$-pointed $k$-algebras

[TABLE]

such that for each $n,$ $(\ker\pi_{n-1})\mathfrak{m}_{n}=0,\ \mathfrak{m}_{n}=\ker\pi_{n-1}.$ That is, $\underset{\underset{n>0}{\leftarrow}}{\lim}\ R_{n}=\hat{R}.$

The following is the most fundamental result in [1] and is proved there.

Theorem 1**.**

Let $k$ be a field, $A$ a $k$-algebra and $\tilde{M}=\{M_{1},\dots,M_{r}\}$ a set of $r$ right $A$-modules such that $\dim_{k}(\operatorname{Ext}^{1}_{A}(M_{i},M_{j}))<\infty$. Then the noncommutative deformation functor has a pro-representing hull $\hat{H}(\tilde{M}),$ i.e. there exists a formal $r$-pointed $k$-algebra $\hat{H}(\tilde{M})$ fitting in a diagram

[TABLE]

Remark 1**.**

Assume that $A$ is a (general) unital, associative ring such that its centre $C(A)$ is a domain with quotient field $Q(A)$. Then for each simple right $A$-module $M$ we have that $\operatorname{End}_{A}(M)$ is a division ring, and there is a homomorphism $\eta:C(A)\rightarrow\operatorname{End}_{A}(M).$ Then $C(A)/\ker\eta=k(M)$ is a field $k(M)\subseteq Q(A).$ Let $k(A)=\cap_{M\in\operatorname{Simp}(A)}k(M)\subseteq Q(A).$ Then Theorem 1 applies to the ring $k(A)[A].$

Theorem 2**.**

Let $A$ be an associative $k$-algebra and $\tilde{M}=\{M_{1},\dots,M_{r}\}$ a set of $r$ right $A$-modules. Then there exists a $k^{r}$-algebra $O^{A}(\tilde{M})$ fitting in the diagram

[TABLE]

such that if $\eta(a)=\operatorname{id}\cdot\alpha,$ then $\rho(a)\in O^{A}(\tilde{M})$ is invertible, and such that for any other $B,\rho^{\prime}$ with this property, there is a unique morphism $\phi:O^{A}(\tilde{M})\rightarrow B$ such that $\phi\circ\rho=\rho^{\prime}.$ Then $O^{A}(\tilde{M})$ is uniquely determined by this universal property.

Proof.

Consider the diagram (1). Let $O^{A}(\tilde{M})=\operatorname{im}(\rho)\subseteq\hat{O}^{A}(\tilde{M})$ together with $\rho(a)^{-1}$ when $\eta(a)=\operatorname{id}\cdot\alpha.$ Then $O^{A}(\tilde{M})$ fits in the diagram, and if $\eta(a)\in\oplus_{i=1}^{r}\operatorname{End}_{k}(M_{i})$ is the identity times a constant $\alpha\in k$, it follows from Proposition 1 that $\rho(a)$ is a unit. If $B$ is any other ring fitting in the diagram via $\rho^{\prime}$, the morphism $\phi:O^{A}(\tilde{M})\rightarrow B$ given by $\phi(\rho(f)\rho(g)^{-1})=\rho^{\prime}(f)\rho^{\prime}(g)^{-1}$ and correspondingly for all other combinations, is uniquely determined. ∎

Example 1**.**

If $A$ is a commutative ring and $\mathfrak{p}\subset A$ a prime ideal, we see that $O^{A}(A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}})\simeq A_{\mathfrak{p}}.$ If $\tilde{P}=\{A_{\mathfrak{p}_{1}}/\mathfrak{p}_{1}A_{\mathfrak{p}_{1}},\dots,A_{\mathfrak{p}_{r}}/\mathfrak{p}_{r}A_{\mathfrak{p}_{r}}\}$ a set of $r$ spectral modules, then $O^{A}(\tilde{P})\simeq\prod_{i=1}^{r}A_{\mathfrak{p}_{i}}$ by the universal property.

Corollary 1**.**

Let $A$ be an associative ring and let $\tilde{M}=\{M_{1},\dots,M_{r}\}$ be a set of $r$ right $A$-modules such that $\operatorname{End}_{k}(M_{i})$ is a division ring contained in the centre of $\hat{O}^{A}(\tilde{M}),$ $1\leq i\leq r.$ Then $O^{A}(\tilde{M})$ is $r$-local with exactly $r$ maximal ideals $\mathfrak{m}_{i},\ 1\leq i\leq r,$ and $O^{A}(\tilde{M})/\mathfrak{m}_{i}\simeq M_{i}.$

Proof.

Let $\mathfrak{m}_{i}=\ker(\eta_{M_{i}}^{O^{A}(\tilde{M})}),\ 1\leq i\leq r.$ Any ideal $\mathfrak{a}\neq(1)=O^{A}(\tilde{M})$ consists of non-units, hence is contained in one of the ideals $\mathfrak{m}_{i}.$ ∎

Proposition 2**.**

For a finite family of $A$-modules $\tilde{M}$, the $k$-algebra $O^{A}(\tilde{M})$ satisfies

[TABLE]

Thus the $O$-construction is a closure operation.

Proof.

This follows by uniqueness of the algebra $O^{A}(\tilde{M})$ from Theorem 2 because $\rho:\hat{O}^{A}(\tilde{M})\rightarrow\hat{O}^{A}(\tilde{M})$ induces an isomorphism $\hat{O}^{A}\rightarrow\hat{O}^{O^{A}(\tilde{M})}.$ ∎

4 Associative Schemes

Let $f:A\rightarrow B$ be a homomorphism between associative rings with $B$ local. Then $B$ as a unique simple module $M$ defined by a unique right maximal ideal $\mathfrak{m}$ which contraction $\mathfrak{p}=f^{-1}(\mathfrak{m})$ is a right ideal. We call the right $A$-module $M^{c}=A/\mathfrak{p}$ the contraction of the $B$-module $M.$

Example 2**.**

When $M$ is a simple right $A$-module, we have the diagram

[TABLE]

with $O^{A}(M)$ local. Then the contraction of the simple $O^{A}(M)$-module $M$ is the simple $A$-module $M.$

Definition 3**.**

A right $A$-module $P$ is called spectral if there exists a local ring $B$ and a ring homomorphism $f:A\rightarrow B$ such that $P=M^{c}.$

When $A$ is a commutative ring, a spectral module is $A/\mathfrak{p}$ for some prime ideal $\mathfrak{p}\subset A.$ In the commutative situation, this gives a bijective correspondence between spectral modules and prime ideals. Also notice that given a ring homomorphism $f:C\rightarrow A,$ we have the diagram

[TABLE]

This proves that the contraction of a spectral module is spectral, and this contraction corresponds to the contraction of prime ideals in the commutative situation.

Definition 4**.**

We let $\operatorname{aSpec}(A)$ denote the set of right spectral $A$-modules. For $f\in A$ we put $D(f)=\{P\in\operatorname{aSpec}(A)|\eta^{A}_{P}(f)\text{ is injective}\}$ and we give $\operatorname{aSpec}(A)$ the topology generated by the sub-basis $\{D(f)\}_{f\in A}.$

Definition 5**.**

We define a sheaf of rings $\mathcal{O}_{X}$ on $X=\operatorname{aSpec}A$ by:

(i) First, define the presheaf

[TABLE]

where the limit is taken over all finite subsets $\tilde{M}$ of simple right $A$-modules.

(ii) Then, define the sheaf

[TABLE]

Let us prove that Definition 5 is equivalent to the classical definition of a sheaf (of abelian groups) in algebraic geometry. So let $U$ be an open set covered by the family $\{V_{i}\}$ and assume that $s\in\mathcal{F}(U)$ maps to $0=s|_{V_{i}}\in\mathcal{F}(V_{i})$ for all $i.$ Then the zero homomorphism $\mathbb{Z}\rightarrow\mathcal{F}(U)$ for all $V\subsetneq U$ lifts uniquely to $\mathcal{F}(U)$ so it has to map $1$ to $s=0\in\mathcal{F}(U).$ Assume that we have elements $s_{i}\in\mathcal{F}(V_{i})$ for all $i$ such that for all $i,j,$ $s_{i}|_{V_{i}\cup V_{j}}=s_{j}|_{V_{i}\cup V_{j}}\in\mathcal{F}(U_{i}\cap U_{j}).$ The algebra $\mathbb{Z}[s]$ maps naturally to all $\mathcal{F}(V_{i})$ by sending $s$ to $s_{i}|V_{i},$ and so (by definition of projective limits), there exists a unique homomorphism $\iota:\mathbb{Z}[s]\rightarrow\mathcal{F}(U).$ Then $s=\iota(s)\in\mathcal{F}(U)$ is the unique lifted element.

Definition 6**.**

For an associative unital ring $A$ we define $\operatorname{aSpecA}$ as the ringed space $(\operatorname{aSpec}A,\mathcal{O}_{\operatorname{aSpec}A}).$

Definition 7**.**

Let $\tilde{P}=\{P_{1},...,P_{r}\}\subseteq X=\operatorname{aSpec}A$ be a finite set. We define the stalk of any presheaf $F_{X}$ on $X$ in $\tilde{P}$ as the inductive limit

[TABLE]

Lemma 2**.**

The stalk in $\tilde{P}$ of the sheaf $\mathcal{O}_{X}$ is isomorphic to the stalk in $\tilde{P}$ of the presheaf $O_{X}.$

Proof.

Inductive limits is a contravariant functor. Thus the morphism $O_{X}\rightarrow\mathcal{O}_{X}$ gives a morphism $\mathcal{O}_{X,\tilde{P}}\rightarrow O_{X,\tilde{P}}.$ Also, for each open $U\supseteq\tilde{P}$ we have by the same reason a homomorphism $O_{X}(U)\rightarrow\mathcal{O}_{X,\tilde{P}}$ giving a homomorphism $O_{X,\tilde{P}}\rightarrow\mathcal{O}_{X,\tilde{P}}.$ By uniqueness, these morphisms are inverses. ∎

Corollary 2**.**

If $\tilde{P}\subseteq X$ is a finite set of spectral right $A$-modules, then $\mathcal{O}_{X,\tilde{P}}\simeq O^{A}({\tilde{P}}).$

Example 3**.**

It follows from Example 1 that for a commutative ring $A$ we have $\operatorname{aSpec}A\simeq\operatorname{Spec}A.$

Corollary 3**.**

Let $A$ be a finite dimensional $\Bbbk$-algebra, $\Bbbk$ algebraically closed. Then the homomorphism

[TABLE]

where $\operatorname{Simp}A$ is the set of all simple (right) $A$-modules, is an isomorphism.

Proof.

When $A$ is finite dimensional $\hat{O}^{A}(\tilde{M})\simeq O^{A}(\tilde{M})$ for any set of simple modules $\tilde{M}.$ Also, because $\Bbbk$ is algebraically closed, $\mathcal{O}^{A}(\operatorname{aSpec}A)\simeq\mathcal{O}^{A}(\operatorname{Simp}A).$ ∎

Definition 8**.**

An associative scheme is a topological space with a sheaf of associative rings $(X,\mathcal{O}_{X})$ which has a cover of open affine subsets $X=\cup_{i\in I}\operatorname{aSpec}(A_{i})$ where each $A_{i}$ is an associative ring. A morphism of associative schemes is a morphism of ringed spaces. The category of schemes over $\Bbbk$ is denoted $\operatorname{aSch}.$

Definition 9**.**

An associative variety over an algebraically closed field $\Bbbk$ is a topological space with a sheaf of associative rings $(X,\mathcal{O}_{X})$ which has a cover of open affine subsets $X=\cup_{i\in I}\operatorname{aSpec}(A_{i})$ where each $A_{i}$ is an associative $\Bbbk$-algebra. A morphism of associative schemes is a morphism of ringed spaces. The category of associative varieties over $\Bbbk$ is denoted $\mathbf{aVar}_{\Bbbk}.$

Corollary 4**.**

Let $\phi:A\rightarrow B$ be homomorphism of associative rings. Then there is a natural morphism $\operatorname{aSpec}(\phi):\operatorname{aSpec}B\rightarrow\operatorname{aSpec}A$

Proof.

The diagram

[TABLE]

gives the map on points sending the spectral $A$-module $P$ to the spectral $B$-module $P.$ The universal property of $A_{P}$ gives that there is a unique morphism $B_{P}\rightarrow A_{P}$ which gives the corresponding morphism of pointed rings, thus a morphism of pre-sheaves, then of sheaves. ∎

5 Sheaves of $\mathcal{O}_{X}$-modules on Associative Varieties $X$

Consider an associative variety $X$ by which we mean an associative scheme $(X,\mathcal{O}_{X})$ over an algebraically closed field $\Bbbk.$ In [1] we prove that the noncommutative deformation functor of sheaves of $\mathcal{O}_{X}$-modules has a pro-representing hull. Combining with Theorem 2 this gives the following.

Corollary 5**.**

For a finite set $\mathcal{F}=\{\mathcal{F}_{1},\dots,\mathcal{F}_{r}\}$ of right $\mathcal{O}_{X}$-modules, there exists a sheaf $\mathcal{A}_{\mathcal{F}}$ of $\Bbbk^{r}$-algebras fitting in the diagram

[TABLE]

such that if $\eta(a)\in\oplus_{i=1}^{r}\operatorname{End}_{\Bbbk}(\mathcal{F}_{i})$ is invertible (locally, i.e. on each open), then $\rho(a)\in\mathcal{A}_{\mathcal{F}}$ is invertible. Also, $\mathcal{A}_{\mathcal{F}}$ is uniquely defined upto unique isomorphism by this property.

Theorem 3**.**

For any associative variety $X$ over $\Bbbk$ there exists a $\Bbbk$-algebra $\mathcal{O}_{X}(X)$ such that $X\simeq\operatorname{aSpec}\mathcal{O}_{X}(X).$

Proof.

An associative variety $X$ is covered by affine associative varieties, that is $X=\bigcup_{i\in I}\operatorname{aSpec}(A_{i})$ for some index set $I$. To each point $M\in X$ we define the skyscraper sheaf $\mathcal{S}_{M}$ by $\mathcal{S}_{M}(U)=\begin{cases}M,\ M\in U\\ 0,\ M\notin U.\end{cases}$ For $U\subseteq X,$ $O_{X}(U)=\underset{\underset{\mathcal{M}\subseteq U}{\leftarrow}}{\lim}A_{\mathcal{M}},$ where the limit is taken over all finite subsets, defines a presheaf on $X.$ The sheaf $\mathcal{O}_{X}$ associated to $O_{X}$ satisfy $\operatorname{aSpec}\mathcal{O}_{X}(X)\simeq X.$ ∎